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Fi(K)570 2006 y Fl(\000)c Fm(e)631 1989 y Fh(\000)p Fi(L)693 2006 y Fk(=)762 1978 y(1)p 742 1997 61 2 v 742 2035 a(2)p Fm(\031)q(i)814 1950 y Fe(Z)837 2044 y Fj(\000)867 2006 y Fm(dz)j(e)940 1989 y Fh(\000)p Fi(z)992 1973 y Fe(\000)1011 2006 y Fk(\()p Fm(z)f Fl(\000)d Fm(K)s Fk(\))1153 1989 y Fh(\000)p Fj(1)1207 2006 y Fl(\000)h Fk(\()p Fm(z)h Fl(\000)f Fm(L)p Fk(\))1381 1989 y Fh(\000)p Fj(1)1426 1973 y Fe(\001)1452 2006 y Fm(;)140 b Fw(\(2.2\))257 2100 y(where)15 b(the)g(con)o(tour)f Fk(\000)g Fw(is)f(giv)o(en)h(b)o (y)f Fk(\000)f(=)g(\000)955 2106 y Fj(0)983 2100 y Fl([)c Fk(\000)1045 2106 y Fh(1)1094 2100 y Fw(with)511 2170 y Fk(\000)537 2176 y Fj(0)613 2170 y Fk(=)42 b Fl(f)p Fm(z)14 b Fl(2)d Fd(C)20 b Fk(:)11 b Fm(z)j Fk(=)e Fm(e)940 2155 y Fi(i')976 2170 y Fm(;)48 b(\031)q(=)p Fk(4)11 b Fl(\024)h Fm(')g Fl(\024)f Fk(2)p Fm(\031)g Fl(\000)e Fm(\031)q(=)p Fk(4)p Fl(g)p Fm(;)511 2245 y Fk(\000)537 2251 y Fh(1)613 2245 y Fk(=)42 b Fl(f)p Fm(z)14 b Fl(2)d Fd(C)20 b Fk(:)11 b Fm(z)j Fk(=)e Fm(r)q(e)960 2230 y Fh(\006)p Fi(i\031)q(=)p Fj(4)1054 2245 y Fm(;)48 b(r)12 b Fl(\025)g Fk(1)p Fl(g)p Fm(:)1604 2208 y Fw(\(2.3\))257 2319 y(F)m(rom)g(\(2.2\))i(w)o(e)g(\034nd)g(the)g(represen)o(tation)414 2413 y Fm(e)433 2396 y Fh(\000)p Fi(K)500 2413 y Fl(\000)c Fm(e)561 2396 y Fh(\000)p Fi(L)623 2413 y Fk(=)692 2385 y(1)p 672 2403 V 672 2441 a(2)p Fm(\031)q(i)744 2356 y Fe(Z)767 2451 y Fj(\000)797 2413 y Fm(dz)j(e)870 2396 y Fh(\000)p Fi(z)916 2413 y Fk(\()p Fm(I)f Fk(+)e Fm(K)s Fk(\)\()p Fm(z)i Fl(\000)d Fm(K)s Fk(\))1200 2396 y Fh(\000)p Fj(1)1257 2413 y Fl(\001)335 b Fw(\(2.4\))615 2503 y Fl(\001)634 2469 y Fe(\000)652 2503 y Fk(\()p Fm(I)13 b Fk(+)d Fm(L)p Fk(\))785 2486 y Fh(\000)p Fj(1)839 2503 y Fl(\000)f Fk(\()p Fm(I)k Fk(+)d Fm(K)s Fk(\))1023 2486 y Fh(\000)p Fj(1)1068 2469 y Fe(\001)1094 2503 y Fk(\()p Fm(I)j Fk(+)c Fm(L)p Fk(\)\()p Fm(z)j Fl(\000)d Fm(L)p Fk(\))1358 2486 y Fh(\000)p Fj(1)1403 2503 y Fm(:)963 2628 y Fw(4)p eop %%Page: 5 5 5 4 bop 257 262 a Fw(Since)573 311 y Fk(\()p Fm(I)13 b Fk(+)c Fm(K)s Fk(\)\()p Fm(z)j Fl(\000)d Fm(K)s Fk(\))857 294 y Fh(\000)p Fj(1)914 311 y Fk(=)j Fl(\000)p Fm(I)h Fk(+)c(\(1)g(+)h Fm(z)r Fk(\)\()p Fm(z)i Fl(\000)d Fm(K)s Fk(\))1329 294 y Fh(\000)p Fj(1)1604 311 y Fw(\(2.5\))257 386 y(one)14 b(gets)h(the)f(estimate)613 464 y Fe(\015)613 489 y(\015)636 499 y Fk(\()p Fm(I)f Fk(+)d Fm(K)s Fk(\)\()p Fm(z)h Fl(\000)f Fm(K)s Fk(\))921 482 y Fh(\000)p Fj(1)966 464 y Fe(\015)966 489 y(\015)1000 499 y Fl(\024)i Fk(1)d(+)1161 471 y(1)g(+)h Fl(j)p Fm(z)r Fl(j)p 1121 489 197 2 v 1121 527 a Fw(dist)p Fk(\()p Fm(z)r(;)d Fd(R)1277 533 y Fj(+)1301 527 y Fk(\))1322 499 y Fm(:)270 b Fw(\(2.6\))257 612 y(Setting)726 684 y Fm(N)759 690 y Fj(\000)793 684 y Fk(:=)11 b(sup)850 718 y Fi(z)q Fh(2)p Fj(\000)963 655 y Fk(1)e(+)h Fl(j)p Fm(z)r Fl(j)p 923 674 V 923 712 a Fw(dist)p Fk(\()p Fm(z)r(;)d Fd(R)1079 718 y Fj(+)1103 712 y Fk(\))1136 684 y Fm(<)12 b Fl(1)382 b Fw(\(2.7\))257 782 y(w)o(e)14 b(\034nd)638 832 y Fk(sup)639 867 y Fi(z)q Fh(2)p Fj(\000)707 797 y Fe(\015)707 822 y(\015)730 832 y Fk(\()p Fm(I)f Fk(+)c Fm(K)s Fk(\)\()p Fm(z)j Fl(\000)e Fm(K)s Fk(\))1015 815 y Fh(\000)p Fj(1)1060 797 y Fe(\015)1060 822 y(\015)1094 832 y Fl(\024)i Fk(\(1)d(+)h Fm(N)1259 838 y Fj(\000)1282 832 y Fk(\))p Fm(;)294 b Fw(\(2.8\))257 931 y(where)18 b(the)f(constan)o(t)h Fm(N)658 937 y Fj(\000)697 931 y Fw(dep)q(ends)g(only)e(on)h Fk(\000)f Fw(but)h(not)g(on)f(the)i (op)q(erator)f Fm(K)s Fw(.)26 b(F)m(rom)257 981 y(\(2.8\))14 b(one)g(also)f(gets)647 1072 y Fk(sup)649 1107 y Fi(z)q Fh(2)p Fj(\000)717 1037 y Fe(\015)717 1062 y(\015)740 1072 y Fk(\()p Fm(I)g Fk(+)c Fm(L)p Fk(\)\()p Fm(z)j Fl(\000)e Fm(L)p Fk(\))1005 1055 y Fh(\000)p Fj(1)1050 1037 y Fe(\015)1050 1062 y(\015)1084 1072 y Fl(\024)i Fk(\(1)d(+)h Fm(N)1249 1078 y Fj(\000)1272 1072 y Fk(\))p Fm(:)304 b Fw(\(2.9\))257 1190 y(Using)14 b(these)h(estimates,)e(w)o(e) i(\034nd)f(from)e(\(2.4\))h(that)570 1246 y Fe(\015)570 1271 y(\015)594 1281 y Fm(e)613 1264 y Fh(\000)p Fi(K)680 1281 y Fl(\000)d Fm(e)741 1264 y Fh(\000)p Fi(L)792 1246 y Fe(\015)792 1271 y(\015)826 1281 y Fl(\024)i Fm(N)915 1246 y Fe(\015)915 1271 y(\015)938 1281 y Fk(\()p Fm(I)h Fk(+)c Fm(K)s Fk(\))1080 1264 y Fh(\000)p Fj(1)1134 1281 y Fl(\000)h Fk(\()p Fm(I)j Fk(+)c Fm(L)p Fk(\))1308 1264 y Fh(\000)p Fj(1)1353 1246 y Fe(\015)1353 1271 y(\015)1583 1281 y Fw(\(2.10\))257 1373 y(with)14 b(a)f(constan)o(t)692 1433 y Fm(N)j Fk(:=)814 1405 y(1)p 802 1424 46 2 v 802 1462 a(2)p Fm(\031)852 1433 y Fk(\(1)10 b(+)f Fm(N)973 1439 y Fj(\000)996 1433 y Fk(\))1012 1416 y Fj(2)1038 1377 y Fe(Z)1061 1471 y Fj(\000)1090 1433 y Fl(j)p Fm(dz)r Fl(j)i(j)p Fm(e)1199 1416 y Fh(\000)p Fi(z)1244 1433 y Fl(j)327 b Fw(\(2.11\))257 1529 y(dep)q(ending)15 b(only)e(on)h(the)g (con)o(tour)g Fk(\000)p Fw(.)791 b Fc(\003)257 1579 y Fw(T)m(o)14 b(pro)o(v)o(e)h(the)g(main)e(theorem)h(w)o(e)h(shall)f (use,)h(as)g(in)f([11)o(])h(and)f([12)o(],)g(an)h(op)q(erator)g(norm) 257 1629 y(v)o(ersion)f(of)f(Cherno\033)600 1613 y Fh(0)613 1629 y Fw(s)h(theorem)f(with)g(error)i(b)q(ound,)e(whic)o(h)h(partly)f (impro)o(v)o(es)f(Lemma)257 1678 y(2.1)h(of)g([11])g(in)g(the)i(case)g Fm(\045)d(<)f Fk(1)p Fw(.)257 1761 y Fg(Theorem)k(2.2)21 b Ff(L)n(et)15 b Fl(f)p Fm(F)6 b Fk(\()p Fm(\034)f Fk(\))p Fl(g)744 1767 y Fi(\034)s Fh(\025)p Fj(0)821 1761 y Ff(b)n(e)16 b(a)f(family)g(of)g(non-ne)n(gative)h(self-adjoint)f(c)n(ontr)n(ac-)257 1811 y(tions)g(on)h Fl(H)f Ff(such)g(that)g Fm(F)6 b Fk(\(0\))11 b(=)h Fm(I)18 b Ff(and)e(let)728 1926 y Fm(S)r Fk(\()p Fm(\034)5 b Fk(\))12 b(:=)883 1898 y Fm(I)g Fl(\000)e Fm(F)c Fk(\()p Fm(\034)f Fk(\))p 883 1917 160 2 v 951 1955 a Fm(\034)1048 1926 y(;)48 b(\034)17 b(>)12 b Fk(0)p Fm(:)363 b Fw(\(2.12\))257 2032 y Ff(Assume)15 b Fm(\045)d Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1\))p Ff(.)18 b(Then)d(ther)n(e)g(is)g(a)g (c)n(onstant)g Fm(M)1106 2038 y Fi(\045)1137 2032 y Fm(>)d Fk(0)j Ff(such)g(that)591 2139 y Fl(k)p Fk(\()p Fm(I)e Fk(+)c Fm(tS)r Fk(\()p Fm(\034)c Fk(\)\))813 2122 y Fh(\000)p Fj(1)868 2139 y Fl(\000)10 b Fk(\()p Fm(I)i Fk(+)e Fm(tH)s Fk(\))1067 2122 y Fh(\000)p Fj(1)1112 2139 y Fl(k)h(\024)h Fm(M)1228 2145 y Fi(\045)1254 2093 y Fe(\020)1284 2111 y Fm(\034)p 1284 2130 23 2 v 1288 2168 a(t)1312 2093 y Fe(\021)1337 2102 y Fi(\045)1583 2139 y Fw(\(2.13\))257 2245 y Ff(holds)17 b(for)g Fm(\034)r(;)7 b(t)14 b Fl(2)f Fk(\(0)p Fm(;)7 b Fk(1])15 b Ff(with)h Fk(0)e Fm(<)g(\034)19 b Fl(\024)c Fm(t)p Ff(,)h(if)g(and)h(only)f(if)g(ther)n(e)h(is)f(a)g(c) n(onstant)h Fm(C)1589 2251 y Fi(\045)1622 2245 y Fm(>)e Fk(0)257 2294 y Ff(such)h(that)707 2348 y Fl(k)p Fm(F)6 b Fk(\()p Fm(\034)f Fk(\))816 2331 y Fi(t=\034)875 2348 y Fl(\000)10 b Fm(e)936 2331 y Fh(\000)p Fi(tH)1006 2348 y Fl(k)h(\024)h Fm(C)1112 2354 y Fi(\045)1138 2302 y Fe(\020)1168 2320 y Fm(\034)p 1168 2339 V 1172 2377 a(t)1196 2302 y Fe(\021)1220 2311 y Fi(\045)1583 2348 y Fw(\(2.14\))257 2437 y Ff(is)j(valid)g(for)g Fm(\034)r(;)7 b(t)k Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])13 b Ff(with)h Fk(0)e Fm(<)f(\034)17 b Fl(\024)12 b Fm(t)p Ff(.)963 2628 y Fw(5)p eop %%Page: 6 6 6 5 bop 257 262 a Fg(Pro)q(of.)18 b Fw(Assume)c(\(2.13\).)j(By)d(Lemma) d(2.1)i(there)i(is)f(a)f(constan)o(t)i Fm(N)h(>)c Fk(0)i Fw(suc)o(h)g(that)502 348 y Fl(k)p Fm(e)542 330 y Fh(\000)p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))659 348 y Fl(\000)9 b Fm(e)719 330 y Fh(\000)p Fi(tH)789 348 y Fl(k)j(\024)g Fm(N)5 b Fl(k)p Fk(\()p Fm(I)12 b Fk(+)e Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))1126 330 y Fh(\000)p Fj(1)1180 348 y Fl(\000)10 b Fk(\()p Fm(I)j Fk(+)c Fm(tH)s Fk(\))1379 330 y Fh(\000)p Fj(1)1424 348 y Fl(k)138 b Fw(\(2.15\))257 434 y(for)14 b Fm(\034)r(;)7 b(t)k(>)h Fk(0)p Fw(.)18 b(Using)c(\(2.13\))e(w)o(e)j(obtain)681 487 y Fe(\015)681 512 y(\015)681 536 y(\015)704 534 y Fm(e)723 517 y Fh(\000)p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))839 534 y Fl(\000)10 b Fm(e)900 517 y Fh(\000)p Fi(tH)970 487 y Fe(\015)970 512 y(\015)970 536 y(\015)1005 534 y Fl(\024)i Fm(N)k(M)1138 540 y Fi(\045)1165 488 y Fe(\020)1194 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y(\015)656 1249 y Fm(F)6 b Fk(\()p Fm(\034)f Fk(\))744 1232 y Fi(t=\034)803 1249 y Fl(\000)10 b Fm(e)864 1232 y Fh(\000)p Fi(tH)934 1201 y Fe(\015)934 1226 y(\015)934 1251 y(\015)969 1249 y Fl(\024)1018 1221 y Fm(\034)p 1018 1240 V 1022 1278 a(t)1055 1249 y Fk(+)f Fm(N)17 b(M)1186 1255 y Fi(\045)1212 1203 y Fe(\020)1242 1221 y Fm(\034)p 1242 1240 V 1246 1278 a(t)1270 1203 y Fe(\021)1294 1212 y Fi(\045)1583 1249 y Fw(\(2.20\))257 1341 y(for)f Fm(\034)r(;)7 b(t)16 b Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1])15 b Fw(with)h Fk(0)g Fm(<)g(\034)21 b Fl(\024)16 b Fm(t)p Fw(.)25 b(Since)978 1325 y Fi(\034)p 978 1332 19 2 v 981 1355 a(t)1018 1341 y Fl(\024)1066 1307 y Fe(\000)1090 1325 y Fi(\034)p 1090 1332 V 1093 1355 a(t)1114 1307 y Fe(\001)1133 1316 y Fi(\045)1168 1341 y Fw(for)17 b Fk(0)e Fm(<)h(\034)21 b Fl(\024)16 b Fm(t)p Fw(,)h Fm(\045)f Fl(2)g Fk([0)p Fm(;)7 b Fk(1])p Fw(,)14 b(w)o(e)257 1391 y(obtain)623 1397 y Fe(\015)623 1422 y(\015)623 1447 y(\015)646 1445 y Fm(F)6 b Fk(\()p Fm(\034)f Fk(\))734 1428 y Fi(t=\034)793 1445 y Fl(\000)10 b Fm(e)854 1428 y Fh(\000)p Fi(tH)924 1397 y Fe(\015)924 1422 y(\015)924 1447 y(\015)959 1445 y Fl(\024)h Fk(\(1)f(+)f Fm(N)16 b(M)1179 1451 y Fi(\045)1199 1445 y Fk(\))1222 1399 y Fe(\020)1252 1417 y Fm(\034)p 1252 1435 23 2 v 1256 1473 a(t)1280 1399 y Fe(\021)1305 1408 y Fi(\045)1583 1445 y Fw(\(2.21\))257 1530 y(for)e Fm(\034)r(;)7 b(t)k Fl(2)g Fk(\(0)p Fm(;)c Fk(1])12 b Fw(with)i Fk(0)d Fm(<)h(\034)k Fl(\024)c Fm(t)p Fw(.)18 b(Setting)13 b Fm(C)992 1536 y Fi(\045)1023 1530 y Fk(:=)e(1)d(+)h Fm(N)17 b(M)1238 1536 y Fi(\045)1271 1530 y Fw(w)o(e)d(ha)o(v)o(e)f(v)o(eri\034ed)h (\(2.14\).)320 1580 y(T)m(o)f(pro)o(v)o(e)h(the)g(con)o(v)o(erse)h(w)o (e)f(use)h(the)g(represen)o(tation)353 1688 y Fk(\()p Fm(I)e Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))555 1670 y Fh(\000)p Fj(1)610 1688 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)d Fm(tH)s Fk(\))809 1670 y Fh(\000)p Fj(1)865 1688 y Fk(=)909 1631 y Fe(Z)950 1641 y Fh(1)932 1725 y Fj(0)1004 1688 y Fm(dx)h(e)1080 1670 y Fh(\000)p Fi(x)1134 1641 y Fe(\020)1158 1688 y Fm(e)1177 1670 y Fh(\000)p Fi(xtS)r Fj(\()p Fi(\034)s Fj(\))1313 1688 y Fl(\000)e Fm(e)1373 1670 y Fh(\000)p Fi(xtH)1463 1641 y Fe(\021)1583 1688 y Fw(\(2.22\))257 1794 y(for)14 b Fm(\034)r(;)7 b(t)k(>)h Fk(0)p Fw(.)18 b(W)m(e)13 b(ha)o(v)o(e)298 1911 y Fk(\()p Fm(I)g Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))500 1894 y Fh(\000)p Fj(1)555 1911 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)c Fm(tH)s Fk(\))753 1894 y Fh(\000)p Fj(1)810 1911 y Fk(=)868 1859 y Fh(1)855 1872 y Fe(X)853 1960 y Fi(n)p Fj(=0)934 1855 y Fe(Z)976 1865 y Fi(n)p Fj(+1)957 1949 y Fi(n)1059 1911 y Fm(dx)i(e)1135 1894 y Fh(\000)p Fi(x)1189 1865 y Fe(\020)1214 1911 y Fm(e)1233 1894 y Fh(\000)p Fi(xtS)r Fj(\()p Fi(\034)s Fj(\))1368 1911 y Fl(\000)f Fm(e)1429 1894 y Fh(\000)p Fi(xtH)1518 1865 y Fe(\021)1583 1911 y Fw(\(2.23\))257 2033 y(for)k Fm(\034)r(;)7 b(t)k(>)h Fk(0)p Fw(.)18 b(By)c(the)h(substitution)f Fm(x)d Fk(=)h Fm(y)f Fk(+)e Fm(n)14 b Fw(w)o(e)g(obtain)379 2119 y Fk(\()p Fm(I)f Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))581 2102 y Fh(\000)p Fj(1)636 2119 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)d Fm(tH)s Fk(\))835 2102 y Fh(\000)p Fj(1)891 2119 y Fk(=)660 b Fw(\(2.24\))595 2163 y Fh(1)582 2175 y Fe(X)581 2263 y Fi(n)p Fj(=0)662 2215 y Fm(e)681 2197 y Fh(\000)p Fi(n)748 2158 y Fe(Z)789 2168 y Fj(1)771 2252 y(0)826 2215 y Fm(dy)13 b(e)900 2197 y Fh(\000)p Fi(y)954 2168 y Fe(\020)978 2215 y Fm(e)997 2197 y Fh(\000)p Fj(\()p Fi(y)q Fj(+)p Fi(n)p Fj(\))p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))1204 2215 y Fl(\000)c Fm(e)1264 2197 y Fh(\000)p Fj(\()p Fi(y)q Fj(+)p Fi(n)p Fj(\))p Fi(tH)1425 2168 y Fe(\021)257 2336 y Fw(for)14 b Fm(\034)r(;)7 b(t)k(>)h Fk(0)p Fw(.)18 b(Since)300 2422 y Fm(e)319 2405 y Fh(\000)p Fj(\()p Fi(y)q Fj(+)p Fi(n)p Fj(\))p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))525 2422 y Fl(\000)10 b Fm(e)586 2405 y Fh(\000)p Fj(\()p Fi(y)q Fj(+)p Fi(n)p Fj(\))p Fi(tH)758 2422 y Fk(=)793 b Fw(\(2.25\))501 2451 y Fe(\020)526 2497 y Fm(e)545 2480 y Fh(\000)p Fi(ntS)r Fj(\()p Fi(\034)s Fj(\))682 2497 y Fl(\000)10 b Fm(e)743 2480 y Fh(\000)p Fi(ntH)833 2451 y Fe(\021)865 2497 y Fm(e)884 2480 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))1019 2497 y Fk(+)f Fm(e)1079 2480 y Fh(\000)p Fi(ntH)1177 2451 y Fe(\020)1202 2497 y Fm(e)1221 2480 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))1355 2497 y Fl(\000)h Fm(e)1416 2480 y Fh(\000)p Fi(y)q(tH)1504 2451 y Fe(\021)963 2628 y Fw(6)p eop %%Page: 7 7 7 6 bop 257 262 a Fw(and)350 382 y Fm(e)369 365 y Fh(\000)p Fi(ntS)r Fj(\()p Fi(\034)s Fj(\))506 382 y Fl(\000)9 b Fm(e)566 365 y Fh(\000)p Fi(ntH)669 382 y Fk(=)712 330 y Fi(n)p Fh(\000)p Fj(1)714 343 y Fe(X)714 432 y Fi(k)q Fj(=0)794 382 y Fm(e)813 365 y Fh(\000)p Fi(k)q(tS)r Fj(\()p Fi(\034)s Fj(\))946 336 y Fe(\020)970 382 y Fm(e)989 365 y Fh(\000)p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))1106 382 y Fl(\000)h Fm(e)1167 365 y Fh(\000)p Fi(tH)1237 336 y Fe(\021)1269 382 y Fm(e)1288 365 y Fh(\000)p Fj(\()p Fi(n)p Fh(\000)p Fi(k)q Fh(\000)p Fj(1\))p Fi(tH)1583 382 y Fw(\(2.26\))257 507 y(w)o(e)k(get)304 599 y Fk(\()p Fm(I)f Fk(+)c Fm(tS)r Fk(\()p Fm(\034)c Fk(\)\))505 581 y Fh(\000)p Fj(1)560 599 y Fl(\000)10 b Fk(\()p Fm(I)j Fk(+)c Fm(tH)s Fk(\))759 581 y Fh(\000)p Fj(1)815 599 y Fk(=)736 b Fw(\(2.27\))319 648 y Fh(1)306 661 y Fe(X)304 749 y Fi(n)p Fj(=0)385 700 y Fm(e)404 683 y Fh(\000)p Fi(n)460 629 y Fe(\()494 648 y Fi(n)p Fh(\000)p Fj(1)495 661 y Fe(X)495 750 y Fi(k)q Fj(=0)563 700 y Fm(e)582 683 y Fh(\000)p Fi(k)q(tS)r Fj(\()p Fi(\034)s Fj(\))715 654 y Fe(\020)740 700 y Fm(e)759 683 y Fh(\000)p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))876 700 y Fl(\000)9 b Fm(e)936 683 y Fh(\000)p Fi(tH)1006 654 y Fe(\021)1038 700 y Fm(e)1057 683 y Fh(\000)p Fj(\()p Fi(n)p Fh(\000)p Fi(k)q Fh(\000)p Fj(1\))p Fi(tH)1268 644 y Fe(Z)1309 654 y Fj(1)1291 738 y(0)1346 700 y Fm(dy)k(e)1420 683 y Fh(\000)p Fi(y)1478 700 y Fm(e)1497 683 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))1622 700 y Fk(+)309 779 y Fe(Z)351 790 y Fj(1)332 874 y(0)388 836 y Fm(dy)g(e)462 819 y Fh(\000)p Fi(y)520 836 y Fm(e)539 819 y Fh(\000)p Fi(ntH)636 790 y Fe(\020)661 836 y Fm(e)680 819 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))815 836 y Fl(\000)c Fm(e)875 819 y Fh(\000)p Fi(y)q(tH)963 790 y Fe(\021)988 777 y(\033)1026 836 y Fm(:)257 950 y Fw(Hence)16 b(w)o(e)e(obtain)f(the)h(estimate)283 1041 y Fl(k)p Fk(\()p Fm(I)f Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))506 1024 y Fh(\000)p Fj(1)560 1041 y Fl(\000)10 b Fk(\()p Fm(I)j Fk(+)c Fm(tH)s Fk(\))759 1024 y Fh(\000)p Fj(1)804 1041 y Fl(k)i(\024)715 b Fw(\(2.28\))440 1085 y Fh(1)427 1097 y Fe(X)426 1185 y Fi(n)p Fj(=0)507 1136 y Fm(e)526 1119 y Fh(\000)p Fi(n)581 1078 y Fe(\032)612 1136 y Fm(n)644 1089 y Fe(\015)644 1114 y(\015)644 1139 y(\015)667 1136 y Fm(e)686 1119 y Fh(\000)p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))803 1136 y Fl(\000)10 b Fm(e)864 1119 y Fh(\000)p Fi(tH)934 1089 y Fe(\015)934 1114 y(\015)934 1139 y(\015)966 1136 y Fk(+)1008 1080 y Fe(Z)1049 1090 y Fj(1)1031 1174 y(0)1086 1136 y Fm(dy)j(e)1160 1119 y Fh(\000)p Fi(y)1225 1089 y Fe(\015)1225 1114 y(\015)1225 1139 y(\015)1248 1136 y Fm(e)1267 1119 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))1401 1136 y Fl(\000)d Fm(e)1462 1119 y Fh(\000)p Fi(y)q(tH)1550 1089 y Fe(\015)1550 1114 y(\015)1550 1139 y(\015)1573 1078 y(\033)257 1266 y Fw(for)k Fm(\034)r(;)7 b(t)k(>)h Fk(0)p Fw(.)18 b(By)c(assumption)f(\(2.14\))f (w)o(e)j(ha)o(v)o(e)e(the)i(estimate)711 1324 y Fe(\015)711 1349 y(\015)711 1374 y(\015)734 1372 y Fm(e)753 1355 y Fh(\000)p Fi(tS)r Fj(\()p Fi(\034)s Fj(\))869 1372 y Fl(\000)10 b Fm(e)930 1355 y Fh(\000)p Fi(tH)1000 1324 y Fe(\015)1000 1349 y(\015)1000 1374 y(\015)1035 1372 y Fl(\024)i Fm(C)1109 1378 y Fi(\045)1135 1326 y Fe(\020)1164 1344 y Fm(\034)p 1164 1362 23 2 v 1168 1400 a(t)1192 1326 y Fe(\021)1217 1334 y Fi(\045)1583 1372 y Fw(\(2.29\))257 1477 y(for)i Fm(\034)r(;)7 b(t)k Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(with)i Fk(0)d Fm(<)h(\034)k Fl(\024)c Fm(t)p Fw(.)18 b(F)m(urther,)c(w)o(e)g(use)h(the)f(decomp)q(osition)259 1542 y Fe(Z)301 1552 y Fj(1)282 1636 y(0)338 1598 y Fm(dy)f(e)412 1581 y Fh(\000)p Fi(y)477 1550 y Fe(\015)477 1575 y(\015)477 1600 y(\015)500 1598 y Fm(e)519 1581 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))653 1598 y Fl(\000)d Fm(e)714 1581 y Fh(\000)p Fi(y)q(tH)802 1550 y Fe(\015)802 1575 y(\015)802 1600 y(\015)837 1598 y Fk(=)714 b Fw(\(2.30\))402 1664 y Fe(Z)443 1674 y Fj(1)425 1758 y Fi(\034)s(=t)493 1721 y Fm(dy)13 b(e)567 1704 y Fh(\000)p Fi(y)632 1673 y Fe(\015)632 1698 y(\015)632 1723 y(\015)655 1721 y Fm(e)674 1704 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))809 1721 y Fl(\000)c Fm(e)869 1704 y Fh(\000)p Fi(y)q(tH)957 1673 y Fe(\015)957 1698 y(\015)957 1723 y(\015)990 1721 y Fk(+)1031 1664 y Fe(Z)1073 1674 y Fi(\034)s(=t)1054 1758 y Fj(0)1141 1721 y Fm(dy)k(e)1215 1704 y Fh(\000)p Fi(y)1280 1673 y Fe(\015)1280 1698 y(\015)1280 1723 y(\015)1303 1721 y Fm(e)1322 1704 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))1457 1721 y Fl(\000)c Fm(e)1517 1704 y Fh(\000)p Fi(y)q(tH)1605 1673 y Fe(\015)1605 1698 y(\015)1605 1723 y(\015)257 1843 y Fw(Setting)k Fm(t)414 1828 y Fh(0)437 1843 y Fk(=)f Fm(y)q(t)i Fw(one)f(has)g Fm(\034)j Fl(\024)c Fm(t)772 1828 y Fh(0)796 1843 y Fw(if)g Fm(\034)5 b(=t)11 b Fl(\024)h Fm(y)q Fw(.)18 b(Hence)d(b)o(y)d(assumption)f(\(2.14\))h(w) o(e)h(obtain)680 1913 y Fe(\015)680 1938 y(\015)680 1963 y(\015)703 1961 y Fm(e)722 1944 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))857 1961 y Fl(\000)c Fm(e)917 1944 y Fh(\000)p Fi(y)q(tH)1005 1913 y Fe(\015)1005 1938 y(\015)1005 1963 y(\015)1040 1961 y Fl(\024)j Fm(C)1114 1967 y Fi(\045)1140 1903 y Fe(\022)1182 1933 y Fm(\034)p 1175 1952 37 2 v 1175 1990 a(ty)1217 1903 y Fe(\023)1248 1911 y Fi(\045)1583 1961 y Fw(\(2.31\))257 2077 y(for)i Fm(\034)r(;)7 b(t;)g(y)13 b Fl(2)e Fk(\(0)p Fm(;)c Fk(1])12 b Fw(and)i Fm(\034)5 b(=t)11 b Fl(\024)h Fm(y)q Fw(.)19 b(This)14 b(yields)f(the)i(estimate)388 2142 y Fe(Z)430 2152 y Fj(1)411 2236 y Fi(\034)s(=t)480 2198 y Fm(dy)e(e)554 2181 y Fh(\000)p Fi(y)619 2151 y Fe(\015)619 2175 y(\015)619 2200 y(\015)642 2198 y Fm(e)661 2181 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))795 2198 y Fl(\000)d Fm(e)856 2181 y Fh(\000)p Fi(y)q(tH)944 2151 y Fe(\015)944 2175 y(\015)944 2200 y(\015)978 2198 y Fl(\024)i Fm(C)1052 2204 y Fi(\045)1078 2142 y Fe(Z)1120 2152 y Fj(1)1101 2236 y(0)1157 2198 y Fm(dy)h(e)1231 2181 y Fh(\000)p Fi(y)1277 2198 y Fm(y)1298 2181 y Fh(\000)p Fi(\045)1351 2152 y Fe(\020)1381 2170 y Fm(\034)p 1381 2189 23 2 v 1385 2227 a(t)1409 2152 y Fe(\021)1433 2161 y Fi(\045)1583 2198 y Fw(\(2.32\))257 2320 y(for)h Fm(\034)r(;)7 b(t)k Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(with)i Fk(0)d Fm(<)h(\034)k Fl(\024)c Fm(t)p Fw(.)18 b(F)m(or)13 b Fm(\045)f(<)g Fk(1)i Fw(one)g(ob)o(viously)e(has)577 2388 y Fe(Z)618 2398 y Fi(\034)s(=t)600 2482 y Fj(0)687 2444 y Fm(dy)h(e)761 2427 y Fh(\000)p Fi(y)826 2397 y Fe(\015)826 2421 y(\015)826 2446 y(\015)849 2444 y Fm(e)868 2427 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))1002 2444 y Fl(\000)d Fm(e)1063 2427 y Fh(\000)p Fi(y)q(tH)1151 2397 y Fe(\015)1151 2421 y(\015)1151 2446 y(\015)1185 2444 y Fl(\024)24 b Fk(2)1269 2398 y Fe(\020)1298 2416 y Fm(\034)p 1298 2435 V 1302 2473 a(t)1326 2398 y Fe(\021)1351 2407 y Fi(\045)1583 2444 y Fw(\(2.33\))963 2628 y(7)p eop %%Page: 8 8 8 7 bop 257 262 a Fw(for)14 b Fm(\034)r(;)7 b(t)k Fl(2)h Fk([0)p Fm(;)7 b Fk(1])12 b Fw(with)h Fk(0)e Fm(<)h(\034)17 b Fl(\024)12 b Fm(t)p Fw(.)18 b(T)m(aking)12 b(in)o(to)h(accoun)o(t)i (\(2.32\))d(and)i(\(2.33\))f(w)o(e)h(obtain)257 311 y(from)e(\(2.30\))h (the)i(estimate)328 370 y Fe(Z)370 380 y Fj(1)351 465 y(0)407 427 y Fm(dy)e(e)481 410 y Fh(\000)p Fi(y)546 379 y Fe(\015)546 404 y(\015)546 429 y(\015)569 427 y Fm(e)588 410 y Fh(\000)p Fi(y)q(tS)r Fj(\()p Fi(\034)s Fj(\))722 427 y Fl(\000)d Fm(e)783 410 y Fh(\000)p Fi(y)q(tH)871 379 y Fe(\015)871 404 y(\015)871 429 y(\015)906 427 y Fl(\024)949 368 y Fe(\022)980 427 y Fm(C)1010 433 y Fi(\045)1036 370 y Fe(Z)1077 380 y Fj(1)1059 465 y(0)1114 427 y Fm(dy)j(e)1188 410 y Fh(\000)p Fi(y)1235 427 y Fm(y)1256 410 y Fh(\000)p Fi(\045)1311 427 y Fk(+)c(2)1373 368 y Fe(\023)1411 381 y(\020)1440 399 y Fm(\034)p 1440 417 23 2 v 1444 455 a(t)1468 381 y Fe(\021)1493 389 y Fi(\045)1583 427 y Fw(\(2.34\))257 537 y(for)17 b Fm(\034)r(;)7 b(t)17 b Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1])15 b Fw(with)i Fk(0)g Fm(<)g(\034)22 b Fl(\024)17 b Fm(t)p Fw(.)28 b(Finally)m(,)15 b(using)i(\(2.29\))f(and)h(\(2.34\))f(w)o(e)i(get)f(from)257 587 y(\(2.28\))c(the)i(estimate)386 673 y Fl(k)p Fk(\()p Fm(I)d Fk(+)e Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))608 656 y Fh(\000)p Fj(1)663 673 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)d Fm(tH)s Fk(\))862 656 y Fh(\000)p Fj(1)906 673 y Fl(k)i(\024)612 b Fw(\(2.35\))602 716 y Fh(1)588 729 y Fe(X)587 817 y Fi(n)p Fj(=0)668 768 y Fm(e)687 751 y Fh(\000)p Fi(n)743 710 y Fe(\032)774 768 y Fm(n)11 b(C)840 774 y Fi(\045)868 768 y Fk(+)f Fm(C)940 774 y Fi(\045)966 712 y Fe(Z)1007 722 y Fj(1)989 806 y(0)1044 768 y Fm(dy)j(e)1118 751 y Fh(\000)p Fi(y)1165 768 y Fm(y)1186 751 y Fh(\000)p Fi(\045)1241 768 y Fk(+)c(2)1303 710 y Fe(\033)1341 722 y(\020)1371 740 y Fm(\034)p 1371 759 V 1375 797 a(t)1399 722 y Fe(\021)1424 731 y Fi(\045)257 892 y Fw(for)14 b Fm(\034)r(;)7 b(t)k Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(with)i Fk(0)d Fm(<)h(\034)k Fl(\024)c Fm(t)p Fw(.)18 b(Setting)536 1011 y Fm(M)576 1017 y Fi(\045)608 1011 y Fk(:=)678 959 y Fh(1)664 972 y Fe(X)663 1060 y Fi(n)p Fj(=0)744 1011 y Fm(e)763 994 y Fh(\000)p Fi(n)819 953 y Fe(\032)850 1011 y Fm(n)11 b(C)916 1017 y Fi(\045)944 1011 y Fk(+)f Fm(C)1016 1017 y Fi(\045)1042 955 y Fe(Z)1083 965 y Fj(1)1065 1049 y(0)1120 1011 y Fm(dy)j(e)1194 994 y Fh(\000)p Fi(y)1241 1011 y Fm(y)1262 994 y Fh(\000)p Fi(\045)1317 1011 y Fk(+)c(2)1379 953 y Fe(\033)1583 1011 y Fw(\(2.36\))257 1135 y(w)o(e)14 b(ha)o(v)o(e)g(v)o(eri\034ed)h (\(2.13\).)978 b Fc(\003)257 1185 y Fw(W)m(e)12 b(note)h(that)g(in)e ([11)o(])h(it)g(w)o(as)h(sho)o(wn)f(that)h(for)f Fm(\045)g Fk(=)f(1)h Fw(condition)g(\(2.13\))g(implies)e(\(2.14\).)257 1235 y(It)i(is)f(unclear)h(whether)g(in)f(this)g(case)h(the)g(con)o(v)o (erse)h(is)e(also)g(true.)18 b(Note)11 b(also)g(that)g(setting)257 1284 y Fm(\034)17 b Fk(=)12 b Fm(t=n)p Fw(,)h Fm(n)e Fk(=)h(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(,)13 b(the)i(estimate)e(\(2.14\))g(transforms)g(in)o(to)724 1390 y Fl(k)p Fm(F)6 b Fk(\()p Fm(t=n)p Fk(\))871 1372 y Fi(n)902 1390 y Fl(\000)j Fm(e)962 1372 y Fh(\000)p Fi(tH)1032 1390 y Fl(k)j(\024)g Fm(C)1139 1396 y Fi(\045)1186 1362 y Fk(1)p 1174 1380 45 2 v 1174 1418 a Fm(n)1199 1406 y Fi(\045)1583 1390 y Fw(\(2.37\))257 1489 y(for)17 b Fm(t)g Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1])15 b Fw(and)i Fm(n)g Fk(=)g(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(.)28 b(Ho)o(w)o(ev)o(er,)17 b(this)h(is)f(nothing)f(else)i(but)f(an) g(op)q(erator)257 1539 y(norm)c(estimate)g(for)h(a)f(c)o(hosen)i (family)c(of)i(con)o(tractions)h Fm(F)6 b Fk(\()p Fm(\034)f Fk(\))p Fw(.)257 1675 y Fn(3)66 b(Auxilia)o(ry)19 b(estimates)257 1766 y Fw(W)m(e)14 b(are)g(going)f(to)h(apply)f(the)h(Theorem)f(2.2)g (to)h(the)h(family)603 1852 y Fm(F)6 b Fk(\()p Fm(\034)f Fk(\))12 b(:=)f Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\))p Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))p Fm(f)t Fk(\()p Fm(\034)g(A=)p Fk(2\))p Fm(;)50 b(\034)16 b(>)c Fk(0)p Fm(:)260 b Fw(\(3.1\))257 1938 y(In)14 b(the)g(follo)o(wing)e(w)o(e)i(use)g(notations)f(whic)o(h) h(essen)o(tially)g(go)f(bac)o(k)h(to)f([11)o(])g(and)h([12)o(])f(but) 257 1987 y(whic)o(h)h(are)g(sligh)o(tly)f(mo)q(di\034ed.)j(W)m(e)e(set) 601 2095 y Fm(A)632 2101 y Fi(\034)695 2095 y Fk(:=)785 2067 y Fm(I)f Fl(\000)c Fm(f)t Fk(\()p Fm(\034)c(A)p Fk(\))p 785 2085 184 2 v 865 2123 a Fm(\034)973 2095 y(;)90 b(\034)16 b(>)c Fk(0)p Fm(;)418 b Fw(\(3.2\))601 2199 y Fm(B)632 2205 y Fi(\034)695 2199 y Fk(:=)785 2171 y Fm(I)13 b Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\))p 785 2190 183 2 v 865 2228 a Fm(\034)972 2199 y(;)90 b(\034)16 b(>)c Fk(0)p Fm(;)419 b Fw(\(3.3\))597 2290 y Fm(K)632 2296 y Fi(\034)695 2290 y Fk(:=)41 b Fm(B)811 2296 y Fi(\034)842 2290 y Fk(+)9 b Fm(A)914 2297 y Fi(\034)s(=)p Fj(2)978 2290 y Fl(\000)1024 2262 y Fm(\034)p 1024 2281 23 2 v 1025 2319 a Fk(4)1052 2290 y Fm(A)1083 2273 y Fj(2)1083 2301 y Fi(\034)s(=)p Fj(2)1138 2290 y Fm(;)89 b(\034)16 b(>)c Fk(0)p Fm(:)254 b Fw(\(3.4\))257 2387 y(One)15 b(has)f Fm(K)454 2393 y Fi(\034)487 2387 y Fl(\025)d Fk(0)j Fw(and)486 2495 y Fm(S)r Fk(\()p Fm(\034)5 b Fk(\))13 b(=)f Fm(K)660 2501 y Fi(\034)690 2495 y Fk(+)736 2467 y Fm(\034)759 2452 y Fj(2)p 736 2486 42 2 v 747 2524 a Fk(4)783 2495 y Fm(A)814 2502 y Fi(\034)s(=)p Fj(2)868 2495 y Fm(B)899 2501 y Fi(\034)921 2495 y Fm(A)952 2502 y Fi(\034)s(=)p Fj(2)1015 2495 y Fl(\000)1062 2467 y Fm(\034)p 1062 2486 23 2 v 1063 2524 a Fk(2)1096 2462 y Fe(\000)1115 2495 y Fm(B)1146 2501 y Fi(\034)1168 2495 y Fm(A)1199 2502 y Fi(\034)s(=)p Fj(2)1262 2495 y Fk(+)e Fm(A)1335 2502 y Fi(\034)s(=)p Fj(2)1389 2495 y Fm(B)1420 2501 y Fi(\034)1441 2462 y Fe(\001)1604 2495 y Fw(\(3.5\))963 2628 y(8)p eop %%Page: 9 9 9 8 bop 257 262 a Fw(for)14 b Fm(\034)i(>)c Fk(0)p Fw(.)18 b(W)m(e)13 b(set)226 366 y Fm(Q)259 372 y Fi(\034)291 366 y Fk(:=)351 338 y Fm(\034)374 323 y Fj(2)p 351 356 42 2 v 361 394 a Fk(4)397 366 y(\()p Fm(I)g Fk(+)d Fm(K)521 372 y Fi(\034)542 366 y Fk(\))558 349 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)636 366 y Fm(A)667 373 y Fi(\034)s(=)p Fj(2)721 366 y Fm(B)752 372 y Fi(\034)774 366 y Fm(A)805 373 y Fi(\034)s(=)p Fj(2)859 366 y Fk(\()p Fm(I)j Fk(+)d Fm(K)983 372 y Fi(\034)1004 366 y Fk(\))1020 349 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1098 366 y Fl(\000)474 b Fw(\(3.6\))491 429 y Fm(\034)p 491 447 23 2 v 492 485 a Fk(2)519 457 y(\()p Fm(I)12 b Fk(+)e Fm(K)642 463 y Fi(\034)663 457 y Fk(\))679 440 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)764 423 y Fe(\000)783 457 y Fm(B)814 463 y Fi(\034)835 457 y Fm(A)866 464 y Fi(\034)s(=)p Fj(2)930 457 y Fk(+)g Fm(A)1003 464 y Fi(\034)s(=)p Fj(2)1057 457 y Fm(B)1088 463 y Fi(\034)1109 423 y Fe(\001)1135 457 y Fk(\()p Fm(I)j Fk(+)c Fm(K)1258 463 y Fi(\034)1279 457 y Fk(\))1295 440 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1374 457 y Fm(;)48 b(\034)16 b(>)c Fk(0)p Fm(;)257 555 y Fw(so)i(that)491 605 y Fm(I)f Fk(+)c Fm(S)r Fk(\()p Fm(\034)c Fk(\))13 b(=)f(\()p Fm(I)h Fk(+)c Fm(K)825 611 y Fi(\034)846 605 y Fk(\))862 588 y Fj(1)p Fi(=)p Fj(2)914 605 y Fk(\()p Fm(I)k Fk(+)d Fm(Q)1036 611 y Fi(\034)1056 605 y Fk(\)\()p Fm(I)j Fk(+)d Fm(K)1196 611 y Fi(\034)1217 605 y Fk(\))1233 588 y Fj(1)p Fi(=)p Fj(2)1285 605 y Fm(;)48 b(\034)17 b(>)11 b Fk(0)p Fm(:)257 677 y Fw(Our)16 b(next)g(aim)e(is)h(to)g(pro)o (v)o(e)h(sev)o(eral)g(estimates)f(whic)o(h)h(w)o(e)f(need)i(for)e(the)h (pro)q(of)f(of)g(the)257 727 y(main)d(theorem.)257 806 y Fg(Lemma)k(3.1)k Ff(L)n(et)e Fm(A)h Ff(and)f Fm(B)j Ff(b)n(e)d(non-ne)n(gative)i(self-adjoint)d(op)n(er)n(ators.)30 b(If)18 b Fm(f)23 b Ff(and)c Fm(g)257 856 y Ff(ar)n(e)d(Kato)f (functions,)g(then)g(one)h(has)476 905 y Fe(\015)476 930 y(\015)476 954 y(\015)499 952 y Fm(B)532 935 y Fj(1)p Fi(=)p Fj(2)530 963 y Fi(\034)584 952 y Fk(\()p Fm(I)d Fk(+)d Fm(K)708 958 y Fi(\034)729 952 y Fk(\))745 935 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)823 905 y Fe(\015)823 930 y(\015)823 954 y(\015)858 952 y Fl(\024)901 905 y Fe(\015)901 930 y(\015)901 954 y(\015)925 952 y Fk(\()p Fm(I)i Fk(+)e Fm(B)1044 958 y Fi(\034)1065 952 y Fk(\))1081 935 y Fj(1)p Fi(=)p Fj(2)1133 952 y Fk(\()p Fm(I)j Fk(+)d Fm(K)1257 958 y Fi(\034)1278 952 y Fk(\))1294 935 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1372 905 y Fe(\015)1372 930 y(\015)1372 954 y(\015)1407 952 y Fl(\024)i Fk(1)132 b Fw(\(3.7\))257 1049 y Ff(and)335 1115 y Fe(\015)335 1140 y(\015)335 1165 y(\015)358 1162 y Fm(A)389 1141 y Fj(1)p Fi(=)p Fj(2)389 1177 y Fi(\034)s(=)p Fj(2)443 1162 y Fk(\()p Fm(I)13 b Fk(+)d Fm(K)567 1168 y Fi(\034)588 1162 y Fk(\))604 1145 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)682 1115 y Fe(\015)682 1140 y(\015)682 1165 y(\015)717 1162 y Fl(\024)760 1126 y(p)p 795 1126 21 2 v 36 x Fk(2)823 1090 y Fe(\015)823 1115 y(\015)823 1140 y(\015)823 1165 y(\015)823 1189 y(\015)846 1104 y(\022)876 1162 y Fm(I)j Fk(+)953 1134 y(1)p 953 1153 V 953 1191 a(2)979 1162 y Fm(A)1010 1169 y Fi(\034)s(=)p Fj(2)1065 1104 y Fe(\023)1095 1113 y Fj(1)p Fi(=)p Fj(2)1154 1162 y Fk(\()p Fm(I)g Fk(+)d Fm(K)1278 1168 y Fi(\034)1299 1162 y Fk(\))1315 1145 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1393 1090 y Fe(\015)1393 1115 y(\015)1393 1140 y(\015)1393 1165 y(\015)1393 1189 y(\015)1428 1162 y Fl(\024)1471 1126 y(p)p 1506 1126 V 36 x Fk(2)77 b Fw(\(3.8\))257 1284 y Ff(for)15 b Fm(\034)h(>)c Fk(0)p Ff(.)19 b(Mor)n(e)n(over,)c(the)g(op)n(er)n(ator)g Fm(I)d Fk(+)c Fm(Q)982 1290 y Fi(\034)1017 1284 y Ff(is)14 b(b)n(ounde)n(d)r(ly)h(invertible)f(for)h(e)n(ach)g Fm(\034)h(>)c Fk(0)257 1334 y Ff(and)k(the)f(norm)h(of)f(its)f(inverse)i(op)n(er)n (ator)g(is)e(uniformly)i(estimate)n(d)e(by)714 1402 y Fe(\015)714 1427 y(\015)737 1438 y Fk(\()p Fm(I)f Fk(+)d Fm(Q)859 1444 y Fi(\034)879 1438 y Fk(\))895 1420 y Fh(\000)p Fj(1)940 1402 y Fe(\015)940 1427 y(\015)975 1438 y Fl(\024)1023 1409 y Fk(1)p 1023 1428 V 1023 1466 a(2)1056 1391 y Fe(\020)1081 1438 y Fk(3)f(+)1152 1401 y Fl(p)p 1187 1401 V 37 x Fk(5)1208 1391 y Fe(\021)1604 1438 y Fw(\(3.9\))257 1536 y Ff(for)16 b Fm(\034)g(>)c Fk(0)p Ff(.)257 1615 y Fg(Pro)q(of.)17 b Fw(The)10 b(pro)q(of)g(can)h(b)q(e)f(obtained)g(from)f([11)o(])g(b)o (y)h(making)e(the)j(replacemen)o(ts)f Fm(A)i Fl($)f Fm(B)257 1665 y Fw(and)j Fm(f)i Fl($)11 b Fm(g)q Fw(.)1197 b Fc(\003)257 1751 y Fg(Lemma)16 b(3.2)k Ff(L)n(et)e Fm(A)h Ff(and)f Fm(B)j Ff(b)n(e)d(non-ne)n(gative)i(self-adjoint)d(op)n(er)n(ators.)30 b(If)18 b Fm(f)23 b Ff(and)c Fm(g)257 1801 y Ff(ar)n(e)e(Kato)e (functions)i(and)f Fm(m)731 1807 y Fi(f)753 1801 y Fk(\()p Fm(x)p Fk(\))d Fm(<)g Fk(1)j Ff(for)g Fm(x)c(>)i Fk(0)p Ff(,)h(then)h(ther)n(e)g(ar)n(e)h(c)n(onstants)f Fm(C)1584 1807 y Fi(A)1623 1801 y Fm(>)e Fk(0)257 1851 y Ff(and)i Fm(C)368 1857 y Fj(0)398 1851 y Fm(>)c Fk(0)i Ff(such)i(that)470 1946 y Fl(k)p Fk(\()p 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%%Page: 10 10 10 9 bop 257 262 a Fw(for)23 b Fm(u)i Fl(2)h(H)d Fw(and)g Fm(\034)30 b(>)d Fk(0)p Fw(.)44 b(Ob)o(viously)m(,)23 b(there)h(is)e(a)g(constan)o(t)h Fm(\016)28 b(>)f Fk(0)22 b Fw(suc)o(h)h(that)257 276 y Fe(\014)257 301 y(\014)271 311 y Fk(\(1)9 b Fl(\000)h Fm(f)t Fk(\()p Fm(x)p Fk(\)\))p Fm(x)479 296 y Fh(\000)p Fj(1)533 311 y Fl(\000)g Fk(1)596 276 y Fe(\014)596 301 y(\014)621 311 y Fm(>)i Fk(1)p Fm(=)p Fk(2)h Fw(for)g Fm(x)f Fl(2)f Fk(\(0)p Fm(;)c(\016)r Fk(\))p Fw(.)17 b(Hence)842 417 y Fk(1)9 b Fl(\000)h Fm(f)t Fk(\()p Fm(x)p Fk(\))i Fl(\025)1055 389 y Fk(1)p 1055 408 21 2 v 1055 446 a(2)1081 417 y Fm(x)478 b Fw(\(3.13\))257 513 y(for)14 b Fm(x)d Fl(2)g Fk(\(0)p Fm(;)c(\016)r Fk(\))p Fw(,)13 b(whic)o(h)h(yields)540 615 y Fk(1)9 b Fl(\000)h Fm(f)t Fk(\()p Fm(x)p Fk(\))i Fl(\025)753 587 y Fk(1)p 753 605 V 753 643 a(2)779 615 y Fm(x)f(\037)840 622 y Fj(\(0)p Fi(;\016)q Fj(\))910 615 y Fk(\()p Fm(x)p Fk(\))f(+)f(\(1)g Fl(\000)h Fm(m)1141 621 y Fi(f)1163 615 y Fk(\()p Fm(\016)r Fk(\)\))i Fm(\037)1269 622 y Fj([)p Fi(\016)o(;)p Fh(1)p Fj(\))1351 615 y Fk(\()p Fm(x)p Fk(\))176 b Fw(\(3.14\))257 711 y(for)17 b Fm(x)f(>)h Fk(0)p Fw(,)g(where)h Fm(\037)612 718 y Fj(\(0)p Fi(;\016)q Fj(\))682 711 y Fk(\()p Fl(\001)p Fk(\))f Fw(and)f Fm(\037)852 718 y Fj([)p Fi(\016)o(;)p Fh(1)p Fj(\))934 711 y Fk(\()p Fl(\001)p Fk(\))h Fw(are)g(the)h(c)o (haracteristic)g(functions)f(of)f(the)257 760 y(in)o(terv)n(als)e Fk(\(0)p Fm(;)7 b(\016)r Fk(\))13 b Fw(and)h Fk([)p Fm(\016)o(;)7 b Fl(1)p Fk(\))p Fw(,)12 b(resp)q(ectiv)o(ely)m(.)19 b(Hence)d(w)o(e)e(\034nd)479 866 y Fm(A)510 873 y Fi(\034)s(=)p Fj(2)575 866 y Fl(\025)624 838 y Fk(1)p 624 857 V 624 895 a(2)650 866 y Fm(A)e(E)724 872 y Fi(A)750 866 y Fk(\([0)p Fm(;)7 b Fk(2)p Fm(\016)r(=\034)e Fk(\)\))j(+)i(2)1011 838 y(1)e Fl(\000)i Fm(m)1118 844 y Fi(f)1140 838 y Fk(\()p Fm(\016)r Fk(\))p 1011 857 182 2 v 1090 895 a Fm(\034)1209 866 y(E)1240 872 y Fi(A)1266 866 y Fk(\([2)p Fm(\016)r(=\034)r(;)d Fl(1)p Fk(\)\))114 b Fw(\(3.15\))257 962 y(for)14 b Fm(\034)i(>)c Fk(0)p Fw(,)h(where)i Fm(E)596 968 y Fi(A)623 962 y Fk(\()p Fl(\001)p Fk(\))f Fw(is)f(the)i(sp)q(ectral)g(measure)e(of)h Fm(A)p Fw(.)k(Therefore)d(w)o(e)f(obtain)412 1044 y Fl(k)p Fk(\()p Fm(I)e Fk(+)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\))619 1027 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)698 1044 y Fm(u)p Fl(k)11 b(\024)797 b Fw(\(3.16\))613 1075 y Fe(q)p 654 1075 180 2 v 654 1129 a Fk(2\(3)9 b(+)763 1093 y Fl(p)p 797 1093 21 2 v 797 1129 a Fk(5\))841 1081 y Fe(\015)841 1106 y(\015)841 1131 y(\015)864 1129 y Fk(\()p Fm(I)k Fk(+)d Fm(A)p Fk(\))1000 1112 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1078 1129 y Fm(E)1109 1135 y Fi(A)1136 1129 y Fk(\([0)p Fm(;)d Fk(2)p Fm(\016)r(=\034)e Fk(\)\))p Fm(u)1345 1081 y Fe(\015)1345 1106 y(\015)1345 1131 y(\015)1375 1129 y Fk(+)613 1176 y Fe(s)p 654 1176 245 2 v 713 1225 a Fk(3)k(+)785 1191 y Fl(p)p 820 1191 21 2 v 34 x Fk(5)p 659 1244 235 2 v 659 1282 a(2\(1)g Fl(\000)h Fm(m)804 1288 y Fi(f)826 1282 y Fk(\()p Fm(\016)r Fk(\)\))906 1253 y Fl(k)p Fm(E)958 1259 y Fi(A)985 1253 y Fk(\([2)p Fm(\016)r(=\034)r(;)d Fl(1)p Fk(\)\))p Fm(u)p Fl(k)17 b Fm(\034)1273 1236 y Fj(1)p Fi(=)p Fj(2)257 1390 y Fw(for)f Fm(u)f Fl(2)g(H)h Fw(and)g Fm(\034)k(>)15 b Fk(0)p Fw(.)24 b(Setting)16 b Fm(C)856 1396 y Fi(A)898 1390 y Fk(:=)957 1338 y Fe(q)p 999 1338 180 2 v 52 x Fk(2\(3)8 b(+)1107 1355 y Fl(p)p 1142 1355 21 2 v 35 x Fk(5)o(\))16 b Fw(and)g Fm(C)1307 1396 y Fj(0)1341 1390 y Fk(:=)1400 1342 y Fe(q)p 1441 1342 186 2 v 1491 1373 a Fj(3+)1533 1349 y Fh(p)p 1560 1349 17 2 v 24 x Fj(5)p 1446 1380 176 2 v 1446 1404 a(2\(1)p Fh(\000)p Fi(m)1548 1408 y Fb(f)1566 1404 y Fj(\()p Fi(\016)q Fj(\)\))1642 1390 y Fw(w)o(e)257 1450 y(pro)o(v)o(e)e(\(3.10\).)1169 b Fc(\003)257 1532 y Fg(Lemma)16 b(3.3)k Ff(L)n(et)13 b Fm(A)g Ff(and)h Fm(B)i Ff(b)n(e)d(non-ne)n (gative)h(self-adjoint)e(op)n(er)n(ators)j(such)e(that)g(their)257 1582 y(form)21 b(sum)f(ob)n(eys)h(\(1.6\))f(for)g(some)h Fm(\013)f Fl(2)h Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))p Ff(.)33 b(F)m(urther,)23 b(let)c Fm(f)25 b Ff(and)c Fm(g)g Ff(b)n(e)f(Kato)257 1632 y(functions)c(which)e(satisfy)h(\(1.7\))g(and) g(\(1.8\).)294 1707 y(\(i\))20 b(If)15 b Fm(p;)7 b(q)12 b Fl(2)f Fk([0)p Fm(;)c(\013)p Fk(])13 b Ff(and)i Fm(p)9 b Fk(+)h Fm(q)i Fl(\025)g Fk(1)p Ff(,)j(then)g(ther)n(e)g(is)g(a)g(c)n (onstant)g Fm(D)1370 1713 y Fi(p;q)1427 1707 y Fm(>)d Fk(0)j Ff(such)g(that)579 1789 y Fl(k)p Fk(\()p Fm(I)e Fk(+)c Fm(H)s Fk(\))742 1772 y Fh(\000)p Fi(p)787 1789 y Fk(\()p Fm(H)k Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(H)s Fk(\))1134 1772 y Fh(\000)p Fi(q)1178 1789 y Fl(k)h(\024)h Fm(D)1288 1795 y Fi(p;q)1345 1789 y Fm(\034)1368 1772 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)1583 1789 y Fw(\(3.17\))361 1871 y Ff(for)j Fm(\034)i(>)12 b Fk(0)p Ff(.)281 1951 y(\(ii\))20 b(Ther)n(e)15 b(is)g(a)g(c)n (onstant)h Fm(D)761 1957 y Fi(\013)796 1951 y Fm(>)c Fk(0)i Ff(such)i(that)760 2033 y Fl(k)p Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))1006 2016 y Fh(\000)p Fi(\013)1056 2033 y Fl(k)i(\024)h Fm(D)1166 2039 y Fi(\013)1202 2033 y Fm(\034)1225 2016 y Fi(\013)p Fh(\000)p Fj(1)1583 2033 y Fw(\(3.18\))361 2115 y Ff(for)j Fm(\034)i(>)12 b Fk(0)p Ff(.)257 2197 y Fg(Pro)q(of.)46 b Fk(\()p Fm(i)p Fk(\))23 b Fw(First)h(w)o(e)f(note)g(that)h(if)e Fm(p)27 b Fl(2)f Fk([0)p Fm(;)7 b(\013)p Fk(])p Fw(,)23 b(then)h(condition)e(\(1.6\))g(implies)257 2247 y(dom)o Fk(\()p Fm(H)390 2232 y Fi(p)409 2247 y Fk(\))12 b Fl(\022)f Fw(dom)o Fk(\()p Fm(A)606 2232 y Fi(p)625 2247 y Fk(\))q Fl(\\)q Fw(dom)n Fk(\()p Fm(B)798 2232 y Fi(p)818 2247 y Fk(\))p Fw(.)17 b(Hence)11 b Fk(\()p Fm(I)t Fk(+)q Fm(A)p Fk(\))1100 2232 y Fi(p)1121 2247 y Fk(\()p Fm(I)t Fk(+)q Fm(H)s Fk(\))1246 2232 y Fh(\000)p Fi(p)1301 2247 y Fw(and)f Fk(\()p Fm(I)t Fk(+)q Fm(B)r Fk(\))1498 2232 y Fi(p)1519 2247 y Fk(\()p Fm(I)t Fk(+)q Fm(H)s Fk(\))1644 2232 y Fh(\000)p Fi(p)257 2297 y Fw(are)15 b(b)q(ounded)f(op)q (erators.)19 b(Since)327 2379 y Fm(I)13 b Fl(\000)d Fm(F)c Fk(\()p Fm(\034)f Fk(\))41 b(=)g Fm(I)13 b Fl(\000)d Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\))827 2361 y Fj(2)855 2379 y Fk(+)10 b(\()p Fm(I)j Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\)\))10 b(+)431 b Fw(\(3.19\))602 2441 y Fk(\()p Fm(I)13 b Fl(\000)d Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fl(\000)d Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\)\()p Fm(I)14 b Fl(\000)9 b Fm(f)t Fk(\()p Fm(\034)c(A=)p Fk(2\)\))10 b Fl(\000)602 2503 y Fk(\()p Fm(I)j Fl(\000)d Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fl(\000)d Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\))10 b Fl(\000)g Fk(\()p Fm(I)j Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\)\)\()p Fm(I)14 b Fl(\000)c Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\))12 b Fm(;)953 2628 y Fw(10)p eop %%Page: 11 11 11 10 bop 257 262 a Fw(w)o(e)14 b(\034nd)257 335 y Fk(\()p Fm(I)e Fk(+)e Fm(H)s Fk(\))399 318 y Fh(\000)p Fi(p)444 335 y Fk(\()p Fm(H)i Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))790 318 y Fh(\000)p Fi(q)846 335 y Fk(=)705 b Fw(\(3.20\))399 424 y Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))541 406 y Fh(\000)p Fi(p)593 365 y Fe(\022)624 424 y Fm(A)g Fl(\000)711 395 y Fm(I)j Fl(\000)e Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\))935 380 y Fj(2)p 711 414 244 2 v 821 452 a Fm(\034)959 365 y Fe(\023)996 424 y Fk(\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))1139 406 y Fh(\000)p Fi(q)1192 424 y Fk(+)399 540 y(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))541 523 y Fh(\000)p Fi(p)593 481 y Fe(\022)624 540 y Fm(B)i Fl(\000)713 512 y Fm(I)i Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\))p 713 530 183 2 v 793 568 a Fm(\034)900 481 y Fe(\023)938 540 y Fk(\()p Fm(I)12 b Fk(+)e Fm(H)s Fk(\))1080 523 y Fh(\000)p Fi(q)1134 540 y Fl(\000)399 632 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))541 615 y Fh(\000)p Fi(p)586 632 y Fk(\()p Fm(I)k Fl(\000)d Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fl(\000)d Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\)\()p Fm(I)14 b Fl(\000)9 b Fm(f)t Fk(\()p Fm(\034)c(A=)p Fk(2\)\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))1457 615 y Fh(\000)p Fi(q)1513 632 y Fm(\034)1536 615 y Fh(\000)p Fj(1)1590 632 y Fk(+)399 694 y(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))541 677 y Fh(\000)p Fi(p)586 694 y Fk(\()p Fm(I)k Fl(\000)d Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fl(\000)d Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))1200 677 y Fh(\000)p Fi(q)1256 694 y Fm(\034)1279 677 y Fh(\000)p Fj(1)1333 694 y Fk(+)399 756 y(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))541 739 y Fh(\000)p Fi(p)586 756 y Fk(\()p Fm(I)k Fl(\000)d Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\)\()p Fm(I)14 b Fl(\000)9 b Fm(f)t Fk(\()p Fm(\034)c(A=)p Fk(2\)\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))1200 739 y Fh(\000)p Fi(q)1256 756 y Fm(\034)1279 739 y Fh(\000)p Fj(1)1335 756 y Fm(;)257 829 y Fw(whic)o(h)14 b(giv)o(es)g(the)g(estimate)317 903 y Fl(k)p Fk(\()p Fm(I)e Fk(+)e Fm(H)s Fk(\))480 885 y Fh(\000)p Fi(p)525 903 y Fk(\()p Fm(H)i Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))871 885 y Fh(\000)p Fi(q)915 903 y Fl(k)j(\024)459 931 y Fe(\015)459 956 y(\015)459 981 y(\015)459 1006 y(\015)482 991 y Fk(\()p Fm(I)h Fk(+)c Fm(H)s Fk(\))624 974 y Fh(\000)p Fi(p)676 933 y Fe(\022)707 991 y Fm(A)g Fl(\000)794 963 y Fm(I)j Fl(\000)e Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\))1018 948 y Fj(2)p 794 982 244 2 v 904 1020 a Fm(\034)1042 933 y Fe(\023)1079 991 y Fk(\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))1222 974 y Fh(\000)p Fi(q)1266 931 y Fe(\015)1266 956 y(\015)1266 981 y(\015)1266 1006 y(\015)1298 991 y Fk(+)253 b Fw(\(3.21\))459 1047 y Fe(\015)459 1072 y(\015)459 1097 y(\015)459 1122 y(\015)482 1108 y Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))624 1090 y Fh(\000)p Fi(p)676 1049 y Fe(\022)707 1108 y Fm(B)i Fl(\000)796 1080 y Fm(I)i Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\))p 796 1098 183 2 v 876 1136 a Fm(\034)983 1049 y Fe(\023)1021 1108 y Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))1163 1090 y Fh(\000)p Fi(q)1207 1047 y Fe(\015)1207 1072 y(\015)1207 1097 y(\015)1207 1122 y(\015)1240 1108 y Fk(+)311 b Fw(\(3.22\))459 1199 y Fl(k)p Fk(\()p Fm(I)12 b Fl(\000)e Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))879 1182 y Fh(\000)p Fi(p)924 1199 y Fl(k)h(k)p Fk(\()p Fm(I)i Fl(\000)c Fm(f)t Fk(\()p Fm(\034)c(A=)p Fk(2\)\)\()p Fm(I)14 b Fk(+)c Fm(H)s Fk(\))1377 1182 y Fh(\000)p Fi(q)1421 1199 y Fl(k)h Fm(\034)1476 1182 y Fh(\000)p Fj(1)1530 1199 y Fk(+)21 b Fw(\(3.23\))459 1262 y Fl(k)p Fk(\()p Fm(I)12 b Fl(\000)e Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))879 1245 y Fh(\000)p Fi(p)924 1262 y Fl(k)h(k)p Fk(\()p Fm(I)i Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(H)s Fk(\))1334 1245 y Fh(\000)p Fi(q)1378 1262 y Fl(k)i Fm(\034)1434 1245 y Fh(\000)p Fj(1)1487 1262 y Fk(+)64 b Fw(\(3.24\))459 1324 y Fl(k)p Fk(\()p Fm(I)12 b Fl(\000)e Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))836 1307 y Fh(\000)p Fi(p)881 1324 y Fl(k)j(k)p Fk(\()p Fm(I)g Fl(\000)e Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))1334 1307 y Fh(\000)p Fi(q)1378 1324 y Fl(k)i Fm(\034)1434 1307 y Fh(\000)p Fj(1)1583 1324 y Fw(\(3.25\))257 1397 y(for)i Fm(\034)i(>)c Fk(0)p Fw(.)18 b(T)m(o)13 b(estimate)g(\(3.21\))g(w)o(e)h (use)405 1437 y Fe(\015)405 1462 y(\015)405 1487 y(\015)405 1511 y(\015)428 1497 y Fk(\()p Fm(I)f Fk(+)d Fm(H)s Fk(\))571 1480 y Fh(\000)p Fi(p)623 1438 y Fe(\022)653 1497 y Fm(A)g Fl(\000)740 1469 y Fm(I)j Fl(\000)c Fm(f)t Fk(\()p Fm(\034)c(A=)p Fk(2\))964 1454 y Fj(2)p 740 1487 244 2 v 850 1525 a Fm(\034)989 1438 y Fe(\023)1026 1497 y Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))1168 1480 y Fh(\000)p Fi(q)1213 1437 y Fe(\015)1213 1462 y(\015)1213 1487 y(\015)1213 1511 y(\015)1247 1497 y Fl(\024)304 b Fw(\(3.26\))606 1553 y Fe(\015)606 1578 y(\015)629 1588 y Fk(\()p Fm(I)13 b Fk(+)d Fm(A)p Fk(\))765 1571 y Fi(p)784 1588 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))927 1571 y Fh(\000)p Fi(p)972 1588 y Fl(k)h(k)p Fk(\()p Fm(I)i Fk(+)c Fm(A)p Fk(\))1160 1571 y Fi(q)1179 1588 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))1321 1571 y Fh(\000)p Fi(q)1366 1553 y Fe(\015)1366 1578 y(\015)1398 1588 y Fl(\001)606 1621 y Fe(\015)606 1646 y(\015)606 1671 y(\015)606 1696 y(\015)629 1681 y Fk(\()p Fm(I)k Fk(+)d Fm(A)p Fk(\))765 1664 y Fh(\000)p Fi(p)817 1623 y Fe(\022)848 1681 y Fm(A)f Fl(\000)935 1653 y Fm(I)j Fl(\000)e Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\))1159 1638 y Fj(2)p 935 1672 V 1045 1710 a Fm(\034)1183 1623 y Fe(\023)1220 1681 y Fk(\()p Fm(I)13 b Fk(+)d Fm(A)p Fk(\))1356 1664 y Fh(\000)p Fi(q)1400 1621 y Fe(\015)1400 1646 y(\015)1400 1671 y(\015)1400 1696 y(\015)257 1777 y Fw(and)k(the)h(functional)d(calculus)i(whic)o(h)g(yield)408 1814 y Fe(\015)408 1839 y(\015)408 1864 y(\015)408 1889 y(\015)431 1874 y Fk(\()p Fm(I)f Fk(+)c Fm(H)s Fk(\))573 1857 y Fh(\000)p Fi(p)625 1816 y Fe(\022)656 1874 y Fm(A)g Fl(\000)743 1846 y Fm(I)j Fl(\000)e Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\))967 1831 y Fj(2)p 743 1865 V 853 1903 a Fm(\034)991 1816 y Fe(\023)1029 1874 y Fk(\()p Fm(I)12 b Fk(+)e Fm(H)s Fk(\))1171 1857 y Fh(\000)p Fi(q)1215 1814 y Fe(\015)1215 1839 y(\015)1215 1864 y(\015)1215 1889 y(\015)1250 1874 y Fl(\024)301 b Fw(\(3.27\))609 1930 y Fe(\015)609 1955 y(\015)632 1966 y Fk(\()p Fm(I)13 b Fk(+)c Fm(A)p Fk(\))767 1948 y Fi(p)787 1966 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))929 1948 y Fh(\000)p Fi(p)974 1966 y Fl(k)j(k)p Fk(\()p Fm(I)g Fk(+)e Fm(A)p Fk(\))1163 1948 y Fi(q)1181 1966 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))1324 1948 y Fh(\000)p Fi(q)1368 1930 y Fe(\015)1368 1955 y(\015)1400 1966 y Fl(\001)609 1998 y Fe(\032)640 2057 y Fl(j)p Fm(f)t Fl(j)688 2063 y Fi(p)p Fj(+)p Fi(q)758 2057 y Fk(+)804 2029 y(1)p 804 2047 21 2 v 804 2085 a(2)830 2057 y Fm(\015)851 2063 y Fi(f)873 2057 y Fk(\(\()p Fm(p)g Fk(+)f Fm(q)q Fk(\))p Fm(=)p Fk(2\))1071 2040 y Fj(2)1090 1998 y Fe(\033)1139 2057 y Fm(\034)1162 2040 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)257 2152 y Fw(for)14 b Fm(\034)i(>)c Fk(0)p Fw(,)h(where)662 2224 y Fm(\015)683 2230 y Fi(h)705 2224 y Fk(\()p Fm(r)q Fk(\))f(:=)f(sup)825 2258 y Fi(x>)p Fj(0)899 2196 y Fk(1)e Fl(\000)g Fm(h)p Fk(\()p Fm(x)p Fk(\))p 899 2214 152 2 v 953 2252 a Fm(x)977 2240 y Fi(r)1055 2224 y Fm(;)48 b(r)12 b Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1])p Fm(;)296 b Fw(\(3.28\))257 2313 y(and)14 b Fm(h)g Fw(is)g(a)f(Kato)h (function.)k(Similarl)o(y)m(,)10 b(one)k(estimates)g(\(3.22\))f(to)g (get)295 2351 y Fe(\015)295 2376 y(\015)295 2401 y(\015)295 2426 y(\015)318 2411 y Fk(\()p Fm(I)g Fk(+)d Fm(H)s Fk(\))461 2394 y Fh(\000)p Fi(p)513 2353 y Fe(\022)543 2411 y Fm(B)i Fl(\000)633 2383 y Fm(I)g Fl(\000)e Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\))p 633 2402 183 2 v 712 2440 a Fm(\034)820 2353 y Fe(\023)857 2411 y Fk(\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))1000 2394 y Fh(\000)p Fi(q)1044 2351 y Fe(\015)1044 2376 y(\015)1044 2401 y(\015)1044 2426 y(\015)1079 2411 y Fl(\024)472 b Fw(\(3.29\))496 2503 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)d Fm(B)r Fk(\))655 2486 y Fi(p)674 2503 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))817 2486 y Fh(\000)p Fi(p)862 2503 y Fl(k)h(k)p Fk(\()p Fm(I)i Fk(+)c Fm(B)r Fk(\))1052 2486 y Fi(q)1071 2503 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))1214 2486 y Fh(\000)p Fi(q)1258 2503 y Fl(k)h(j)p Fm(g)q Fl(j)1335 2509 y Fi(p)p Fj(+)p Fi(q)1407 2503 y Fm(\034)1430 2486 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)953 2628 y Fw(11)p eop %%Page: 12 12 12 11 bop 257 262 a Fw(for)14 b Fm(\034)i(>)c Fk(0)p Fw(.)18 b(Since)347 347 y Fl(k)p Fk(\(1)9 b Fl(\000)g Fm(f)t Fk(\()p Fm(\034)c(A=)p Fk(2\)\)\()p Fm(I)14 b Fk(+)c Fm(H)s Fk(\))767 329 y Fh(\000)p Fi(r)811 347 y Fl(k)h(\024)h Fk(2)908 329 y Fh(\000)p Fi(r)952 347 y Fl(k)p Fk(\()p Fm(I)h Fk(+)c Fm(A)p Fk(\))1108 329 y Fi(r)1127 347 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))1269 329 y Fh(\000)p Fi(r)1314 347 y Fl(k)i Fm(\015)1367 353 y Fi(f)1389 347 y Fk(\()p Fm(r)q Fk(\))h Fm(\034)1476 329 y Fi(r)1583 347 y Fw(\(3.30\))257 432 y(and)400 517 y Fl(k)p Fk(\()p Fm(I)h Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\)\)\()p Fm(I)15 b Fk(+)9 b Fm(H)s Fk(\))778 499 y Fh(\000)p Fi(r)822 517 y Fl(k)j(\024)f(k)p Fk(\()p Fm(I)i Fk(+)d Fm(B)r Fk(\))1057 499 y Fi(r)1076 517 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1218 499 y Fh(\000)p Fi(r)1262 517 y Fl(k)j Fm(\015)1316 523 y Fi(g)1336 517 y Fk(\()p Fm(r)q Fk(\))f Fm(\034)1422 499 y Fi(r)1583 517 y Fw(\(3.31\))257 602 y(for)j Fm(\034)i(>)c Fk(0)i Fw(and)f Fm(r)g Fl(2)e Fk(\(0)p Fm(;)c(\013)p Fk(])p Fw(,)12 b(w)o(e)i(\034nally)f(obtain)g (the)h(estimate)227 687 y Fl(k)p Fk(\()p Fm(I)f Fk(+)c Fm(H)s Fk(\))390 670 y Fh(\000)p Fi(p)436 687 y Fk(\()p Fm(H)j Fl(\000)d Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(H)s Fk(\))782 670 y Fh(\000)p Fi(q)826 687 y Fl(k)23 b(\024)g(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(A)p Fk(\))1081 670 y Fi(p)1101 687 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))1243 670 y Fh(\000)p Fi(p)1288 687 y Fl(k)j(\001)250 b Fw(\(3.32\))369 774 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(A)p Fk(\))525 757 y Fi(q)544 774 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))686 757 y Fh(\000)p Fi(q)731 774 y Fl(k)759 715 y Fe(\032)790 774 y Fl(j)p Fm(f)t Fl(j)838 780 y Fi(p)p Fj(+)p Fi(q)908 774 y Fk(+)954 746 y(1)p 954 764 21 2 v 954 802 a(2)980 774 y Fm(\015)1001 780 y Fi(f)1023 774 y Fk(\(\()p Fm(p)g Fk(+)h Fm(q)q Fk(\))p Fm(=)p Fk(2\))1221 757 y Fj(2)1239 715 y Fe(\033)1277 774 y Fm(\034)1300 757 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)1413 774 y Fk(+)369 866 y Fl(k)p Fk(\()p Fm(I)j Fk(+)c Fm(B)r Fk(\))527 849 y Fi(p)547 866 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))690 849 y Fh(\000)p Fi(p)735 866 y Fl(k)h(k)p Fk(\()p Fm(I)i Fk(+)c Fm(B)r Fk(\))925 849 y Fi(q)944 866 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))1087 849 y Fh(\000)p Fi(q)1131 866 y Fl(k)h(j)p Fm(g)q Fl(j)1208 872 y Fi(p)p Fj(+)p Fi(q)1280 866 y Fm(\034)1303 849 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)1416 866 y Fk(+)369 933 y(2)390 916 y Fh(\000)p Fj(\()p Fi(p)p Fj(+)p Fi(q)q Fj(\))503 933 y Fm(\015)524 939 y Fi(f)546 933 y Fk(\()p Fm(p)p Fk(\))p Fm(\015)620 939 y Fi(f)642 933 y Fk(\()p Fm(q)q Fk(\))p Fl(k)p Fk(\()p Fm(I)i Fk(+)d Fm(A)p Fk(\))851 916 y Fi(p)870 933 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1012 916 y Fh(\000)p Fi(p)1058 933 y Fl(k)i(k)p Fk(\()p Fm(I)h Fk(+)e Fm(A)p Fk(\))1246 916 y Fi(q)1265 933 y Fk(\()p Fm(I)i Fk(+)e Fm(H)s Fk(\))1407 916 y Fh(\000)p Fi(q)1451 933 y Fl(k)i Fm(\034)1507 916 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)1619 933 y Fk(+)369 995 y(2)390 978 y Fh(\000)p Fi(p)435 995 y Fm(\015)456 1001 y Fi(f)478 995 y Fk(\()p Fm(p)p Fk(\))p Fm(\015)552 1001 y Fi(g)572 995 y Fk(\()p Fm(q)q Fk(\))p Fl(k)p Fk(\()p Fm(I)h Fk(+)d Fm(A)p Fk(\))781 978 y Fi(p)800 995 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))943 978 y Fh(\000)p Fi(p)988 995 y Fl(k)h(k)p Fk(\()p Fm(I)i Fk(+)c Fm(B)r Fk(\))1178 978 y Fi(q)1197 995 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))1340 978 y Fh(\000)p Fi(q)1384 995 y Fl(k)h Fm(\034)1439 978 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)1552 995 y Fk(+)369 1058 y(2)390 1041 y Fh(\000)p Fi(q)434 1058 y Fm(\015)455 1064 y Fi(f)477 1058 y Fk(\()p Fm(q)q Fk(\))p Fm(\015)550 1064 y Fi(g)570 1058 y Fk(\()p Fm(p)p Fk(\))p Fl(k)p Fk(\()p Fm(I)i Fk(+)d Fm(A)p Fk(\))780 1041 y Fi(q)798 1058 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))941 1041 y Fh(\000)p Fi(q)985 1058 y Fl(k)h(k)p Fk(\()p Fm(I)i Fk(+)d Fm(B)r Fk(\))1176 1041 y Fi(p)1195 1058 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))1338 1041 y Fh(\000)p Fi(p)1383 1058 y Fl(k)h Fm(\034)1438 1041 y Fi(p)p Fj(+)p Fi(q)q Fh(\000)p Fj(1)257 1143 y Fw(for)j Fm(\034)i(>)c Fk(0)p Fw(.)18 b(Setting)237 1228 y Fm(D)271 1234 y Fi(p;q)328 1228 y Fk(:=)12 b Fl(k)p Fk(\()p Fm(I)g Fk(+)e Fm(A)p Fk(\))540 1211 y Fi(p)559 1228 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))702 1211 y Fh(\000)p Fi(p)747 1228 y Fl(k)h(\001)792 b Fw(\(3.33\))438 1316 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(A)p Fk(\))594 1299 y Fi(q)613 1316 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))755 1299 y Fh(\000)p Fi(q)800 1316 y Fl(k)828 1258 y Fe(\032)858 1316 y Fl(j)p Fm(f)t Fl(j)906 1322 y Fi(p)p Fj(+)p Fi(q)976 1316 y Fk(+)1023 1288 y(1)p 1023 1307 V 1023 1345 a(2)1049 1316 y Fm(\015)1070 1322 y Fi(f)1092 1316 y Fk(\(\()p Fm(p)g Fk(+)h Fm(q)q Fk(\))p Fm(=)p Fk(2\))1290 1299 y Fj(2)1308 1258 y Fe(\033)1348 1316 y Fk(+)438 1408 y Fl(k)p Fk(\()p Fm(I)j Fk(+)c Fm(B)r Fk(\))596 1390 y Fi(p)616 1408 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))759 1390 y Fh(\000)p Fi(p)804 1408 y Fl(k)h(k)p Fk(\()p Fm(I)i Fk(+)c Fm(B)r Fk(\))994 1390 y Fi(q)1013 1408 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))1156 1390 y Fh(\000)p Fi(q)1200 1408 y Fl(k)h(j)p Fm(g)q Fl(j)1277 1414 y Fi(p)p Fj(+)p Fi(q)1347 1408 y Fk(+)438 1475 y(2)459 1458 y Fh(\000)p Fj(\()p Fi(p)p Fj(+)p Fi(q)q Fj(\))572 1475 y Fm(\015)593 1481 y Fi(f)615 1475 y Fk(\()p Fm(p)p Fk(\))p Fm(\015)689 1481 y Fi(f)711 1475 y Fk(\()p Fm(q)q Fk(\))p Fl(k)p Fk(\()p Fm(I)i Fk(+)c Fm(A)p Fk(\))919 1458 y Fi(p)939 1475 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))1081 1458 y Fh(\000)p Fi(p)1126 1475 y Fl(k)j(k)p Fk(\()p Fm(I)g Fk(+)e Fm(A)p Fk(\))1315 1458 y Fi(q)1333 1475 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))1476 1458 y Fh(\000)p Fi(q)1520 1475 y Fl(k)f Fk(+)438 1537 y(2)459 1520 y Fh(\000)p Fi(p)504 1537 y Fm(\015)525 1543 y Fi(f)547 1537 y Fk(\()p Fm(p)p Fk(\))p Fm(\015)621 1543 y Fi(g)641 1537 y Fk(\()p Fm(q)q Fk(\))p Fl(k)p Fk(\()p Fm(I)k Fk(+)d Fm(A)p Fk(\))850 1520 y Fi(p)869 1537 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1011 1520 y Fh(\000)p Fi(p)1057 1537 y Fl(k)i(k)p Fk(\()p Fm(I)h Fk(+)e Fm(B)r Fk(\))1247 1520 y Fi(q)1266 1537 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1408 1520 y Fh(\000)p Fi(q)1453 1537 y Fl(k)g Fk(+)438 1600 y(2)459 1582 y Fh(\000)p Fi(q)503 1600 y Fm(\015)524 1606 y Fi(f)546 1600 y Fk(\()p Fm(q)q Fk(\))p Fm(\015)619 1606 y Fi(g)639 1600 y Fk(\()p Fm(p)p Fk(\))p Fl(k)p Fk(\()p Fm(I)k Fk(+)d Fm(A)p Fk(\))849 1582 y Fi(q)867 1600 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))1010 1582 y Fh(\000)p Fi(q)1054 1600 y Fl(k)h(k)p Fk(\()p Fm(I)i Fk(+)c Fm(B)r Fk(\))1244 1582 y Fi(p)1264 1600 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))1407 1582 y Fh(\000)p Fi(p)1452 1600 y Fl(k)257 1685 y Fw(for)k Fm(\034)i(>)c Fk(0)i Fw(w)o(e)g(pro)o(v)o(e)g(the)g (estimate)f(\(3.17\).)320 1734 y Fk(\()p Fm(ii)p Fk(\))h Fw(Using)g(the)g(decomp)q(osition)f(\(3.19\))g(w)o(e)h(\034nd)g(the)g (estimate)280 1784 y Fe(\015)280 1809 y(\015)303 1819 y Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))528 1802 y Fh(\000)p Fi(\013)578 1784 y Fe(\015)578 1809 y(\015)612 1819 y Fl(\024)939 b Fw(\(3.34\))422 1886 y Fk(4)450 1851 y Fe(\015)450 1876 y(\015)472 1886 y Fk(\()p Fm(I)13 b Fl(\000)d Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\)\)\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))872 1869 y Fh(\000)p Fi(\013)922 1851 y Fe(\015)922 1876 y(\015)963 1886 y Fm(\034)986 1869 y Fh(\000)p Fj(1)1040 1886 y Fk(+)f Fl(k)p Fk(\()p Fm(I)k Fl(\000)c Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(H)s Fk(\))1459 1869 y Fh(\000)p Fi(\013)1508 1886 y Fl(k)i Fm(\034)1564 1869 y Fh(\000)p Fj(1)257 1971 y Fw(for)i Fm(\034)i(>)c Fk(0)p Fw(.)18 b(T)m(aking)12 b(in)o(to)h(accoun)o(t)i (\(3.30\))e(and)g(\(3.31\))g(w)o(e)h(obtain)442 2021 y Fe(\015)442 2046 y(\015)465 2056 y Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))690 2039 y Fh(\000)p Fi(\013)740 2021 y Fe(\015)740 2046 y(\015)774 2056 y Fl(\024)777 b Fw(\(3.35\))643 2123 y Fk(2)664 2106 y Fj(2)p Fh(\000)p Fi(\013)730 2123 y Fl(k)p Fk(\()p Fm(I)12 b Fk(+)e Fm(A)p Fk(\))886 2106 y Fi(\013)910 2123 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1052 2106 y Fh(\000)p Fi(\013)1102 2123 y Fl(k)i Fm(\015)1155 2129 y Fi(f)1177 2123 y Fk(\()p Fm(\013)p Fk(\))g Fm(\034)1270 2106 y Fi(\013)p Fh(\000)p Fj(1)1346 2123 y Fk(+)643 2186 y Fl(k)p Fk(\()p Fm(I)h Fk(+)e Fm(B)r Fk(\))801 2169 y Fi(\013)825 2186 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))968 2169 y Fh(\000)p Fi(\013)1017 2186 y Fl(k)h Fm(\015)1070 2192 y Fi(g)1090 2186 y Fk(\()p Fm(\013)p Fk(\))h Fm(\034)1184 2169 y Fi(\013)p Fh(\000)p Fj(1)257 2271 y Fw(for)i Fm(\034)i(>)c Fk(0)p Fw(.)18 b(Setting)544 2356 y Fm(D)578 2362 y Fi(\013)614 2356 y Fk(:=)11 b(2)690 2339 y Fj(2)p Fh(\000)p Fi(\013)756 2356 y Fl(k)p Fk(\()p Fm(I)i Fk(+)c Fm(A)p Fk(\))912 2339 y Fi(\013)936 2356 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))1079 2339 y Fh(\000)p Fi(\013)1128 2356 y Fl(k)i Fm(\015)1182 2362 y Fi(f)1204 2356 y Fk(\()p Fm(\013)p Fk(\))320 b Fw(\(3.36\))746 2418 y Fk(+)p Fl(k)p Fk(\()p Fm(I)12 b Fk(+)e Fm(B)r Fk(\))936 2401 y Fi(\013)960 2418 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))1103 2401 y Fh(\000)p Fi(\013)1152 2418 y Fl(k)i Fm(\015)1206 2424 y Fi(g)1225 2418 y Fk(\()p Fm(\013)p Fk(\))257 2503 y Fw(w)o(e)i(pro)o(v)o(e)g(\(3.18\))f(for)h Fm(\034)i(>)c Fk(0)p Fw(.)932 b Fc(\003)953 2628 y Fw(12)p eop %%Page: 13 13 13 12 bop 257 262 a Fn(4)66 b(Error)21 b(estimate)257 352 y Fw(In)14 b(order)h(to)f(pro)o(v)o(e)f(the)i(main)d(theorem)h(w)o (e)h(need)h(the)g(follo)o(wing)c(t)o(w)o(o)i(lemmata.)257 432 y Fg(Lemma)j(4.1)k Ff(L)n(et)c Fm(A)h Ff(and)g Fm(B)i Ff(b)n(e)d(non-ne)n(gative)h(self-adjoint)f(op)n(er)n(ators)i(such)f (that)f(for)257 482 y(some)22 b Fm(\013)g Fl(2)h Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))20 b Ff(the)h(c)n(ondition)h(\(1.6\))f (is)g(satis\034e)n(d.)38 b(F)m(urther,)24 b(let)c Fm(f)26 b Ff(and)c Fm(g)g Ff(b)n(e)257 532 y(Kato)c(functions)h(which)e(ob)n (ey)h(c)n(onditions)h(\(1.7\))f(and)g(\(1.8\).)28 b(If)18 b(in)f(addition)i(one)f(has)257 582 y Fw(dom)o Fk(\()p Fm(A)383 567 y Fj(1)p Fi(=)p Fj(2)435 582 y Fk(\))g Fl(\022)f Fw(dom)o Fk(\()p Fm(B)646 567 y Fj(1)p Fi(=)p Fj(2)699 582 y Fk(\))h Ff(and)h Fm(m)853 588 y Fi(f)875 582 y Fk(\()p Fm(x)p Fk(\))e Fm(<)h Fk(1)g Ff(for)g Fm(x)g(>)f Fk(0)p Ff(,)i(then)f(ther)n(e)h(is)f(a)g(c)n(onstant)257 632 y Fm(S)282 639 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)394 632 y Fm(>)12 b Fk(0)i Ff(such)i(that)375 748 y Fm(t)p Fl(k)p Fk(\()p Fm(I)c Fk(+)e Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))612 730 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)691 748 y Fk(\()p Fm(H)12 b Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1052 730 y Fh(\000)p Fj(1)1097 748 y Fl(k)i(\024)h Fm(S)1198 755 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)1305 702 y Fe(\020)1335 719 y Fm(\034)p 1335 738 23 2 v 1339 776 a(t)1363 702 y Fe(\021)1387 710 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)1604 748 y Fw(\(4.1\))257 849 y Ff(for)k Fm(t;)7 b(\034)15 b Fl(2)d Fk(\(0)p Fm(;)7 b Fk(1])13 b Ff(with)h Fk(0)e Fm(<)f(\034)17 b Fl(\024)12 b Fm(t)p Ff(.)257 929 y Fg(Pro)q(of.)18 b Fw(By)c(the)h(functional)d(calculus)i (one)h(has)424 1017 y Fm(t)p Fl(k)p Fk(\()p Fm(I)d Fk(+)e Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))661 999 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)740 1017 y Fk(\()p Fm(H)12 b Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1101 999 y Fh(\000)p Fj(1)1146 1017 y Fl(k)i(\024)394 b Fw(\(4.2\))625 1079 y Fm(t)640 1062 y Fj(1)p Fi(=)p Fj(2)692 1079 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\))899 1062 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)978 1079 y Fk(\()p Fm(H)13 b Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1340 1062 y Fh(\000)p Fj(1)1384 1079 y Fl(k)257 1166 y Fw(for)k Fm(t)d Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(and)i Fm(\034)i(>)c Fk(0)p Fw(.)18 b(By)c(Lemma)d(3.2)i(w)o(e)h(\034nd)384 1254 y Fm(t)p Fl(k)p Fk(\()p Fm(I)f Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))622 1237 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)701 1254 y Fk(\()p Fm(H)12 b Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1062 1237 y Fh(\000)p Fj(1)1107 1254 y Fl(k)i(\024)433 b Fw(\(4.3\))586 1316 y Fm(C)616 1322 y Fi(A)654 1316 y Fl(k)p Fk(\()p Fm(I)12 b Fk(+)e Fm(A)p Fk(\))810 1299 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)888 1316 y Fk(\()p Fm(H)j Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1250 1299 y Fh(\000)p Fj(1)1294 1316 y Fl(k)h Fm(t)1341 1299 y Fj(1)p Fi(=)p Fj(2)1403 1316 y Fk(+)586 1378 y Fm(C)616 1384 y Fj(0)645 1378 y Fl(k)p Fk(\()p Fm(H)h Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1027 1361 y Fh(\000)p Fj(1)1072 1378 y Fl(k)i Fm(\034)1127 1361 y Fj(1)p Fi(=)p Fj(2)1191 1378 y Fm(t)1206 1361 y Fj(1)p Fi(=)p Fj(2)257 1466 y Fw(for)j Fm(t)d Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(and)i Fm(\034)i(>)c Fk(0)p Fw(.)18 b(Since)c(dom)n Fk(\()p Fm(H)938 1451 y Fj(1)p Fi(=)p Fj(2)991 1466 y Fk(\))d(=)h Fw(dom)o Fk(\()p Fm(A)1188 1451 y Fj(1)p Fi(=)p Fj(2)1240 1466 y Fk(\))p Fw(,)h(w)o(e)i(get)249 1554 y Fl(k)p Fk(\()p Fm(I)e Fk(+)c Fm(A)p Fk(\))405 1536 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)484 1554 y Fk(\()p Fm(H)j Fl(\000)d Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))845 1536 y Fh(\000)p Fj(1)890 1554 y Fl(k)h(\024)650 b Fw(\(4.4\))450 1616 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))613 1599 y Fj(1)p Fi(=)p Fj(2)666 1616 y Fk(\()p Fm(I)k Fk(+)c Fm(A)p Fk(\))801 1599 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)879 1616 y Fl(k)j(k)p Fk(\()p Fm(I)g Fk(+)e Fm(H)s Fk(\))1075 1599 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1153 1616 y Fk(\()p Fm(H)i Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1514 1599 y Fh(\000)p Fj(1)1559 1616 y Fl(k)257 1703 y Fw(for)14 b Fm(t)d Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(and)i Fm(\034)i(>)c Fk(0)p Fw(.)18 b(Using)13 b(again)g(the)i(functional)d (calculus)i(w)o(e)g(obtain)275 1791 y Fl(k)p Fk(\()p Fm(I)f Fk(+)c Fm(A)p Fk(\))431 1774 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)510 1791 y Fk(\()p Fm(H)j Fl(\000)d Fm(S)r Fk(\()p Fm(\034)c Fk(\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))855 1774 y Fh(\000)p Fj(1)899 1791 y Fl(k)i(\024)640 b Fw(\(4.5\))417 1853 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))580 1836 y Fj(1)p Fi(=)p Fj(2)633 1853 y Fk(\()p Fm(I)j Fk(+)e Fm(A)p Fk(\))768 1836 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)846 1853 y Fl(k)i(k)p Fk(\()p Fm(I)g Fk(+)e Fm(H)s Fk(\))1042 1836 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1120 1853 y Fk(\()p Fm(H)i Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))1466 1836 y Fh(\000)p Fi(\013)1516 1853 y Fl(k)i Fm(t)1563 1836 y Fh(\000)p Fi(\013)257 1941 y Fw(for)16 b Fm(t)f Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1])15 b Fw(and)h Fm(\034)k(>)c Fk(0)p Fw(.)24 b(By)16 b(Lemma)e(3.3)p Fk(\()p Fm(i)p Fk(\))h Fw(there)j(is)e(a)g(constan)o(t)g Fm(D)1457 1940 y Fa(1)p 1458 1945 15 2 v 1458 1961 a(2)1477 1951 y Fi(;\013)1525 1941 y Fm(>)g Fk(0)g Fw(suc)o(h)257 1990 y(that)426 2078 y Fl(k)p Fk(\()p Fm(I)d Fk(+)c Fm(A)p Fk(\))582 2061 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)661 2078 y Fk(\()p Fm(H)j Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\()p Fm(I)13 b Fk(+)d Fm(tH)s Fk(\))1006 2061 y Fh(\000)p Fj(1)1050 2078 y Fl(k)i(\024)489 b Fw(\(4.6\))627 2140 y Fm(D)666 2139 y Fa(1)p 667 2144 V 667 2161 a(2)686 2151 y Fi(;\013)731 2140 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))894 2123 y Fj(1)p Fi(=)p Fj(2)946 2140 y Fk(\()p Fm(I)k Fk(+)d Fm(A)p Fk(\))1082 2123 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1160 2140 y Fl(k)h Fm(\034)1215 2123 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)1326 2140 y Fm(t)1341 2123 y Fh(\000)p Fi(\013)1391 2140 y Fm(;)257 2228 y Fw(whic)o(h)j(yields)437 2315 y Fl(k)p Fk(\()p Fm(I)e Fk(+)e Fm(A)p Fk(\))593 2298 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)671 2315 y Fk(\()p Fm(H)j Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1033 2298 y Fh(\000)p Fj(1)1077 2315 y Fl(k)i Fm(t)1125 2298 y Fj(1)p Fi(=)p Fj(2)1188 2315 y Fl(\024)384 b Fw(\(4.7\))638 2402 y Fm(D)677 2401 y Fa(1)p 677 2406 V 677 2422 a(2)696 2412 y Fi(;\013)741 2402 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)d Fm(H)s Fk(\))905 2384 y Fj(1)p Fi(=)p Fj(2)957 2402 y Fk(\()p Fm(I)j Fk(+)c Fm(A)p Fk(\))1092 2384 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1171 2402 y Fl(k)1210 2356 y Fe(\020)1240 2374 y Fm(\034)p 1240 2392 23 2 v 1244 2430 a(t)1267 2356 y Fe(\021)1292 2364 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)257 2503 y Fw(for)14 b Fm(t)d Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(and)i Fm(\034)i(>)c Fk(0)p Fw(.)953 2628 y(13)p eop %%Page: 14 14 14 13 bop 320 262 a Fw(T)m(o)12 b(estimate)g(the)h(second)h(term)e(of)h (the)g(righ)o(t)f(hand)h(side)g(of)f(\(4.3\))h(w)o(e)g(use)g(again)f (the)257 311 y(functional)h(calculus)h(to)g(\034nd)g(that)485 406 y Fl(k)p Fk(\()p Fm(H)e Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))867 389 y Fh(\000)p Fj(1)912 406 y Fl(k)i(\024)993 378 y Fk(1)p 993 396 21 2 v 996 434 a Fm(t)1028 406 y Fk(+)e Fl(k)p Fm(S)r Fk(\()p Fm(\034)c Fk(\)\()p Fm(I)14 b Fk(+)9 b Fm(H)s Fk(\))1315 389 y Fh(\000)p Fi(\013)1365 406 y Fl(k)i Fm(t)1412 389 y Fh(\000)p Fi(\013)1604 406 y Fw(\(4.8\))257 505 y(for)j Fm(t)d Fl(2)g Fk(\(0)p Fm(;)c Fk(1])13 b Fw(and)g Fm(\034)k(>)12 b Fk(0)p Fw(.)17 b(By)d(virtue)g(of) f(Lemma)e(3.3)p Fk(\()p Fm(ii)p Fk(\))j Fw(there)h(is)f(a)f(constan)o (t)h Fm(D)1589 511 y Fi(\013)1625 505 y Fm(>)e Fk(0)257 555 y Fw(suc)o(h)j(that)665 605 y Fl(k)p Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))926 588 y Fh(\000)p Fj(1)971 605 y Fl(k)i(\024)h Fm(D)1081 611 y Fi(\013)1117 605 y Fm(\034)1140 588 y Fi(\013)p Fh(\000)p Fj(1)1217 605 y Fm(t)1232 588 y Fh(\000)p Fi(\013)1604 605 y Fw(\(4.9\))257 676 y(for)i Fm(t)d Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(and)i Fm(\034)i(>)c Fk(0)p Fw(,)h(whic)o(h)h (yields)571 786 y Fl(k)p Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\()p Fm(I)13 b Fk(+)d Fm(tH)s Fk(\))832 768 y Fh(\000)p Fj(1)876 786 y Fl(k)i Fm(\034)932 768 y Fj(1)p Fi(=)p Fj(2)995 786 y Fm(t)1010 768 y Fj(1)p Fi(=)p Fj(2)1074 786 y Fl(\024)g Fm(D)1152 792 y Fi(\013)1194 739 y Fe(\020)1224 757 y Fm(\034)p 1224 776 23 2 v 1228 814 a(t)1252 739 y Fe(\021)1277 748 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)1583 786 y Fw(\(4.10\))257 885 y(for)i Fm(t)d Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(and)i Fm(\034)i(>)c Fk(0)p Fw(.)18 b(Hence)d(w)o(e)f(obtain)405 994 y Fl(k)p Fk(\()p Fm(H)e Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))787 977 y Fh(\000)p Fj(1)832 994 y Fl(k)i Fm(\034)887 977 y Fj(1)p Fi(=)p Fj(2)951 994 y Fm(t)966 977 y Fj(1)p Fi(=)p Fj(2)1029 994 y Fl(\024)h Fk(\(1)d(+)h Fm(D)1195 1000 y Fi(\013)1219 994 y Fk(\))1253 948 y Fe(\020)1283 966 y Fm(\034)p 1283 985 V 1287 1023 a(t)1311 948 y Fe(\021)1336 957 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)1583 994 y Fw(\(4.11\))257 1094 y(for)15 b Fm(t;)7 b(\034)17 b Fl(2)12 b Fk(\(0)p Fm(;)7 b Fk(1])13 b Fw(and)i Fk(0)d Fm(<)h(\034)18 b Fl(\024)13 b Fm(t)p Fw(.)20 b(Applying)13 b(the)i(estimates)g(\(4.7\))f(and)g(\(4.10\))g(w) o(e)g(\034nd)257 1144 y(from)e(\(4.3\))i(that)250 1229 y Fm(t)p Fl(k)p Fk(\()p Fm(I)f Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))488 1212 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)567 1229 y Fk(\()p Fm(H)12 b Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))928 1212 y Fh(\000)p Fj(1)973 1229 y Fl(k)i(\024)546 b Fw(\(4.12\))452 1269 y Fe(n)479 1315 y Fm(C)509 1321 y Fi(A)547 1315 y Fm(D)586 1314 y Fa(1)p 587 1319 15 2 v 587 1335 a(2)606 1325 y Fi(;\013)651 1315 y Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(H)s Fk(\))814 1298 y Fj(1)p Fi(=)p Fj(2)866 1315 y Fk(\()p Fm(I)k Fk(+)d Fm(A)p Fk(\))1002 1298 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1080 1315 y Fl(k)f Fk(+)h Fm(C)1182 1321 y Fj(0)1200 1315 y Fk(\(1)f(+)g Fm(D)1321 1321 y Fi(\013)1346 1315 y Fk(\))1362 1269 y Fe(o)d(\020)1426 1287 y Fm(\034)p 1426 1306 23 2 v 1430 1344 a(t)1454 1269 y Fe(\021)1479 1278 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)257 1415 y Fw(for)14 b Fm(t;)7 b(\034)16 b Fl(2)11 b Fk(\(0)p Fm(;)c Fk(1])12 b Fw(with)i Fk(0)d Fm(<)h(\034)k Fl(\024)c Fm(t)p Fw(.)18 b(Setting)383 1500 y Fm(S)408 1507 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)520 1500 y Fk(:=)11 b Fm(C)605 1506 y Fi(A)643 1500 y Fm(D)682 1499 y Fa(1)p 683 1504 15 2 v 683 1520 a(2)702 1510 y Fi(;\013)747 1500 y Fl(k)p Fk(\()p Fm(I)i Fk(+)c Fm(H)s Fk(\))910 1483 y Fj(1)p Fi(=)p Fj(2)962 1500 y Fk(\()p Fm(I)k Fk(+)d Fm(A)p Fk(\))1098 1483 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)1176 1500 y Fl(k)f Fk(+)h Fm(C)1278 1506 y Fj(0)1296 1500 y Fk(\(1)f(+)g Fm(D)1417 1506 y Fi(\013)1442 1500 y Fk(\))125 b Fw(\(4.13\))257 1591 y(w)o(e)14 b(get)h(\(4.1\).)1171 b Fc(\003)257 1677 y Fg(Lemma)16 b(4.2)k Ff(L)n(et)c Fm(A)h Ff(and)g Fm(B)i Ff(b)n(e)d(non-ne)n(gative)h(self-adjoint)f(op)n(er)n(ators)i(such)f (that)f(for)257 1726 y(some)22 b Fm(\013)g Fl(2)h Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))20 b Ff(the)h(c)n(ondition)h(\(1.6\))f (is)g(satis\034e)n(d.)38 b(F)m(urther,)24 b(let)c Fm(f)26 b Ff(and)c Fm(g)g Ff(b)n(e)257 1776 y(Kato)15 b(functions)f(which)h(ob) n(ey)f(c)n(onditions)h(\(1.7\))f(and)h(\(1.8\).)k(Then)c(ther)n(e)f(is) g(a)h(c)n(onstant)257 1826 y Fm(G)290 1832 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)384 1826 y Fm(>)d Fk(0)j Ff(such)g(that)411 1932 y Fm(t)p Fl(k)p Fk(\()p Fm(I)e Fk(+)c Fm(tH)s Fk(\))604 1915 y Fh(\000)p Fj(1)649 1932 y Fk(\()p Fm(H)j Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1010 1915 y Fh(\000)p Fj(1)1055 1932 y Fl(k)i(\024)h Fm(G)1164 1938 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1265 1886 y Fe(\020)1294 1904 y Fm(\034)p 1294 1922 23 2 v 1298 1960 a(t)1322 1886 y Fe(\021)1347 1894 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1583 1932 y Fw(\(4.14\))257 2031 y Ff(for)k Fm(t)11 b Fl(2)g Fk(\(0)p Fm(;)c Fk(1])14 b Ff(and)h Fm(\034)i(>)12 b Fk(0)p Ff(.)257 2116 y Fg(Pro)q(of.)18 b Fw(By)c(the)h(functional)d(calculus)i(w)o(e)h(get)419 2202 y Fm(t)p Fl(k)p Fk(\()p Fm(I)d Fk(+)e Fm(tH)s Fk(\))612 2185 y Fh(\000)p Fi(\013)661 2202 y Fk(\()p Fm(H)j Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1023 2185 y Fh(\000)p Fi(\013)1072 2202 y Fl(k)h(\024)447 b Fw(\(4.15\))620 2264 y Fl(k)p Fk(\()p Fm(I)12 b Fk(+)e Fm(H)s Fk(\))783 2247 y Fh(\000)p Fi(\013)832 2264 y Fk(\()p Fm(H)j Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(H)s Fk(\))1179 2247 y Fh(\000)p Fi(\013)1228 2264 y Fl(k)i Fm(t)1276 2247 y Fh(\000)p Fj(\(2)p Fi(\013)p Fh(\000)p Fj(1\))257 2350 y Fw(for)k Fm(t)g Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1])15 b Fw(and)h Fm(\034)21 b(>)16 b Fk(0)p Fw(.)25 b(Applying)16 b(Lemma)e(3.3)h(w)o(e)i(\034nd)f(a)g(constan)o(t)h Fm(G)1522 2356 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1621 2350 y Fm(>)f Fk(0)257 2399 y Fw(suc)o(h)f(that)406 2495 y Fm(t)p Fl(k)p Fk(\()p Fm(I)e Fk(+)c Fm(tH)s Fk(\))599 2478 y Fh(\000)p Fi(\013)649 2495 y Fk(\()p Fm(H)j Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1010 2478 y Fh(\000)p Fi(\013)1060 2495 y Fl(k)i(\024)h Fm(G)1169 2501 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1270 2449 y Fe(\020)1299 2467 y Fm(\034)p 1299 2486 V 1303 2524 a(t)1327 2449 y Fe(\021)1352 2458 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1583 2495 y Fw(\(4.16\))953 2628 y(14)p eop %%Page: 15 15 15 14 bop 257 262 a Fw(for)14 b Fm(t)d Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1])12 b Fw(and)i Fm(\034)i(>)c Fk(0)p Fw(,)h(pro)o(ving)g (\(4.14\).)695 b Fc(\003)257 311 y Fw(W)m(e)14 b(are)g(no)o(w)g(going)e (to)i(pro)o(v)o(e)g(the)g(main)e(theorem)i(men)o(tioned)e(in)i(the)g (In)o(tro)q(duction.)257 411 y Fg(Pro)q(of)h(of)h(Theorem)e(1.1.)19 b Fw(By)14 b(the)h(resolv)o(en)o(t)f(iden)o(tities)326 482 y Fk(\()p Fm(I)f Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))528 465 y Fh(\000)p Fj(1)582 482 y Fl(\000)10 b Fk(\()p Fm(I)j Fk(+)c Fm(tH)s Fk(\))781 465 y Fh(\000)p Fj(1)867 482 y Fk(=)42 b Fm(t)12 b Fk(\()p Fm(I)h Fk(+)c Fm(tS)r Fk(\()p Fm(\034)c Fk(\)\))1169 465 y Fh(\000)p Fj(1)1215 482 y Fk(\()p Fm(H)12 b Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1576 465 y Fh(\000)p Fj(1)867 544 y Fk(=)42 b Fm(t)12 b Fk(\()p Fm(I)h Fk(+)c Fm(tH)s Fk(\))1125 527 y Fh(\000)p Fj(1)1170 544 y Fk(\()p Fm(H)j Fl(\000)d Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))1575 527 y Fh(\000)p Fj(1)257 615 y Fw(one)14 b(gets)268 686 y Fk(\()p Fm(I)f Fk(+)c Fm(tS)r Fk(\()p Fm(\034)c Fk(\)\))469 669 y Fh(\000)p Fj(1)524 686 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)d Fm(tH)s Fk(\))723 669 y Fh(\000)p Fj(1)779 686 y Fk(=)i Fm(t)p Fk(\()p Fm(I)h Fk(+)c Fm(tH)s Fk(\))995 669 y Fh(\000)p Fj(1)1040 686 y Fk(\()p Fm(H)j Fl(\000)d Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1401 669 y Fh(\000)p Fj(1)1446 686 y Fk(+)105 b Fw(\(4.17\))469 748 y Fm(t)484 731 y Fj(2)502 748 y Fk(\()p Fm(I)13 b Fk(+)d Fm(tH)s Fk(\))660 731 y Fh(\000)p Fj(1)704 748 y Fk(\()p Fm(H)j Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))1110 731 y Fh(\000)p Fj(1)1155 748 y Fk(\()p Fm(H)13 b Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1517 731 y Fh(\000)p Fj(1)257 819 y Fw(for)k Fm(\034)r(;)7 b(t)k(>)h Fk(0)p Fw(.)18 b(Hence)d(w)o(e)f(\034nd)g(the)h(estimate)247 890 y Fl(k)p Fk(\()p Fm(I)e Fk(+)c Fm(tS)r Fk(\()p Fm(\034)c Fk(\)\))469 873 y Fh(\000)p Fj(1)524 890 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)d Fm(tH)s Fk(\))723 873 y Fh(\000)p Fj(1)767 890 y Fl(k)i(\024)751 b Fw(\(4.18\))448 952 y Fm(t)p Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(tH)s Fk(\))641 935 y Fh(\000)p Fj(1)686 952 y Fk(\()p Fm(H)j Fl(\000)d Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1047 935 y Fh(\000)p Fj(1)1092 952 y Fl(k)f Fk(+)448 1014 y Fm(t)463 997 y Fj(2)482 1014 y Fl(k)p Fk(\()p Fm(I)j Fk(+)e Fm(tH)s Fk(\))660 997 y Fh(\000)p Fj(1)704 1014 y Fk(\()p Fm(H)j Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))1110 997 y Fh(\000)p Fj(1)1155 1014 y Fk(\()p Fm(H)13 b Fl(\000)c Fm(S)r Fk(\()p Fm(\034)c Fk(\)\)\()p Fm(I)14 b Fk(+)c Fm(tH)s Fk(\))1517 997 y Fh(\000)p Fj(1)1561 1014 y Fl(k)257 1085 y Fw(for)k Fm(\034)r(;)7 b(t)k(>)h Fk(0)p Fw(,)h(whic)o(h)h(can)g(b)q(e)h(written)f(as)418 1156 y Fl(k)p Fk(\()p Fm(I)f Fk(+)c Fm(tS)r Fk(\()p Fm(\034)c Fk(\)\))640 1139 y Fh(\000)p Fj(1)695 1156 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)d Fm(tH)s Fk(\))894 1139 y Fh(\000)p Fj(1)938 1156 y Fl(k)i(\024)580 b Fw(\(4.19\))619 1218 y Fm(t)p Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(tH)s Fk(\))812 1201 y Fh(\000)p Fj(1)857 1218 y Fk(\()p Fm(H)j Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1218 1201 y Fh(\000)p Fj(1)1263 1218 y Fl(k)g Fk(+)619 1280 y Fm(t)634 1263 y Fj(2)653 1280 y Fl(k)p Fk(\()p Fm(I)j Fk(+)e Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))875 1263 y Fh(\000)p Fj(1)p Fi(=)p Fj(2)954 1280 y Fk(\()p Fm(H)12 b Fl(\000)e Fm(S)r Fk(\()p Fm(\034)5 b Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(tH)s Fk(\))1315 1263 y Fh(\000)p Fj(1)1360 1280 y Fl(k)1381 1263 y Fj(2)1399 1280 y Fm(:)257 1351 y Fw(T)m(aking)k(in)o(to)g(accoun)o(t)h(Lemma)d (4.1)i(and)h(Lemma)d(4.2)i(w)o(e)h(get)304 1441 y Fl(k)p Fk(\()p Fm(I)f Fk(+)d Fm(tS)r Fk(\()p Fm(\034)5 b Fk(\)\))527 1423 y Fh(\000)p Fj(1)582 1441 y Fl(\000)k Fk(\()p Fm(I)k Fk(+)c Fm(tH)s Fk(\))780 1423 y Fh(\000)p Fj(1)825 1441 y Fl(k)i(\024)h Fm(G)934 1447 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1023 1395 y Fe(\020)1053 1413 y Fm(\034)p 1053 1431 23 2 v 1057 1469 a(t)1081 1395 y Fe(\021)1106 1403 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1198 1441 y Fk(+)d Fm(S)1266 1423 y Fj(2)1264 1452 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)1371 1395 y Fe(\020)1401 1413 y Fm(\034)p 1401 1431 V 1405 1469 a(t)1429 1395 y Fe(\021)1454 1403 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1583 1441 y Fw(\(4.20\))257 1528 y(for)16 b Fm(t;)7 b(\034)19 b Fl(2)14 b Fk(\(0)p Fm(;)7 b Fk(1])14 b Fw(with)i Fk(0)e Fm(<)h(\034)20 b Fl(\024)15 b Fm(t)p Fw(.)23 b(Setting)16 b Fm(M)1038 1534 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1136 1528 y Fk(:=)e Fm(G)1227 1534 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1320 1528 y Fk(+)d Fm(S)1390 1513 y Fj(2)1388 1541 y Fi(\013)p Fh(\000)p Fj(1)p Fi(=)p Fj(2)1503 1528 y Fw(one)16 b(gets)h(a)257 1584 y(constan)o(t)c(suc)o(h)g(that)g(the)g(condition)e(\(2.13\))h (holds)g(for)g Fm(\034)r(;)7 b(t)k Fl(2)g Fk(\(0)p Fm(;)c Fk(1])k Fw(with)h Fk(0)g Fm(<)g(\034)k Fl(\024)c Fm(t)p Fw(.)17 b(By)257 1634 y(Theorem)10 b(2.2)g(there)i(is)e(a)g(constan)o (t)h Fm(C)859 1640 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)953 1634 y Fm(>)h Fk(0)e Fw(suc)o(h)h(\(2.14\))f(is)g(v)n(alid)f(for)h Fm(\045)i Fk(=)g(2)p Fm(\013)s Fl(\000)r Fk(1)e Fw(and)257 1684 y Fm(\034)r(;)d(t)12 b Fl(2)g Fk(\(0)p Fm(;)7 b Fk(1])13 b Fw(and)h Fk(0)d Fm(<)i(\034)j Fl(\024)d Fm(t)p Fw(.)18 b(Setting)d Fm(C)921 1690 y Fj(1)p Fi(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1041 1684 y Fk(:=)d Fm(C)1127 1690 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1224 1684 y Fw(and)i Fm(\034)i Fk(=)d Fm(t=n)p Fw(,)g Fm(n)f Fk(=)g(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(,)257 1733 y(w)o(e)14 b(immediately)d(v)o(erify)i(\(1.10\))g(for)h Fm(t)d Fl(2)g Fk([0)p Fm(;)c Fk(1])p Fw(.)320 1783 y(T)m(o)18 b(extend)j(the)f (result)g(to)f Fm(t)i Fl(2)f Fk([0)p Fm(;)7 b(T)f Fk(])p Fw(,)18 b Fk(0)j Fm(<)g(T)26 b(<)21 b Fl(1)p Fw(,)f(w)o(e)g(set)g Fm(A)1429 1789 y Fi(s)1468 1783 y Fk(:=)g Fm(s)h(A)f Fw(and)257 1838 y Fm(B)288 1844 y Fi(s)319 1838 y Fk(:=)12 b Fm(s)g(B)17 b Fw(where)e Fm(s)e(>)f Fk(0)p Fw(.)19 b(Ob)o(viously)m(,)12 b(one)j(has)f Fm(H)1091 1844 y Fi(s)1120 1838 y Fk(:=)e Fm(A)1207 1844 y Fi(s)1249 1805 y(:)1237 1838 y Fk(+)h Fm(B)1313 1844 y Fi(s)1343 1838 y Fk(=)f Fm(s)h(H)s Fw(.)19 b(If)14 b Fm(A)g Fw(and)g Fm(B)257 1888 y Fw(are)e(non-negativ)o(e)f(self-adjoin)o(t)f(op)q (erators)i(suc)o(h)g(that)f(for)g(some)g Fm(\013)g Fl(2)g Fk(\(1)p Fm(=)p Fk(2)p Fm(;)c Fk(1\))j Fw(condition)257 1937 y(\(1.6\))k(is)g(satis\034ed,)h(then,)g(of)f(course,)h(one)f(has)h (dom)n Fk(\()p Fm(H)1148 1922 y Fi(\013)1145 1948 y(s)1172 1937 y Fk(\))e Fl(\022)f Fw(dom)o Fk(\()p Fm(A)1371 1922 y Fi(\013)1371 1948 y(s)1395 1937 y Fk(\))d Fl(\\)h Fw(dom)n Fk(\()p Fm(B)1585 1922 y Fi(\013)1583 1948 y(s)1609 1937 y Fk(\))15 b Fw(for)257 1995 y Fm(s)g(>)g Fk(0)p Fw(.)23 b(Similarly)m(,)12 b(the)17 b(condition)e(dom)n Fk(\()p Fm(A)964 1980 y Fj(1)p Fi(=)p Fj(2)1016 1995 y Fk(\))g Fl(\022)g Fw(dom)n Fk(\()p Fm(B)1221 1980 y Fj(1)p Fi(=)p Fj(2)1274 1995 y Fk(\))h Fw(implies)e(dom)n Fk(\()p Fm(A)1574 1974 y Fj(1)p Fi(=)p Fj(2)1574 2000 y Fi(s)1627 1995 y Fk(\))g Fl(\022)257 2053 y Fw(dom)o Fk(\()p Fm(B)385 2032 y Fj(1)p Fi(=)p Fj(2)383 2058 y Fi(s)438 2053 y Fk(\))g Fw(for)f Fm(s)f(>)g Fk(0)p Fw(.)18 b(Hence,)d(there)g(is)e(a)h (constan)o(t)g Fm(C)1171 2059 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1254 2053 y Fk(\()p Fm(s)p Fk(\))e Fm(>)g Fk(0)h Fw(suc)o(h)i(that)355 2108 y Fe(\015)355 2133 y(\015)378 2144 y Fk(\()q Fm(f)t Fk(\()p Fm(tA)481 2150 y Fi(s)499 2144 y Fm(=)p Fk(2)p Fm(n)p Fk(\))p Fm(g)q Fk(\()p Fm(tB)665 2150 y Fi(s)684 2144 y Fm(=n)p Fk(\))p Fm(f)t Fk(\()p Fm(tA)832 2150 y Fi(s)850 2144 y Fm(=)p Fk(2)p Fm(n)p Fk(\)\))949 2123 y Fi(n)981 2144 y Fl(\000)9 b Fm(e)1041 2126 y Fh(\000)p Fi(tH)1107 2130 y Fb(s)1125 2108 y Fe(\015)1125 2133 y(\015)1160 2144 y Fl(\024)j Fm(C)1234 2150 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1316 2144 y Fk(\()p Fm(s)p Fk(\))1416 2115 y(1)p 1372 2134 108 2 v 1372 2172 a Fm(n)1397 2160 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1583 2144 y Fw(\(4.21\))257 2228 y(for)i Fm(t)d Fl(2)h Fk([0)p Fm(;)7 b Fk(1])k Fw(and)j Fm(n)e Fk(=)f(1)p Fm(;)c Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(,)14 b(whic)o(h)f(yields)388 2283 y Fe(\015)388 2308 y(\015)411 2319 y Fk(\()p Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\))p Fm(g)q Fk(\()p Fm(tB)r(=n)p Fk(\))p Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\))930 2298 y Fi(n)962 2319 y Fl(\000)d Fm(e)1023 2302 y Fh(\000)p Fi(tH)1093 2283 y Fe(\015)1093 2308 y(\015)1127 2319 y Fl(\024)i Fm(C)1201 2325 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1284 2319 y Fk(\()p Fm(s)p Fk(\))1384 2291 y(1)p 1340 2309 V 1340 2347 a Fm(n)1365 2335 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1583 2319 y Fw(\(4.22\))257 2403 y(for)g Fm(t)g Fl(2)f Fk([0)p Fm(;)c(s)p Fk(])k Fw(and)g Fm(n)h Fk(=)g(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(.)17 b(Cho)q(osing)11 b Fm(s)h Fk(=)g Fm(T)18 b Fw(and)12 b(setting)g Fm(C)1333 2409 y Fi(T)s(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1459 2403 y Fk(:=)g Fm(C)1545 2409 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1627 2403 y Fk(\()p Fm(T)6 b Fk(\))257 2453 y Fw(w)o(e)17 b(get)g(that)g(for)f(an)o(y)g(\034nite)h(in)o(terv)n(al)e Fk([0)p Fm(;)7 b(T)f Fk(])15 b Fw(there)j(is)e(a)h(constan)o(t)g Fm(C)1394 2459 y Fi(T)s(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1524 2453 y Fm(>)f Fk(0)h Fw(suc)o(h)257 2503 y(that)d(\(1.10\))f (holds.)1083 b Fc(\003)953 2628 y Fw(15)p eop %%Page: 16 16 16 15 bop 257 262 a Fn(5)66 b(Related)20 b(families)257 352 y Fw(Let)h(us)f(sho)o(w)g(that)g(the)g(estimate)f(\(1.10\))g(holds) h(not)f(only)g(for)h(the)g(family)d Fm(F)6 b Fk(\()p Fm(\034)f Fk(\))21 b(=)257 402 y Fm(f)t Fk(\()p Fm(\034)5 b(A=)p Fk(2\))p Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))p Fm(f)t Fk(\()p Fm(\034)g(A=)p Fk(2\))16 b Fw(but)e(also)g(for)f(the)i (famili)o(es)632 490 y Fm(F)659 496 y Fj(1)678 490 y Fk(\()p Fm(\034)5 b Fk(\))41 b(:=)h Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))970 473 y Fj(1)p Fi(=)p Fj(2)1023 490 y Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))p Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))1242 473 y Fj(1)p Fi(=)p Fj(2)1296 490 y Fm(;)296 b Fw(\(5.1\))632 552 y Fm(F)659 558 y Fj(2)678 552 y Fk(\()p Fm(\034)5 b Fk(\))41 b(:=)h Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\))969 535 y Fj(1)p Fi(=)p Fj(2)1022 552 y Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))p Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))1241 535 y Fj(1)p Fi(=)p Fj(2)1295 552 y Fm(;)297 b Fw(\(5.2\))632 615 y Fm(F)659 621 y Fj(3)678 615 y Fk(\()p Fm(\034)5 b Fk(\))41 b(:=)h Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\))p Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))p Fm(;)513 b Fw(\(5.3\))632 677 y Fm(F)659 683 y Fj(4)678 677 y Fk(\()p Fm(\034)5 b Fk(\))41 b(:=)h Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))p Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))p Fm(;)513 b Fw(\(5.4\))632 739 y Fm(F)659 745 y Fj(5)678 739 y Fk(\()p Fm(\034)5 b Fk(\))41 b(:=)h Fm(g)q Fk(\()p Fm(\034)5 b(B)r(=)p Fk(2\))p Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))p Fm(g)q Fk(\()p Fm(\034)g(B)r(=)p Fk(2\))p Fm(;)320 b Fw(\(5.5\))257 827 y(where)15 b Fm(\034)i Fl(\025)12 b Fk(0)p Fw(.)17 b(T)m(o)c(this)h(end)h(w)o(e)f(pro)o(v)o(e)g(the)g (follo)o(wing:)257 908 y Fg(Lemma)i(5.1)k Ff(L)n(et)c Fm(A)h Ff(and)g Fm(B)i Ff(b)n(e)d(non-ne)n(gative)h(self-adjoint)f(op)n (er)n(ators)i(such)f(that)f(for)257 957 y(some)g Fm(\013)c Fl(2)g Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))14 b Ff(the)i(c)n (ondition)g(\(1.6\))f(is)g(satis\034e)n(d.)21 b(F)m(urther,)16 b(let)f Fm(f)20 b Ff(and)c Fm(g)h Ff(b)n(e)e(Kato)257 1007 y(functions)k(which)e(ob)n(ey)h(\(1.7\))g(and)h(\(1.8\).)28 b(Then)18 b(ther)n(e)g(is)g(a)g(c)n(onstant)g Fm(C)1478 1013 y Fi(F)1522 1007 y Fm(>)f Fk(0)h Ff(such)257 1057 y(that)705 1087 y Fe(\015)705 1112 y(\015)728 1122 y Fk(\()p Fm(I)13 b Fl(\000)d Fm(F)c Fk(\()p Fm(t=n)p Fk(\)\))p Fm(e)978 1105 y Fh(\000)p Fi(tH)1047 1087 y Fe(\015)1047 1112 y(\015)1082 1122 y Fl(\024)12 b Fm(C)1156 1128 y Fi(F)1195 1094 y Fm(e)1214 1079 y Fi(t)p 1188 1113 49 2 v 1188 1151 a Fm(n)1213 1139 y Fi(\013)1604 1122 y Fw(\(5.6\))257 1206 y Ff(for)k Fm(t)11 b Fl(\025)h Fk(0)j Ff(and)g Fm(n)d Fk(=)f(1)p Fm(;)c Fk(2)p Fm(;)g(:)g(:)g(:)m Ff(.)257 1287 y Fg(Pro)q(of.)18 b Fw(W)m(e)13 b(use)i(the)g(represen)o (tation)245 1375 y Fk(\()p Fm(I)e Fl(\000)c Fm(F)d Fk(\()p Fm(t=n)p Fk(\)\))p Fm(e)494 1357 y Fh(\000)p Fi(tH)576 1375 y Fk(=)996 b Fw(\(5.7\))387 1437 y Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\))p Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\()p Fm(I)15 b Fl(\000)9 b Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\))p Fm(e)959 1420 y Fh(\000)p Fi(tH)1039 1437 y Fk(+)h Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\()p Fm(I)j Fl(\000)d Fm(g)q Fk(\()p Fm(tB)r(=n)p Fk(\)\))p Fm(e)1521 1420 y Fh(\000)p Fi(tH)1601 1437 y Fk(+)387 1499 y(\()p Fm(I)j Fl(\000)c Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\))p Fm(e)679 1482 y Fh(\000)p Fi(tH)750 1499 y Fm(;)257 1587 y Fw(whic)o(h)14 b(yields)g(the)g (estimate)378 1675 y Fl(k)p Fk(\()p Fm(I)f Fl(\000)c Fm(F)d Fk(\()p Fm(t=n)p Fk(\)\))p Fm(e)648 1658 y Fh(\000)p Fi(tH)718 1675 y Fl(k)12 b(\024)821 b Fw(\(5.8\))579 1737 y Fk(2)p Fl(k)p Fk(\()p Fm(I)13 b Fl(\000)c Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\))p Fm(e)913 1720 y Fh(\000)p Fi(tH)984 1737 y Fl(k)g Fk(+)h Fl(k)p Fk(\()p Fm(I)i Fl(\000)e Fm(g)q Fk(\()p Fm(tB)r(=n)p Fk(\)\))p Fm(e)1347 1720 y Fh(\000)p Fi(tH)1418 1737 y Fl(k)p Fm(:)257 1825 y Fw(Since)277 1913 y Fk(\()p Fm(I)j Fl(\000)c Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\))p Fm(e)569 1896 y Fh(\000)p Fi(tH)652 1913 y Fk(=)920 b Fw(\(5.9\))419 1976 y Fk(\()p Fm(I)13 b Fl(\000)d Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\)\()p Fm(I)j Fk(+)d Fm(A)p Fk(\))829 1958 y Fh(\000)p Fi(\013)879 1976 y Fk(\()p Fm(I)i Fk(+)e Fm(A)p Fk(\))1014 1958 y Fi(\013)1038 1976 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1180 1958 y Fh(\000)p Fi(\013)1230 1976 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))1372 1958 y Fi(\013)1396 1976 y Fm(e)1415 1958 y Fh(\000)p Fj(\()p Fi(I)r Fj(+)p Fi(H)r Fj(\))p Fi(t)1554 1976 y Fm(e)1573 1958 y Fi(t)1599 1976 y Fm(;)257 2064 y Fw(w)o(e)14 b(\034nd)g(the)h (estimate)245 2152 y Fl(k)p Fk(\()p Fm(I)d Fl(\000)e Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\))p Fm(e)558 2134 y Fh(\000)p Fi(tH)629 2152 y Fl(k)h(\024)890 b Fw(\(5.10\))351 2214 y Fl(k)p Fk(\()p Fm(I)13 b Fl(\000)c Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(A)p Fk(\))781 2197 y Fh(\000)p Fi(\013)831 2214 y Fl(k)j(k)p Fk(\()p Fm(I)g Fk(+)e Fm(A)p Fk(\))1020 2197 y Fi(\013)1044 2214 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1186 2197 y Fh(\000)p Fi(\013)1236 2214 y Fl(k)i(k)p Fk(\()p Fm(I)i Fk(+)c Fm(H)s Fk(\))1431 2197 y Fi(\013)1455 2214 y Fm(e)1474 2197 y Fh(\000)p Fj(\()p Fi(I)r Fj(+)p Fi(H)r Fj(\))p Fi(t)1612 2214 y Fl(k)p Fm(e)1652 2197 y Fi(t)1667 2214 y Fm(:)257 2302 y Fw(Since)337 2390 y Fl(k)p Fk(\()p Fm(I)k Fl(\000)c Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\)\()p Fm(I)14 b Fk(+)9 b Fm(A)p Fk(\))767 2373 y Fh(\000)p Fi(\013)817 2390 y Fl(k)j(\024)701 b Fw(\(5.11\))479 2479 y Fk(sup)479 2514 y Fi(\025)p Fh(\025)p Fj(0)554 2451 y Fk(1)9 b Fl(\000)g Fm(f)t Fk(\()p Fm(t\025=)p Fk(2)p Fm(n)p Fk(\))p 554 2470 235 2 v 595 2508 a(\(1)g(+)g Fm(\025)p Fk(\))722 2496 y Fi(\013)804 2479 y Fl(\024)j Fk(sup)848 2514 y Fi(\025>)p Fj(0)922 2451 y Fk(1)d Fl(\000)h Fm(f)t Fk(\()p Fm(\025)p Fk(\))p 922 2470 153 2 v 975 2508 a Fm(\025)999 2496 y Fi(\013)1087 2421 y Fe(\022)1138 2451 y Fm(t)p 1123 2470 46 2 v 1123 2508 a Fk(2)p Fm(n)1173 2421 y Fe(\023)1204 2429 y Fi(\013)1239 2479 y Fl(\024)i Fm(\015)1304 2485 y Fi(f)1326 2479 y Fk(\()p Fm(\013)p Fk(\))1392 2421 y Fe(\022)1443 2451 y Fm(t)p 1427 2470 V 1427 2508 a Fk(2)p Fm(n)1478 2421 y Fe(\023)1509 2429 y Fi(\013)1539 2479 y Fm(;)953 2628 y Fw(16)p eop %%Page: 17 17 17 16 bop 257 262 a Fw(w)o(e)14 b(obtain)g(the)g(estimate)340 368 y Fl(k)p Fk(\()p Fm(I)f Fl(\000)c Fm(f)t Fk(\()p Fm(tA=)p Fk(2)p Fm(n)p Fk(\)\))p Fm(e)653 351 y Fh(\000)p Fi(tH)724 368 y Fl(k)i(\024)h Fk(2)821 351 y Fh(\000)p Fi(\013)871 368 y Fm(\015)892 374 y Fi(f)914 368 y Fk(\()p Fm(\013)p Fk(\))p Fm(b)p Fk(\()p Fm(\013)p Fk(\))p Fl(k)p Fk(\()p Fm(I)g Fk(+)e Fm(A)p Fk(\))1206 351 y Fi(\013)1230 368 y Fk(\()p Fm(I)i Fk(+)e Fm(H)s Fk(\))1372 351 y Fh(\000)p Fi(\013)1422 368 y Fl(k)1455 340 y Fm(e)1474 325 y Fi(t)p 1447 359 49 2 v 1447 397 a Fm(n)1472 385 y Fi(\013)1583 368 y Fw(\(5.12\))257 478 y(for)k Fm(n)d Fk(=)h(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)t Fw(,)13 b(where)i Fm(b)p Fk(\()p Fm(\013)p Fk(\))d(:=)f(sup)886 488 y Fi(\025>)p Fj(0)958 478 y Fm(\025)982 463 y Fi(\013)1006 478 y Fm(e)1025 463 y Fh(\000)p Fi(\025)1073 478 y Fw(.)18 b(Simila)o(rly)m(,)10 b(w)o(e)k(pro)o(v)o(e)g(that)386 597 y Fl(k)p Fk(\()p Fm(I)f Fl(\000)c Fm(g)q Fk(\()p Fm(tB)r(=n)p Fk(\)\))p Fm(e)677 579 y Fh(\000)p Fi(tH)748 597 y Fl(k)j(\024)f Fm(\015)845 603 y Fi(g)865 597 y Fk(\()p Fm(\013)p Fk(\))p Fm(b)p Fk(\()p Fm(\013)p Fk(\))p Fl(k)p Fk(\()p Fm(I)i Fk(+)c Fm(B)r Fk(\))1159 579 y Fi(\013)1184 597 y Fk(\()p Fm(I)j Fk(+)e Fm(H)s Fk(\))1326 579 y Fh(\000)p Fi(\013)1376 597 y Fl(k)1409 569 y Fm(e)1428 553 y Fi(t)p 1401 587 V 1401 625 a Fm(n)1426 613 y Fi(\013)1583 597 y Fw(\(5.13\))257 699 y(for)k Fm(n)d Fk(=)h(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(.)18 b(Setting)291 791 y Fm(C)321 797 y Fi(F)360 791 y Fk(:=)1179 b Fw(\(5.14\))374 853 y Fk(2)395 836 y Fj(1)p Fh(\000)p Fi(\013)461 853 y Fm(\015)482 859 y Fi(f)504 853 y Fk(\()p Fm(\013)p Fk(\))p Fm(b)p Fk(\()p Fm(\013)p Fk(\))p Fl(k)p Fk(\()p Fm(I)13 b Fk(+)c Fm(A)p Fk(\))796 836 y Fi(\013)820 853 y Fk(\()p Fm(I)k Fk(+)d Fm(H)s Fk(\))963 836 y Fh(\000)p Fi(\013)1012 853 y Fl(k)f Fk(+)h Fm(\015)1105 859 y Fi(g)1124 853 y Fk(\()p Fm(\013)p Fk(\))p Fm(b)p Fk(\()p Fm(\013)p Fk(\))p Fl(k)p Fk(\()p Fm(I)j Fk(+)c Fm(B)r Fk(\))1418 836 y Fi(\013)1443 853 y Fk(\()p Fm(I)k Fk(+)c Fm(H)s Fk(\))1585 836 y Fh(\000)p Fi(\013)1635 853 y Fl(k)257 944 y Fw(one)14 b(gets)h(the)f(estimate)g (\(5.6\).)902 b Fc(\003)257 1036 y Fg(Theorem)15 b(5.2)21 b Ff(L)n(et)f Fm(A)g Ff(and)h Fm(B)h Ff(b)n(e)f(non-ne)n(gative)g (self-adjoint)e(op)n(er)n(ators)j(such)e(that)257 1085 y(for)e(some)g Fm(\013)d Fl(2)h Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))16 b Ff(the)h(c)n(ondition)h(\(1.6\))g(is)f(satis\034e) n(d.)26 b(F)m(urther,)19 b(let)e Fm(f)22 b Ff(and)c Fm(g)g Ff(b)n(e)257 1135 y(Kato)e(functions)f(which)g(satisfy)g(c)n(onditions) h(\(1.7\))f(and)h(\(1.8\).)k(If)15 b(in)g(addition)g(one)h(has)257 1185 y Fw(dom)o Fk(\()p Fm(A)383 1170 y Fj(1)p Fi(=)p Fj(2)435 1185 y Fk(\))d Fl(\022)f Fw(dom)o Fk(\()p Fm(B)636 1170 y Fj(1)p Fi(=)p Fj(2)689 1185 y Fk(\))j Ff(and)h Fm(m)837 1191 y Fi(f)859 1185 y Fk(\()p Fm(x)p Fk(\))d Fm(<)f Fk(1)j Ff(for)h Fm(x)c(>)h Fk(0)p Ff(,)i(then)h(for)g(any)g (\034nite)f(interval)257 1244 y Fk([0)p Fm(;)7 b(T)f Fk(])14 b Ff(ther)n(e)h(ar)n(e)g(c)n(onstants)h Fm(C)754 1223 y Fj(\()p Fi(j)r Fj(\))751 1257 y Fi(T)s(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)877 1244 y Fm(>)c Fk(0)p Ff(,)i Fm(j)g Fk(=)e(1)p Fm(;)7 b Fk(2)p Fm(;)g Fk(3)p Fm(;)g Fk(4)p Fm(;)g Fk(5)o Ff(,)12 b(such)j(that)485 1327 y Fe(\015)485 1352 y(\015)508 1363 y Fm(F)535 1369 y Fi(j)552 1363 y Fk(\()p Fm(t=n)p Fk(\))645 1346 y Fi(n)677 1363 y Fl(\000)10 b Fm(e)738 1346 y Fh(\000)p Fi(tH)808 1327 y Fe(\015)808 1352 y(\015)843 1363 y Fl(\024)h Fm(C)919 1341 y Fj(\()p Fi(j)r Fj(\))916 1375 y Fi(T)s(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1079 1335 y Fk(1)p 1036 1353 108 2 v 1036 1391 a Fm(n)1061 1379 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1148 1363 y Fm(;)49 b(j)14 b Fk(=)e(1)p Fm(;)7 b Fk(2)p Fm(;)g Fk(3)p Fm(;)g Fk(4)p Fm(;)g Fk(5)118 b Fw(\(5.15\))257 1468 y Ff(for)16 b Fm(t)11 b Fl(2)g Fk([0)p Fm(;)c(T)f Fk(])14 b Ff(and)h Fm(n)d Fk(=)f(1)p Fm(;)c Fk(2)p Fm(;)g(:)g(:)g(:)m Ff(.)257 1559 y Fg(Pro)q(of.)18 b Fw(W)m(e)13 b(set)730 1609 y Fm(f)750 1615 y Fj(0)769 1609 y Fk(\()p Fm(x)p Fk(\))e(:=)h Fm(f)t Fk(\(2)p Fm(x)p Fk(\))993 1592 y Fj(1)p Fi(=)p Fj(2)1046 1609 y Fm(;)48 b(x)11 b Fl(\025)h Fk(0)p Fm(:)365 b Fw(\(5.16\))257 1684 y(The)22 b(function)e Fm(f)539 1690 y Fj(0)558 1684 y Fk(\()p Fm(x)p Fk(\))h Fw(is)f(also)g(a)h(Kato)f(function)h(whic)o(h) f(satis\034es)i Fl(j)p Fm(f)1412 1690 y Fj(0)1430 1684 y Fl(j)1442 1690 y Fj(2)p Fi(\013)1505 1684 y Fm(<)h Fl(1)e Fw(and)257 1734 y Fm(m)293 1740 y Fi(f)309 1744 y Fa(0)328 1734 y Fk(\()p Fm(x)p Fk(\))12 b Fm(<)g Fk(1)h Fw(for)h Fm(x)d(>)h Fk(0)p Fw(.)18 b(One)c(has)696 1830 y Fm(f)716 1836 y Fj(0)735 1830 y Fk(\()p Fm(\034)5 b(A=)p Fk(2\))11 b(=)h Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))1028 1813 y Fj(1)p Fi(=)p Fj(2)1081 1830 y Fm(;)48 b(\034)16 b Fl(\025)c Fk(0)p Fm(:)331 b Fw(\(5.17\))257 1921 y(W)m(e)14 b(set)583 1971 y Fm(F)610 1977 y Fj(0)628 1971 y Fk(\()p Fm(\034)5 b Fk(\))12 b(:=)f Fm(f)770 1977 y Fj(0)789 1971 y Fk(\()p Fm(\034)5 b(A=)p Fk(2\))p Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))p Fm(f)1046 1977 y Fj(0)1066 1971 y Fk(\()p Fm(\034)g(A=)p Fk(2\))p Fm(;)48 b(\034)16 b Fl(\025)c Fk(0)p Fm(:)218 b Fw(\(5.18\))257 2054 y(By)15 b(Theorem)e(1.1)g(there)i(is)f(a)f(constan)o(t)i Fm(C)948 2033 y Fj(\(0\))945 2066 y Fi(T)s(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1071 2054 y Fm(>)c Fk(0)j Fw(suc)o(h)h(that)644 2172 y Fl(k)p Fm(F)692 2178 y Fj(0)710 2172 y Fk(\()p Fm(t=n)p Fk(\))803 2155 y Fi(n)834 2172 y Fl(\000)10 b Fm(e)895 2155 y Fh(\000)p Fi(tH)965 2172 y Fl(k)h(\024)h Fm(C)1074 2151 y Fj(\(0\))1071 2185 y Fi(T)s(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1234 2144 y Fk(1)p 1191 2163 V 1191 2201 a Fm(n)1216 2189 y Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1583 2172 y Fw(\(5.19\))257 2278 y(for)g Fm(t)g Fl(2)f Fk([0)p Fm(;)c(T)f Fk(])k Fw(and)i Fm(n)f Fk(=)h(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(.)17 b(By)c(\(5.17\))e(one)h(has) g Fm(F)1159 2284 y Fj(0)1177 2278 y Fk(\()p Fm(\034)5 b Fk(\))12 b(=)g Fm(F)1315 2284 y Fj(1)1333 2278 y Fk(\()p Fm(\034)5 b Fk(\))12 b Fw(for)g Fm(\034)k Fl(\025)c Fk(0)p Fw(,)g(whic)o(h)257 2328 y(pro)o(v)o(es)j(the)f(assertion)h(for)e Fm(j)h Fk(=)e(1)p Fw(.)18 b(W)m(e)13 b(note)h(that)308 2419 y Fm(F)335 2425 y Fj(4)353 2419 y Fk(\()p Fm(\034)5 b Fk(\))408 2402 y Fi(n)443 2419 y Fk(=)11 b Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))596 2402 y Fj(1)p Fi(=)p Fj(2)650 2419 y Fm(F)677 2425 y Fj(1)695 2419 y Fk(\()p Fm(\034)g Fk(\))750 2402 y Fi(n)p Fh(\000)p Fj(1)815 2419 y Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))925 2402 y Fj(1)p Fi(=)p Fj(2)978 2419 y Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))p Fm(;)49 b(\034)17 b Fl(\025)12 b Fk(0)p Fm(;)47 b(n)12 b Fk(=)g(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)s(:)50 b Fw(\(5.20\))953 2628 y(17)p eop %%Page: 18 18 18 17 bop 257 262 a Fw(Using)14 b(the)h(represen)o(tation)367 338 y Fm(F)394 344 y Fj(4)412 338 y Fk(\()p Fm(\034)5 b Fk(\))467 321 y Fi(n)499 338 y Fl(\000)10 b Fm(e)560 321 y Fh(\000)p Fi(tH)642 338 y Fk(=)909 b Fw(\(5.21\))568 400 y Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))678 383 y Fj(1)p Fi(=)p Fj(2)738 367 y Fe(\000)757 400 y Fm(F)784 406 y Fj(1)803 400 y Fk(\()p Fm(\034)g Fk(\))858 383 y Fi(n)p Fh(\000)p Fj(1)923 400 y Fk(\()p Fm(I)13 b Fl(\000)c Fm(F)1038 406 y Fj(1)1057 400 y Fk(\()p Fm(\034)c Fk(\))1112 367 y Fe(\001)1138 400 y Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))1248 383 y Fj(1)p Fi(=)p Fj(2)1301 400 y Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))10 b(+)568 471 y Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))678 453 y Fj(1)p Fi(=)p Fj(2)738 437 y Fe(\000)757 471 y Fm(F)784 477 y Fj(1)803 471 y Fk(\()p Fm(\034)g Fk(\))858 453 y Fi(n)890 471 y Fl(\000)k Fm(e)950 453 y Fh(\000)p Fi(tH)1020 437 y Fe(\001)1046 471 y Fm(f)t Fk(\()p Fm(\034)c(A)p Fk(\))1156 453 y Fj(1)p Fi(=)p Fj(2)1210 471 y Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))10 b(+)568 503 y Fe(\020)593 549 y Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))703 532 y Fj(1)p Fi(=)p Fj(2)765 549 y Fl(\000)10 b Fm(I)828 503 y Fe(\021)860 549 y Fm(e)879 532 y Fh(\000)p Fi(tH)949 549 y Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))1059 532 y Fj(1)p Fi(=)p Fj(2)1113 549 y Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))1223 532 y Fj(1)p Fi(=)p Fj(2)1276 549 y Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))10 b(+)568 641 y Fm(e)587 624 y Fh(\000)p Fi(tH)664 595 y Fe(\020)689 641 y Fm(f)t Fk(\()p Fm(\034)5 b(A)p Fk(\))799 624 y Fj(1)p Fi(=)p Fj(2)862 641 y Fl(\000)k Fm(I)924 595 y Fe(\021)956 641 y Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\))11 b(+)e Fm(e)1136 624 y Fh(\000)p Fi(tH)1213 641 y Fk(\()p Fm(g)q Fk(\()p Fm(\034)c(B)r Fk(\))11 b Fl(\000)e Fm(I)s Fk(\))257 730 y Fw(where)15 b Fm(\034)i Fk(=)12 b Fm(t=n)p Fw(,)h Fm(t)e Fl(\025)h Fk(0)i Fw(and)f Fm(n)f Fk(=)f(1)p Fm(;)c Fk(2)g Fm(:)g(:)g(:)n Fw(,)13 b(w)o(e)h(\034nd)g(the)h(estimate)268 806 y Fl(k)p Fm(F)316 812 y Fj(4)334 806 y Fk(\()p Fm(\034)5 b Fk(\))389 789 y Fi(n)421 806 y Fl(\000)k Fm(e)481 789 y Fh(\000)p Fi(tH)552 806 y Fl(k)i(\024)628 771 y Fe(\015)628 796 y(\015)651 806 y Fm(F)678 812 y Fj(1)696 806 y Fk(\()p Fm(\034)5 b Fk(\))751 789 y Fi(n)p Fh(\000)p Fj(1)816 806 y Fk(\()p Fm(I)13 b Fl(\000)d Fm(F)932 812 y Fj(1)950 806 y Fk(\()p Fm(\034)5 b Fk(\))1005 771 y Fe(\015)1005 796 y(\015)1035 806 y Fk(+)516 b Fw(\(5.22\))410 850 y Fe(\015)410 875 y(\015)433 885 y Fm(F)460 891 y Fj(1)478 885 y Fk(\()p Fm(\034)5 b Fk(\))533 868 y Fi(n)565 885 y Fl(\000)10 b Fm(e)626 868 y Fh(\000)p Fi(tH)696 850 y Fe(\015)696 875 y(\015)728 885 y Fk(+)g(2)798 838 y Fe(\015)798 862 y(\015)798 887 y(\015)821 885 y Fk(\()p Fm(I)j Fl(\000)c Fm(f)t Fk(\()p Fm(\034)c(A)p Fk(\))1019 868 y Fj(1)p Fi(=)p Fj(2)1072 885 y Fk(\))p Fm(e)1107 868 y Fh(\000)p Fi(tH)1178 838 y Fe(\015)1178 862 y(\015)1178 887 y(\015)1210 885 y Fk(+)1252 850 y Fe(\015)1252 875 y(\015)1275 885 y Fk(\()p Fm(I)12 b Fl(\000)e Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\)\))p Fm(e)1507 868 y Fh(\000)p Fi(tH)1578 850 y Fe(\015)1578 875 y(\015)1608 885 y Fm(:)257 972 y Fw(Since)376 1016 y Fe(\015)376 1041 y(\015)376 1066 y(\015)399 1064 y Fk(\()p Fm(I)13 b Fl(\000)c Fm(f)t Fk(\()p Fm(\034)c(A)p Fk(\))597 1047 y Fj(1)p Fi(=)p Fj(2)651 1064 y Fk(\))p Fm(e)686 1047 y Fh(\000)p Fi(tH)756 1016 y Fe(\015)756 1041 y(\015)756 1066 y(\015)791 1064 y Fl(\024)12 b Fm(\015)856 1070 y Fi(f)878 1064 y Fk(\()p Fm(\013)p Fk(\))p Fm(b)p Fk(\()p Fm(\013)p Fk(\))p Fl(k)p Fk(\()p Fm(I)g Fk(+)e Fm(A)p Fk(\))1170 1047 y Fi(\013)1193 1064 y Fk(\()p Fm(I)j Fk(+)d Fm(H)s Fk(\))1336 1047 y Fh(\000)p Fi(\013)1385 1064 y Fl(k)1419 1036 y Fm(e)1438 1021 y Fi(t)p 1411 1054 49 2 v 1411 1092 a Fm(n)1436 1080 y Fi(\013)1583 1064 y Fw(\(5.23\))257 1152 y(and)376 1196 y Fe(\015)376 1221 y(\015)376 1246 y(\015)400 1244 y Fk(\()p Fm(I)i Fl(\000)e Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\))597 1227 y Fj(1)p Fi(=)p Fj(2)650 1244 y Fk(\))p Fm(e)685 1227 y Fh(\000)p Fi(tH)756 1196 y Fe(\015)756 1221 y(\015)756 1246 y(\015)790 1244 y Fl(\024)12 b Fm(\015)855 1250 y Fi(g)875 1244 y Fk(\()p Fm(\013)p Fk(\))p Fm(b)p Fk(\()p Fm(\013)p Fk(\))p Fl(k)p Fk(\()p Fm(I)g Fk(+)e Fm(B)r Fk(\))1169 1227 y Fi(\013)1193 1244 y Fk(\()p Fm(I)j Fk(+)c Fm(H)s Fk(\))1335 1227 y Fh(\000)p Fi(\013)1385 1244 y Fl(k)1418 1216 y Fm(e)1437 1201 y Fi(t)p 1411 1234 V 1411 1272 a Fm(n)1436 1260 y Fi(\013)1583 1244 y Fw(\(5.24\))257 1338 y(as)14 b(w)o(ell)f(as)g(the)h(estimate)f Fl(k)p Fm(F)727 1344 y Fj(1)745 1338 y Fk(\()p Fm(\034)5 b Fk(\))800 1323 y Fi(n)p Fh(\000)p Fj(1)865 1338 y Fk(\()p Fm(I)12 b Fl(\000)d Fm(F)979 1344 y Fj(1)997 1338 y Fk(\()p Fm(\034)c Fk(\))p Fl(k)12 b(\024)1135 1322 y Fj(1)p 1133 1329 21 2 v 1133 1353 a Fi(n)1159 1338 y Fw(,)h Fm(n)e Fk(=)h(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(,)13 b(w)o(e)h(obtain)e(from)257 1400 y(the)h(statemen)o(t)e(for)h Fm(j)i Fk(=)d(1)h Fw(the)g(existence)h(of)e(a)h(constan)o(t)g Fm(C)1205 1379 y Fj(\(4\))1202 1413 y Fi(T)s(;)p Fj(2)p Fi(\013)p Fh(\000)p Fj(1)1328 1400 y Fm(>)g Fk(0)f Fw(suc)o(h)i(that)e (\(5.15\))257 1450 y(holds)j(for)f Fm(j)h Fk(=)e(4)p Fw(,)h Fm(t)f Fl(2)f Fk([0)p Fm(;)c(T)f Fk(])12 b Fw(and)i Fm(n)d Fk(=)h(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(.)18 b(Next,)c(using)691 1527 y Fm(F)718 1533 y Fj(3)736 1527 y Fk(\()p Fm(\034)5 b Fk(\))791 1510 y Fi(n)826 1527 y Fk(=)11 b Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\))p Fm(F)1005 1533 y Fj(4)1025 1527 y Fk(\()p Fm(\034)g Fk(\))1080 1510 y Fi(n)p Fh(\000)p Fj(1)1145 1527 y Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))328 b Fw(\(5.25\))257 1603 y(and)584 1653 y Fm(F)611 1659 y Fj(2)629 1653 y Fk(\()p Fm(\034)5 b Fk(\))684 1636 y Fi(n)718 1653 y Fk(=)12 b Fm(g)q Fk(\()p Fm(\034)5 b(B)r Fk(\))871 1636 y Fj(1)p Fi(=)p Fj(2)924 1653 y Fm(F)951 1659 y Fj(4)970 1653 y Fk(\()p Fm(\034)g Fk(\))1025 1636 y Fi(n)p Fh(\000)p Fj(1)1090 1653 y Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))p Fm(g)q Fk(\()p Fm(\034)g(B)r Fk(\))1309 1636 y Fj(1)p Fi(=)p Fj(2)1583 1653 y Fw(\(5.26\))257 1719 y Fm(\034)17 b Fl(\025)12 b Fk(0)p Fw(,)g Fm(n)g Fk(=)g(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)l Fw(,)13 b(as)h(w)o(ell)e(as)i(the)g(estimates)f(\(5.23\))f(and)h(\(5.24\))g (one)g(v)o(eri\034es)h(\(5.15\))257 1769 y(for)j Fm(j)i Fk(=)e(2)p Fm(;)7 b Fk(3)16 b Fw(in)g(the)i(same)e(w)o(a)o(y)g(as)h(ab) q(o)o(v)o(e.)27 b(T)m(o)16 b(pro)o(v)o(e)h(the)h(statemen)o(t)f(for)f Fm(j)k Fk(=)d(5)f Fw(w)o(e)257 1819 y(in)o(tro)q(duce)f(the)f(function) 738 1895 y Fm(g)758 1901 y Fj(0)777 1895 y Fk(\()p Fm(x)p Fk(\))d(:=)h Fm(g)q Fk(\()p Fm(x=)p Fk(2\))1019 1878 y Fj(2)1037 1895 y Fm(;)48 b(x)11 b Fl(\025)h Fk(0)p Fm(:)374 b Fw(\(5.27\))257 1972 y(Then)15 b(one)f(gets)h Fm(F)555 1978 y Fj(5)573 1972 y Fk(\()p Fm(\034)5 b Fk(\))12 b(=)g Fm(g)704 1978 y Fj(0)722 1972 y Fk(\()p Fm(\034)5 b(B)r Fk(\))810 1957 y Fj(1)p Fi(=)p Fj(2)863 1972 y Fm(f)t Fk(\()p Fm(\034)g(A)p Fk(\))p Fm(g)993 1978 y Fj(0)1013 1972 y Fk(\()p Fm(\034)g(B)r Fk(\))1101 1957 y Fj(1)p Fi(=)p Fj(2)1168 1972 y Fw(for)13 b Fm(\034)k Fl(\025)12 b Fk(0)p Fw(.)18 b(The)d(function)e Fm(g)1629 1978 y Fj(0)1662 1972 y Fw(is)257 2022 y(again)g(a)h(Kato)g(function)g (whic)o(h)g(satis\034es)h(\(1.8\).)i(Hence,)f(applying)c(here)j(the)g (result)g(for)257 2072 y Fm(F)284 2078 y Fj(2)303 2072 y Fk(\()p Fm(\034)5 b Fk(\))14 b Fw(w)o(e)g(pro)o(v)o(e)g(the)g(case)h Fm(j)f Fk(=)e(5)p Fw(.)845 b Fc(\003)257 2206 y Fn(6)66 b(Example)257 2297 y Fw(Let)12 b Fl(H)g Fk(:=)g Fm(L)460 2303 y Fj(2)478 2297 y Fk(\(\012\))g Fw(where)g Fk(\012)g Fw(is)f(a)g(b)q(ounded)h(domain)d(in)i Fd(R)1173 2282 y Fi(l)1182 2297 y Fw(,)h Fm(l)g Fk(=)g(1)p Fm(;)7 b Fk(2)p Fm(;)g(:)g(:)g(:)m Fw(,)k(with)g(b)q(oundary)257 2347 y Fm(@)r Fk(\012)19 b Fw(of)e Fm(C)414 2332 y Fh(1)449 2347 y Fw(-class.)31 b(By)18 b Fm(A)g Fw(w)o(e)g(denote)h(the)g (negativ)o(e)f(half)f(Laplacian)f(with)i(Diric)o(hlet)257 2397 y(b)q(oundary)c(conditions)g(on)f Fm(L)728 2403 y Fj(2)747 2397 y Fk(\(\012\))p Fw(,)h(i.e)f Fm(A)f Fk(=)g Fl(\000)1014 2381 y Fj(1)p 1014 2388 17 2 v 1014 2411 a(2)1036 2397 y Fk(\001)1071 2403 y Fi(D)1100 2397 y Fw(.)18 b(The)c(domain)e(is)i(giv)o(en)f(b)o(y)721 2503 y(dom)o Fk(\()p Fm(A)p Fk(\))f(:=)f Fm(H)968 2486 y Fj(2)965 2513 y(2)987 2503 y Fk(\(\012\))e Fl(\\)1121 2459 y Fj(o)1107 2503 y Fm(H)1145 2486 y Fj(1)1142 2513 y(2)1163 2503 y Fk(\(\012\))379 b Fw(\(6.1\))953 2628 y(18)p eop %%Page: 19 19 19 18 bop 257 279 a Fw(where)404 237 y Fj(o)390 279 y Fm(H)428 264 y Fj(1)425 289 y(2)459 279 y Fk(\(\012\))14 b Fw(is)h(the)g(closure)g(of)f Fm(C)870 264 y Fh(1)867 289 y Fj(0)905 279 y Fk(\(\012\))h Fw(in)f(the)h(Leb)q(esgue)h(space)g Fm(H)1433 264 y Fj(1)1430 289 y(2)1451 279 y Fk(\(\012\))p Fw(,)e(cf.)20 b(De\034-)257 329 y(nitions)14 b(4.2.1/1)d(and)j(4.2.1/2) e(of)h([18)o(].)k(Using)d(the)h(space)f Fm(H)1234 314 y Fj(2)1231 340 y(2)p Fi(;B)1283 344 y Fb(D)1310 329 y Fk(\(\012\))p Fw(,)602 425 y Fm(H)640 408 y Fj(2)637 435 y(2)p Fi(;B)689 439 y Fb(D)716 425 y Fk(\(\012\))e(:=)f Fl(f)p Fm(u)g Fl(2)h Fm(H)979 408 y Fj(2)976 435 y(2)997 425 y Fk(\(\012\))g(:)f Fm(B)1125 431 y Fi(D)1156 425 y Fm(u)p Fl(j)1192 431 y Fi(@)r Fj(\012)1248 425 y Fk(=)h(0)p Fl(g)p Fm(;)258 b Fw(\(6.2\))257 514 y(cf.)18 b(De\034nition)c(4.3.3/2) d(of)i([18)o(],)g(where)i Fm(B)940 520 y Fi(D)985 514 y Fw(is)f(giv)o(en)f(b)o(y)857 603 y Fm(B)888 609 y Fi(D)919 603 y Fm(u)e Fk(:=)g Fm(u)p Fl(j)1045 609 y Fi(@)r Fj(\012)1604 603 y Fw(\(6.3\))257 692 y(for)j Fm(u)d Fl(2)g Fm(H)433 677 y Fj(2)430 702 y(2)452 692 y Fk(\(\012\))p Fw(,)j(w)o(e)g(obtain) 775 781 y(dom)o Fk(\()p Fm(A)p Fk(\))e(:=)f Fm(H)1022 764 y Fj(2)1019 791 y(2)p Fi(;B)1071 795 y Fb(D)1098 781 y Fk(\(\012\))p Fm(:)432 b Fw(\(6.4\))257 870 y(F)m(urther,)17 b(let)e Fm(B)k Fw(b)q(e)d(the)g(negativ)o(e)f(half)g(Laplacian)f(with)i (Neumann)e(b)q(oundary)i(condi-)257 920 y(tions,)e(i.e)f Fm(B)h Fk(:=)d Fl(\000)563 903 y Fj(1)p 564 910 17 2 v 564 934 a(2)585 920 y Fk(\001)620 926 y Fi(N)651 920 y Fw(.)18 b(One)d(has)779 1013 y(dom)o Fk(\()p Fm(B)r Fk(\))d(:=)f Fm(H)1028 996 y Fj(2)1025 1023 y(2)p Fi(;B)1077 1027 y Fb(N)1105 1013 y Fk(\(\012\))437 b Fw(\(6.5\))257 1102 y(where)601 1152 y Fm(H)639 1135 y Fj(2)636 1162 y(2)p Fi(;B)688 1166 y Fb(N)716 1152 y Fk(\(\012\))12 b(:=)f Fl(f)p Fm(u)g Fl(2)g Fm(H)978 1135 y Fj(2)975 1162 y(2)997 1152 y Fk(\(\012\))h(:)f Fm(B)1125 1158 y Fi(N)1157 1152 y Fm(u)p Fl(j)1193 1158 y Fi(@)r Fj(\012)1249 1152 y Fk(=)h(0)p Fl(g)p Fm(;)257 b Fw(\(6.6\))257 1225 y(cf.)18 b(De\034nition)c(4.3.3/2)d(of)i([18)o(].)18 b(The)c(b)q(oundary)g(op)q(erator)h Fm(B)1265 1231 y Fi(N)1310 1225 y Fw(is)f(giv)o(en)f(b)o(y)661 1336 y Fk(\()p Fm(B)708 1342 y Fi(N)741 1336 y Fm(u)p Fk(\)\()p Fm(x)p Fk(\))e(=)936 1308 y Fm(@)p 897 1326 104 2 v 897 1364 a(@)r(\027)s Fk(\()p Fm(x)p Fk(\))1005 1336 y Fm(u)p Fk(\()p Fm(x)p Fk(\))p Fm(;)48 b(x)11 b Fl(2)h Fm(@)r Fk(\012)p Fm(;)318 b Fw(\(6.7\))257 1452 y(for)15 b Fm(u)d Fl(2)h Fm(H)437 1437 y Fj(2)434 1462 y(2)455 1452 y Fk(\(\012\))i Fw(where)h Fm(\027)s Fk(\()p Fm(x)p Fk(\))d Fw(is)i(the)g(outer)g(unit) g(normal)d(to)j(the)g(b)q(oundary)f Fm(@)r Fk(\012)h Fw(at)g(the)257 1525 y(p)q(oin)o(t)f Fm(x)d Fl(2)g Fm(@)r Fk(\012)p Fw(.)18 b(Since)26 b(dom)o Fk(\()p Fm(A)770 1510 y Fj(1)p Fi(=)p Fj(2)822 1525 y Fk(\))12 b(=)919 1483 y Fj(o)905 1525 y Fm(H)943 1510 y Fj(1)940 1535 y(2)962 1525 y Fk(\(\012\))i Fw(and)g(dom)n Fk(\()p Fm(B)1246 1510 y Fj(1)p Fi(=)p Fj(2)1299 1525 y Fk(\))d(=)h Fm(H)1408 1510 y Fj(1)1405 1535 y(2)1427 1525 y Fk(\(\012\))i Fw(one)g(gets)745 1614 y(dom)n Fk(\()p Fm(A)870 1597 y Fj(1)p Fi(=)p Fj(2)923 1614 y Fk(\))d Fl(\022)h Fw(dom)o Fk(\()p Fm(B)1122 1597 y Fj(1)p Fi(=)p Fj(2)1174 1614 y Fk(\))p Fm(:)402 b Fw(\(6.8\))257 1703 y(Hence)529 1768 y(dom)n Fk(\()p Fm(H)661 1751 y Fj(1)p Fi(=)p Fj(2)713 1768 y Fk(\))12 b(=)g Fw(dom)n Fk(\()p Fm(A)910 1751 y Fj(1)p Fi(=)p Fj(2)962 1768 y Fk(\))e Fl(\\)f Fw(dom)n Fk(\()p Fm(B)1152 1751 y Fj(1)p Fi(=)p Fj(2)1205 1768 y Fk(\))j(=)1302 1724 y Fj(o)1288 1768 y Fm(H)1326 1751 y Fj(1)1323 1778 y(2)1345 1768 y Fk(\(\012\))p Fm(:)185 b Fw(\(6.9\))257 1846 y(and)15 b Fm(H)g Fk(:=)e Fm(A)501 1813 y Fi(:)489 1846 y Fk(+)g Fm(B)j Fk(=)d(2)p Fm(A)f Fk(=)h Fl(\000)p Fk(\001)802 1852 y Fi(D)832 1846 y Fw(.)20 b(Therefore)c(dom)n Fk(\()p Fm(H)1185 1831 y Fi(\013)1209 1846 y Fk(\))d(=)f Fw(dom)o Fk(\()p Fm(A)1408 1831 y Fi(\013)1432 1846 y Fk(\))i Fw(for)h(arbitrary)257 1896 y Fm(\013)d Fl(2)f Fk([0)p Fm(;)c Fk(1])p Fw(.)320 1946 y(No)o(w)13 b(w)o(e)i(are)f(going)f(to)h (calculate)h(the)f(domains)f(dom)n Fk(\()p Fm(A)1242 1931 y Fi(\013)1266 1946 y Fk(\))h Fw(and)g(dom)o Fk(\()p Fm(B)1505 1931 y Fi(\013)1529 1946 y Fk(\))g Fw(for)g Fm(\013)e Fl(2)257 1995 y Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))p Fw(.)20 b(By)15 b(Theorem)g(1)f(and)h(Theorem)f(2)h(of)f ([4])g(and)h(Theorem)f(8.1)g(of)g([5)o(])h(\(see)h(also)257 2045 y(Theorem)e(4.3.3)e(of)h([18)o(]\))h(w)o(e)g(\034nd)634 2134 y(dom)n Fk(\()p Fm(A)759 2117 y Fi(\013)783 2134 y Fk(\))e(=)f Fm(H)892 2117 y Fj(2)p Fi(\013)889 2144 y Fj(2)p Fi(;B)941 2148 y Fb(D)968 2134 y Fk(\(\012\))p Fm(;)49 b(\013)11 b Fl(2)g Fk(\(1)p Fm(=)p Fk(2)p Fm(;)c Fk(1\))p Fm(;)268 b Fw(\(6.10\))257 2223 y(and)575 2300 y(dom)n Fk(\()p Fm(B)702 2283 y Fi(\013)726 2300 y Fk(\))12 b(=)798 2229 y Fe(\()852 2262 y Fm(H)890 2247 y Fj(2)p Fi(\013)887 2272 y Fj(2)930 2262 y Fk(\(\012\))p Fm(;)78 b(\013)11 b Fl(2)h Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(3)p Fm(=)p Fk(4\))p Fm(;)852 2336 y(H)890 2321 y Fj(2)p Fi(\013)887 2347 y Fj(2)p Fi(;B)939 2351 y Fb(N)967 2336 y Fk(\(\012\))p Fm(;)41 b(\013)11 b Fl(2)h Fk(\(3)p Fm(=)p Fk(4)p Fm(;)7 b Fk(1\))p Fm(:)1583 2300 y Fw(\(6.11\))257 2414 y(Since)15 b Fm(H)404 2399 y Fj(2)p Fi(\013)401 2426 y Fj(2)p Fi(;B)453 2430 y Fb(D)480 2414 y Fk(\(\012\))c Fl(\022)h Fm(H)635 2399 y Fj(2)p Fi(\013)632 2424 y Fj(2)675 2414 y Fk(\(\012\))i Fw(one)h(gets)665 2503 y(dom)o Fk(\()p Fm(H)798 2486 y Fi(\013)821 2503 y Fk(\))d(=)g Fw(dom)n Fk(\()p Fm(A)1018 2486 y Fi(\013)1042 2503 y Fk(\))g Fl(\022)g Fw(dom)n Fk(\()p Fm(B)1241 2486 y Fi(\013)1265 2503 y Fk(\))302 b Fw(\(6.12\))953 2628 y(19)p eop %%Page: 20 20 20 19 bop 257 262 a Fw(for)14 b Fm(\013)d Fl(2)g Fk(\(1)p Fm(=)p Fk(2)p Fm(;)c Fk(3)p Fm(=)p Fk(4\))p Fw(.)17 b(Applying)12 b(no)o(w)i(Theorem)f(5.2)g(w)o(e)h(\034nd)g(that)491 311 y Fe(\015)491 336 y(\015)491 361 y(\015)514 313 y(\020)539 359 y Fm(e)558 342 y Fh(\000)p Fi(t)p Fj(\()p Fh(\000)p Fj(\001)663 346 y Fb(D)688 342 y Fj(\))p Fi(=)p Fj(2)p Fi(n)758 359 y Fm(e)777 342 y Fh(\000)p Fi(t)p Fj(\()p Fh(\000)p Fj(\001)882 346 y Fb(N)908 342 y Fj(\))p Fi(=)p Fj(2)p Fi(n)977 313 y Fe(\021)1002 321 y Fi(n)1034 359 y Fl(\000)9 b Fm(e)1094 342 y Fh(\000)p Fi(t)p Fj(\()p Fh(\000)p Fj(\001)1199 346 y Fb(D)1225 342 y Fj(\))1240 311 y Fe(\015)1240 336 y(\015)1240 361 y(\015)1275 359 y Fk(=)j Fm(O)q Fk(\()p Fm(n)1393 342 y Fh(\000)p Fi(\024)1440 359 y Fk(\))127 b Fw(\(6.13\))257 458 y(for)14 b(an)o(y)f Fm(\024)f Fk(:=)f(2)p Fm(\013)e Fl(\000)g Fk(1)j Fm(<)f(\024)688 464 y Fj(0)718 458 y Fk(:=)779 441 y Fj(1)p 779 448 17 2 v 779 472 a(2)814 458 y Fw(\()p Fm(\013)h(<)917 441 y Fj(3)p 917 448 V 917 472 a(4)939 458 y Fw(\))i(uniformly)d(in)i Fm(t)f Fl(2)f Fk([0)p Fm(;)c(T)f Fk(])12 b Fw(as)i Fm(n)e Fl(!)f(1)p Fw(.)320 508 y(If)i Fm(\013)e Fl(2)h Fk(\(3)p Fm(=)p Fk(4)p Fm(;)7 b Fk(1\))p Fw(,)12 b(then)407 590 y(dom)o Fk(\()p Fm(A)533 573 y Fi(\013)557 590 y Fk(\))d Fl(\\)g Fw(dom)n Fk(\()p Fm(B)746 573 y Fi(\013)771 590 y Fk(\))i(=)h Fm(H)880 573 y Fj(2)p Fi(\013)877 602 y Fj(2)p Fi(;)p Fh(f)p Fi(B)946 606 y Fb(D)971 602 y Fi(;B)1006 606 y Fb(N)1032 602 y Fh(g)1052 590 y Fk(\(\012\))f(:=)549 661 y Fl(f)p Fm(u)g Fl(2)h Fm(H)683 644 y Fj(2)p Fi(\013)680 672 y Fj(2)723 661 y Fk(\(\012\))f(:)g Fm(B)850 667 y Fi(D)881 661 y Fm(u)p Fl(j)917 667 y Fi(@)r Fj(\012)973 661 y Fk(=)h(0)p Fm(;)7 b(B)1088 667 y Fi(N)1119 661 y Fm(u)p Fl(j)1155 667 y Fi(@)r Fj(\012)1211 661 y Fk(=)12 b(0)p Fl(g)f(\022)h Fm(H)1390 644 y Fj(2)p Fi(\013)1387 672 y Fj(2)p Fi(;B)1439 676 y Fb(D)1466 661 y Fk(\(\012\))p Fm(:)257 744 y Fw(whic)o(h)18 b(yields)f(that)g(dom)n Fk(\()p Fm(A)718 729 y Fi(\013)742 744 y Fk(\))12 b Fl(\\)f Fw(dom)o Fk(\()p Fm(B)937 729 y Fi(\013)961 744 y Fk(\))17 b Fw(is)g(a)g(prop)q(er)i(subset)f(of)f(dom)o Fk(\()p Fm(H)1528 729 y Fi(\013)1551 744 y Fk(\))h Fw(whic)o(h)257 794 y(do)q(es)d(not)f(allo)o(w)e(to)i(apply)f(Theorem)g(5.2,)g(either.) 320 844 y(If)f Fm(\013)f Fk(=)h(1)p Fw(,)g(then)h(one)f(gets)h(that)g (dom)n Fk(\()p Fm(A)p Fk(\))6 b Fl(\\)g Fw(dom)o Fk(\()p Fm(B)r Fk(\))13 b Fw(is)f(a)g(prop)q(er)h(subset)h(of)e(dom)n Fk(\()p Fm(H)s Fk(\))257 893 y Fw(to)q(o)18 b(whic)o(h)g(yields)g Fm(H)k Fl(6)p Fk(=)d Fm(A)12 b Fk(+)g Fm(B)r Fw(.)32 b(Therefore)19 b(the)g(results)g(of)f([11)o(])f(and)h([12)o(])g(are)h (not)257 943 y(applicable.)320 993 y(Notice)e(that)g(in)g(con)o(trast)h (to)e([14)o(])h(and)g(to)g(examples)f(giv)o(en)g(there,)j(here)f(w)o(e) g(ha)o(v)o(e)257 1043 y(giv)o(en)g(an)f(example)g(when)h(the)h(op)q (erator)f Fm(B)990 1028 y Fi(\013)1032 1043 y Fw(is)g(not)f(small)f (with)h(resp)q(ect)k(to)c Fm(A)1598 1028 y Fi(\013)1640 1043 y Fw(for)257 1093 y Fm(\013)12 b Fl(2)f Fk(\(1)p Fm(=)p Fk(2)p Fm(;)c Fk(3)p Fm(=)p Fk(4\))p Fw(.)257 1228 y Fn(7)66 b(Remarks)257 1319 y Fw(Let)15 b(us)f(mak)o(e)e(the)j (follo)o(wing)c(remarks:)297 1402 y(\(i\))20 b(By)g(an)g(example)f(giv) o(en)g(in)h([17)o(])f(the)i(error)g(b)q(ound)f(estimate)f Fm(O)q Fk(\()p Fm(n)1484 1387 y Fh(\000)p Fj(\(2)p Fi(\013)p Fh(\000)p Fj(1\))1619 1402 y Fk(\))h Fw(in)361 1452 y(Theorem)14 b(1.1)f(and)h(Theorem)g(5.2)f(cannot)h(b)q(e)h(impro)o(v)o(ed,)d(i.e.) 18 b(it)c(is)g(the)h Ff(ultimate)361 1502 y(optimal)f Fw(one.)285 1581 y(\(ii\))20 b(This)14 b(optimal)e(error)j(b)q(ound)g (w)o(as)f(already)g(found)g(in)g([14)o(])g(see)h(Theorem)f(5.3)g(and) 361 1631 y(Theorem)e(5.5.)17 b(Comparing)10 b(with)j([14)o(])f(the)h (conditions)f(on)h(the)g(op)q(erators)h Fm(A)f Fw(and)361 1681 y Fm(B)18 b Fw(are)d(relaxed)h(here.)23 b(There)16 b(it)f(w)o(as)g(assumed)g(that)g Fm(B)1273 1666 y Fi(\013)1312 1681 y Fw(is)g(small)e(with)i(resp)q(ect)361 1730 y(to)f Fm(A)443 1715 y Fi(\013)467 1730 y Fw(,)f(i.e.)k(dom)o Fk(\()p Fm(A)689 1715 y Fi(\013)713 1730 y Fk(\))11 b Fl(\022)h Fw(dom)o Fk(\()p Fm(B)912 1715 y Fi(\013)936 1730 y Fk(\))i Fw(and)643 1813 y Fl(k)p Fm(B)697 1796 y Fi(\013)721 1813 y Fm(u)p Fl(k)d(\024)h Fm(a)p Fl(k)p Fm(A)895 1796 y Fi(\013)918 1813 y Fm(u)p Fl(k)d Fk(+)g Fm(b)p Fl(k)p Fm(u)p Fl(k)p Fm(;)47 b(u)12 b Fl(2)f Fw(dom)n Fk(\()p Fm(A)1356 1796 y Fi(\013)1380 1813 y Fk(\))p Fm(;)196 b Fw(\(7.1\))361 1896 y(for)18 b(some)e Fm(\013)i Fl(2)g Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))16 b Fw(and)i Fm(a)f Fl(2)h Fk(\(0)p Fm(;)7 b Fk(1\))p Fw(,)17 b Fm(b)h(>)g Fk(0)p Fw(.)30 b(This)17 b(condition)g(is)h(reduced)361 1946 y(in)g(the)h(presen)o(t)h(pap)q(er)e(to)h(the)f(mild)e(sub)q (ordination)i(condition)f(dom)o Fk(\()p Fm(A)1570 1930 y Fj(1)p Fi(=)p Fj(2)1622 1946 y Fk(\))i Fl(\022)361 1995 y Fw(dom)o Fk(\()p Fm(B)489 1980 y Fj(1)p Fi(=)p Fj(2)541 1995 y Fk(\))p Fw(,)g(whic)o(h)f(\(7.1\))g(ob)o(viously)f (implies.)29 b(The)19 b(y)o(et)f(op)q(en)h(problem)e(is)h(to)361 2045 y(eliminate)12 b(completely)h(this)h(sub)q(ordination)f (condition.)274 2125 y(\(iii\))19 b(The)13 b(conditions)e(on)h(the)h (Kato)f(functions)g Fm(f)17 b Fw(and)12 b Fm(g)h Fw(are)g(essen)o (tially)e(relaxed)i(com-)361 2175 y(paring)20 b(with)f([14)o(].)37 b(Indeed,)22 b(in)e([14)o(])f(the)i(Kato)f(functions)g Fm(f)25 b Fw(and)20 b Fm(g)i Fw(ha)o(v)o(e)e(to)361 2224 y(satisfy)14 b(at)g(zero)h(a)f(smo)q(othness)h(condition)e(of)h(the)h (t)o(yp)q(e)f Fl(j)p Fm(f)t Fl(j)1330 2230 y Fj(2)1349 2224 y Fm(;)7 b Fl(j)p Fm(g)q Fl(j)1413 2230 y Fj(2)1442 2224 y Fm(<)13 b Fl(1)g Fw(whic)o(h)i(is)361 2274 y(replaced)j(here)f (b)o(y)g Fl(j)p Fm(f)t Fl(j)729 2280 y Fj(2)p Fi(\013)769 2274 y Fm(;)7 b Fl(j)p Fm(g)q Fl(j)833 2280 y Fj(2)p Fi(\013)888 2274 y Fm(<)16 b Fl(1)p Fw(,)g Fm(\013)g Fl(2)f Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))p Fw(.)25 b(A)17 b(b)q(eha)o(viour)f(at)g(in\034nit)o(y)361 2324 y(lik)o(e)d Fm(f)t Fk(\()p Fm(x)p Fk(\))g Fl(\030)e Fm(x)597 2309 y Fh(\000)p Fj(2)p Fi(\013)677 2324 y Fw(is)j(also)f(demanded)g (there.)275 2403 y(\(iv\))20 b(Theorem)d(1.1)f(sho)o(ws)i(that)f(only)f (relations)h(b)q(et)o(w)o(een)i(certain)f(domains)d(related)361 2453 y(to)g Fm(A)p Fw(,)h Fm(B)i Fw(and)d Fm(H)j Fw(are)e(decisiv)o(e)g (for)f(the)i Ff(c)n(onver)n(genc)n(e)g(r)n(ate)f Fw(of)f(the)h(T)m (rotter-Kato)361 2503 y(pro)q(duct)f(form)o(ula.)953 2628 y(20)p eop %%Page: 21 21 21 20 bop 286 262 a Fw(\(v\))21 b(Theorem)c(1.1)f(holds)h(for)g Fm(\013)g Fl(2)g Fk(\(1)p Fm(=)p Fk(2)p Fm(;)7 b Fk(1\))p Fw(.)27 b(The)18 b(metho)q(d)e(of)h(the)h(pro)q(of)f(do)q(es)h(not)361 311 y(allo)o(w)f(to)h(include)g(the)g(case)i Fm(\013)e Fk(=)h(1)p Fw(.)30 b(Ho)o(w)o(ev)o(er,)19 b(this)f(case)h(w)o(as)f (considered)i(in)361 361 y([11)o(])15 b(and)f([12)o(].)21 b(It)15 b(is)g(remark)n(able)f(that)h(one)g(do)q(es)h(not)f(need)h(an)o (y)e(sub)q(ordination)361 411 y(condition)f(there.)275 494 y(\(vi\))20 b(F)m(or)g Fm(\013)i Fk(=)g(1)p Fm(=)p Fk(2)e Fw(w)o(e)g(cannot)h(exp)q(ect)h(op)q(erator-norm)d(con)o(v)o (ergence)j(in)e(general,)361 544 y(see)f([17)o(].)27 b(Ho)o(w)o(ev)o(er,)18 b(if)e(there)j(is)e(a)g(sub)q(ordination)f(suc)o (h)i(that)f(the)h(op)q(erator)g Fm(B)361 594 y Fw(is)d(relativ)o(ely)f (compact)g(with)h(resp)q(ect)i(to)e Fm(A)p Fw(,)f(then)i(op)q (erator-norm)e(con)o(v)o(ergence)361 643 y(holds,)f(see)i([15)o(].)320 735 y(The)e(\034rst)h(v)o(ersion)f(of)g(this)g(result)h(w)o(as)f (announced)h(in)f([8)o(])g(with)f(a)h(sk)o(etc)o(h)h(of)f(a)g(pro)q(of) 257 785 y(whic)o(h)i(relies)g(on)f(an)g(op)q(erator-norm)f(estimate)h (pro)o(v)o(ed)g(b)o(y)h(Birman)d(and)j(Solom)o(y)o(ak)c(in)257 834 y([1].)25 b(In)17 b(the)g(presen)o(t)h(pap)q(er)g(w)o(e)f(ha)o(v)o (e)f(impro)o(v)o(ed)f(this)h(previous)h(result,)h(b)o(y)e(remo)o(ving) 257 884 y(some)d(restrictiv)o(e)i(condition)e(on)g(the)h(Kato)f (functions)h Fm(f)k Fw(and)c Fm(g)g Fw(imp)q(osed)f(there)i(as)e(w)o (ell)257 934 y(as)e(b)o(y)g(giving)e(a)i(di\033eren)o(t,)g(simpler)f (pro)q(of)g(whic)o(h)h(do)q(es)h(not)e(mak)o(e)g(use)h(of)f(op)q (erator-norm)257 984 y(inequalities)j(\340)h(la)f(Birman-Solom)n(y)o (ak)o(.)257 1121 y Fn(A)n(c)n(kno)n(wledgemen)n(ts)257 1212 y Fw(T.I.)i(and)f(H.N.)h(are)g(grateful)g(to)g(the)h(Univ)o (ersit\351)g(de)f(la)g(M\351diterran\351e)i(and)e(Cen)o(tre)h(de)257 1262 y(Ph)o(ysique)11 b(Th\351orique-CNRS-Lumin)o(y)m(,)d(Marseille)j (\(F)m(rance\),)g(for)g(hospitalit)o(y)e(extended)257 1312 y(to)k(them)f(and)g(\034nancial)g(supp)q(ort.)18 b(H.N.)12 b(and)g(V.Z.)g(thank)h(Departmen)o(t)f(of)g(Mathemat-)257 1361 y(ics,)g(F)m(acult)o(y)f(of)g(Science,)i(Kanaza)o(w)o(a)f(Univ)o (ersit)o(y)m(,)f(Kanaza)o(w)o(a)g(\(Japan\),)h(for)f(hospitalit)o(y)257 1411 y(and)20 b(for)f(\034nancial)g(supp)q(ort)h(extended)i(to)d(them)g (during)g(their)h(visit)f(there.)37 b(T.I.)19 b(is)257 1461 y(supp)q(orted)d(b)o(y)d(the)i(Japan)f(So)q(ciet)o(y)g(of)f(the)i (Promotion)d(of)h(Science)i(\(Grand-in-Aid)e(for)257 1511 y(Scien)o(ti\034c)19 b(Researc)o(hes)h(\(B\))f(No.)31 b(11440040,)16 b(13440044\).)29 b(W)m(e)18 b(w)o(ould)f(lik)o(e)h(to)g (thank)257 1561 y(Prof.)h(D.)13 b(F)m(ujiw)o(ara)f(and)i(Prof.)k(H.)c (T)m(rieb)q(el)g(who)f(help)q(ed)i(us)f(to)g(clarify)f(the)i(relation)e (of)257 1610 y(the)i(domains)d(of)h(fractional)g(p)q(o)o(w)o(ers)h(of)g (Diric)o(hlet)f(and)h(Neumann)e(Laplacians.)257 1748 y Fn(References)278 1839 y Fw([1])20 b(M.)d(S.)h(Birman)e(and)i(M.)f (Z.)h(Solom)o(y)o(ak,)e(Estimates)h(for)h(the)g(di\033erence)i(of)e (frac-)343 1888 y(tional)11 b(p)q(o)o(w)o(ers)j(of)e(self-adjoin)o(t)g (op)q(erators)i(in)e(case)i(of)e(un)o(b)q(ounded)i(p)q(erturbations,) 343 1938 y Ff(Zap.)e(Nauchn.)h(Semin.)g(L)n(eningr.)f(Otd.)g(Mat.)g (Inst.)g(Steklova)f Fg(178)p Fw(\(1989\),)f(120-145)343 1988 y(\(Russian\),)j(translated)h(in)g Ff(J.)g(Soviet)h(Math.)f Fg(61)g Fw(\(1992\),)e(No.)i(2,)f(2018-2034.)278 2071 y([2])20 b(P)m(.)9 b(R.)f(Cherno\033,)j(Note)f(on)f(pro)q(duct)i(form)o (ulas)c(for)j(op)q(erator)g(semigroups,)f Ff(J.)h(F)m(unct.)343 2121 y(A)o(nal.)j Fg(2)p Fw(\(1968\),)g(238-242.)278 2204 y([3])20 b(P)m(.)d(R.)g(Cherno\033,)i(Semigroup)d(pro)q(duct)j (form)o(ulas)d(and)i(addition)f(of)g(un)o(b)q(ounded)343 2254 y(op)q(erators,)d Ff(Bul)r(l.)g(A)o(mer.)h(Math.)g(So)n(c.)f Fg(76)g Fw(\(1970\),)f(395-398.)278 2337 y([4])20 b(D.)11 b(F)m(ujiw)o(ara,)g(Concrete)j(c)o(haracterization)g(of)e(the)h (domains)d(of)i(fractional)g(p)q(o)o(w)o(ers)343 2387 y(of)e(some)f(elliptic)h(di\033eren)o(tial)g(op)q(erators)i(of)e(the)h (second)g(order.)g Ff(Pr)n(o)n(c.)h(Jap)n(an)h(A)n(c)n(ad.)343 2436 y Fg(43)g Fw(\(1967\),)g(82-86.)953 2628 y(21)p eop %%Page: 22 22 22 21 bop 278 262 a Fw([5])20 b(P)m(.)11 b(Grisv)n(ard,)g (Caract\351risation)h(de)g(quelques)h(espaces)g(d'in)o(terp)q(olation.) d Ff(A)o(r)n(ch.)j(R)n(a-)343 311 y(tional)h(Me)n(ch.)h(A)o(nal.)f Fg(25)g Fw(\(1967\),)e(40-63.)278 394 y([6])20 b(T.)10 b(Kato,)h(On)g(the)h(T)m(rotter-Lie)f(pro)q(duct)g(form)o(ula.)d Ff(Pr)n(o)n(c.)13 b(Jap)n(an)f(A)n(c)n(ad.)f Fg(50)g Fw(\(1974\),)343 444 y(694-698.)278 527 y([7])20 b(T.)f(Kato,)j(T)m (rotter's)e(pro)q(duct)i(form)o(ula)17 b(for)j(an)g(arbitrary)g(pair)g (of)g(self-adjoin)o(t)343 577 y(con)o(traction)15 b(semigroups.)e Ff(T)m(opics)j(in)g(F)m(unct.)g(A)o(nal.,)f(A)n(d.)h(Math.)g(Suppl.)g (Studies)343 627 y Fw(V)m(ol)9 b Fg(3)p Fw(,)i(185-195)d(\(I.Goh)o(b)q (erg)i(and)h(Mc.Kac,)g(eds.\).)f(A)o(cad.)g(Press,)j(New)e(Y)m(ork,)f (1978.)278 710 y([8])20 b(T.)d(Ic)o(hinose,)i(H.)e(Neidhardt)h(and)g (V.)f(A.)h(Zagrebno)o(v,)g(Op)q(erator)h(norm)d(con)o(v)o(er-)343 760 y(gence)j(of)f(T)m(rotter-Kato)g(pro)q(duct)h(form)o(ula.)c(Pro)q (ceedings)20 b(of)d(the)i(In)o(ternational)343 809 y(Conference)c(on)f (F)m(unctional)e(Analysis.)h(Kiev,)h(2001.)e(T)m(o)h(app)q(ear.)278 892 y([9])20 b(T.)15 b(Ic)o(hinose)i(and)f(Hideo)g(T)m(am)o(ura,)e 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b(L.)g(Roga)o(v)n(a,)f(Error)j(b)q(ounds)f(for)f (the)i(T)m(rotter-t)o(yp)q(e)g(form)o(ulas)c(for)j(self-adjoin)o(t)343 2171 y(op)q(erators.)e Ff(F)m(unct.)h(A)o(nal.)f(Applic)n(ation)g Fg(27)g Fw(\(1993\),)e(No.)i(3,)f(217-219.)257 2254 y([17])20 b(Hiroshi)14 b(T)m(am)o(ura,)e(A)i(remark)g(on)g(op)q(erator-norm)f (con)o(v)o(ergence)j(of)e(T)m(rotter-Kato)343 2304 y(pro)q(duct)g(form) o(ula.)d Ff(Inte)n(gr.)k(Equ.)h(Op)n(er.)f(The)n(ory)f Fg(37)g Fw(\(2000\),)e(350-356.)257 2387 y([18])20 b(H.)f(T)m(rieb)q (el,)h(In)o(terp)q(olation)e(theory)m(.)h(F)m(unction)g(spaces.)h (Di\033eren)o(tial)f(op)q(erators.)343 2437 y(Deutsc)o(her)c(V)m(erlag) f(des)g(Wissensc)o(haften,)h(Berlin,)e(1978.)953 2628 y(22)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF