This is a multi-part message in MIME format. ---------------0502091026424 Content-Type: text/plain; name="05-60.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="05-60.keywords" Strichartz estimates, decay of energy, monodromy operator ---------------0502091026424 Content-Type: application/postscript; name="stri17.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="stri17.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.92b Copyright 2002 Radical Eye Software %%Title: stri17.dvi %%Pages: 16 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: CMBX10 CMR9 CMCSC10 CMMI9 CMR6 CMMI6 CMSY9 CMSY6 MSBM10 %%+ CMR10 CMEX10 CMMI10 CMR8 CMMI8 CMSY10 CMSY8 MSBM7 CMTI10 CMTI9 CMBX9 %%+ CMTT9 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips -o stri17.ps stri17 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2005.01.31:1057 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX9 %!PS-AdobeFont-1.1: CMBX9 1.0 %%CreationDate: 1991 Aug 20 16:36:25 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-58 -250 1195 750}readonly def /UniqueID 5000767 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI9 %!PS-AdobeFont-1.1: CMTI9 1.0 %%CreationDate: 1991 Aug 18 21:08:07 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-35 -250 1148 750}readonly def /UniqueID 5000827 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E3948FFB3DF7BFF10C9BDA4EFE5F68A8CB1526990D1357AE6D2F7C2D2EF8496 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM7 %!PS-AdobeFont-1.1: MSBM7 2.1 %%CreationDate: 1992 Oct 17 08:30:50 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM7) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 82 /R put readonly def /FontBBox{0 -504 2615 1004}readonly def /UniqueID 5032014 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -955 1185 779}readonly def /UniqueID 5000818 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 67 /C put dup 78 /N put dup 82 /R put dup 90 /Z put readonly def /FontBBox{-55 -420 2343 920}readonly def /UniqueID 5031982 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6A66A4000A13D5F68BFF326D 1D432B0D064B56C598F4338C319309181D78E1629A31ECA5DD8536379B03C383 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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cleartomark %%EndFont %%BeginFont: CMR8 %!PS-AdobeFont-1.1: CMR8 1.0 %%CreationDate: 1991 Aug 20 16:39:40 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-4 -948 1329 786}readonly def /UniqueID 5000816 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY9 %!PS-AdobeFont-1.1: CMSY9 1.0 %%CreationDate: 1991 Aug 15 07:22:27 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-30 -958 1146 777}readonly def /UniqueID 5000819 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI6 %!PS-AdobeFont-1.1: CMMI6 1.100 %%CreationDate: 1996 Jul 23 07:53:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{11 -250 1241 750}readonly def /UniqueID 5087381 def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D6A8F05B47AF95EF28A9C561DBDC98C47CF5 5250011D19E9366EB6FD153D3A100CAA6212E3D5D93990737F8D326D347B7EDC 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /.notdef put readonly def /FontBBox{-20 -250 1193 750}readonly def /UniqueID 5000789 def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI9 %!PS-AdobeFont-1.1: CMMI9 1.100 %%CreationDate: 1996 Jul 23 07:53:55 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. 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b(fact)j(in)e([20)q(],)j(Lemma)e(7,)i(the)d(transformation)h Fk(\037F)13 b Fn(\()p Fk(\013)p Fn(\()p Fk(t)p Fn(\))p Fk(U)2885 3527 y Fj(0)2926 3513 y Fn(\()p Fk(t)p Fn(\)\()p Fk( )3123 3527 y Fj(1)3164 3513 y Fk(f)d Fn(\)\)\()p Fk(\022)s(;)15 b(s)p Fn(\))21 b(is)h(treated,)0 3621 y(where)31 b Fk(\013)p Fn(\()p Fk(t)p Fn(\))d Fh(2)e Fk(C)612 3588 y Ff(1)686 3621 y Fn(\()p Fg(R)p Fn(\))32 b(is)f(suc)m(h)g(that)h Fk(\013)p Fn(\()p Fk(t)p Fn(\))c(=)e(0)32 b(for)f Fk(t)c Fh(\024)f Fk(t)2043 3635 y Fj(0)2082 3621 y Fk(;)32 b(\013)p Fn(\()p Fk(t)p Fn(\))c(=)e(1)32 b(for)f Fk(t)26 b Fh(\025)h Fk(t)2832 3635 y Fj(0)2892 3621 y Fn(+)21 b(1)p Fk(;)36 b(t)3123 3635 y Fj(0)3189 3621 y Fk(>)27 b Fn(0.)44 b(The)30 b(analysis)0 3729 y(of)43 b(the)f(term)h Fk(\037F)13 b Fn(\(\(1)29 b Fh(\000)f Fk(\013)p Fn(\()p Fk(t)p Fn(\)\))p Fk(U)1143 3743 y Fj(0)1184 3729 y Fn(\()p Fk(t)p Fn(\)\()p Fk( )1381 3743 y Fj(1)1421 3729 y Fk(f)10 b Fn(\)\)\()p Fk(\022)s(;)15 b(s)p Fn(\))43 b(is)f(trivial)i(and)d(w)m(e)i(obtain)g(the)g(result.)77 b(Moreo)m(v)m(er,)47 b(in)c(a)0 3837 y(neigh)m(b)s(orho)s(o)s(d)29 b(of)h(0)h(w)m(e)g(ha)m(v)m(e)h(the)e(represen)m(tation)992 4019 y Fk(\037)p Fn(\()p Fk(U)1146 4033 y Fj(0)1185 4019 y Fn(\()p Fk(T)13 b Fn(\))21 b Fh(\000)f Fk(e)1475 3982 y Ff(\000)p Fi(i\022)1593 4019 y Fk(I)7 b Fn(\))1675 3982 y Ff(\000)p Fj(1)1770 4019 y Fk( )1829 4033 y Fj(1)1894 4019 y Fn(=)25 b Fk(B)2059 4033 y Fj(0)2098 4019 y Fn(\()p Fk(\022)s Fn(\))p Fk(\022)2260 3982 y Fi(n)p Ff(\000)p Fj(1)2412 4019 y Fn(ln)15 b Fk(\022)22 b Fn(+)e Fk(B)2728 4033 y Fj(1)2767 4019 y Fn(\()p Fk(\022)s Fn(\))p Fk(;)805 b Fn(\(2.2\))0 4211 y(where)32 b Fk(B)334 4225 y Fj(0)373 4211 y Fn(\()p Fk(\022)s Fn(\))g(and)f Fk(B)768 4225 y Fj(1)808 4211 y Fn(\()p Fk(\022)s Fn(\))h(are)g(analytic)i(for)d Fh(j)p Fk(\022)s Fh(j)d(\024)g Fk(\017)1857 4225 y Fj(0)1928 4211 y Fn(and)k Fk(@)2160 4167 y Fi(j)2155 4241 y(\022)2197 4211 y Fk(B)2266 4225 y Fj(0)2305 4211 y Fn(\()p Fk(\022)s 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b(ole)34 b(of)g Fk(\037)p Fn(\()p Fk(U)10 b Fn(\()p Fk(T)j Fn(\))21 b Fh(\000)f Fk(z)t(I)7 b Fn(\))3049 4711 y Ff(\000)p Fj(1)3145 4744 y Fk( )3204 4758 y Fj(1)3276 4744 y Fd(for)34 b Fk(n)f Fd(o)-5 b(dd)35 b Fn(\()p Fd(r)-5 b(esp.)0 4852 y Fk(n)32 b Fd(even)p Fn(\))p Fd(,)g(if)752 5061 y Fn(dim)14 b(Ker)1085 4960 y Fl(\020)1140 5061 y Fk(I)27 b Fh(\000)20 b Fk( )1375 4937 y Fl(Z)1466 4963 y Fi(T)1426 5143 y Fj(0)1537 5061 y Fk(U)10 b Fn(\()p Fk(T)e(;)15 b(s)p Fn(\))p Fk(Q)p Fn(\()p Fk(s)p Fn(\))p Fk(ds )s Fn(\()p Fk(U)2257 5075 y Fj(0)2297 5061 y Fn(\()p Fk(T)e Fn(\))21 b Fh(\000)f Fk(z)2587 5075 y Fj(0)2627 5061 y Fk(I)7 b Fn(\))2709 5023 y Ff(\000)p Fj(1)2803 5061 y Fk( )2862 5075 y Fj(1)2902 4960 y Fl(\021)2982 5061 y Fk(>)25 b Fn(0)p Fk(:)p eop end %%Page: 7 7 TeXDict begin 7 6 bop 1330 66 a Fj(GLOBAL)31 b(STRICHAR)-6 b(TZ)31 b(ESTIMA)-6 b(TES)1295 b(7)150 266 y Fn(Finally)-8 b(,)28 b(to)e(study)e(the)h(in)m(v)m(ertibilit)m(y)i(of)e(the)h(op)s (erator)f(\()p Fk(I)17 b Fh(\000)10 b 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Fk(g)s Fn(\()p Fk(\022)s(;)g(s)p Fn(\))p Fk(d\022)s(ds;)0 1988 y Fn(where)30 b Fk(d)310 2002 y Fi(\013)385 1988 y Fn(=)25 b([)p Fk(i\013)c Fh(\000)f Fk(\031)s(;)15 b(i\013)22 b Fn(+)d Fk(\031)s Fn(])31 b(and)f Fk(\013)c(>)f Fn(0)30 b(will)h(b)s(e)f(c)m(hosen)h(large)g (enough)f(b)s(elo)m(w.)150 2204 y(Cho)s(ose)41 b(an)g(in)m(teger)i Fk(m)h Fh(2)f Fg(Z)e Fn(so)g(that)h Fk(t)1619 2171 y Ff(0)1686 2204 y Fn(=)h Fk(t)28 b Fh(\000)f Fk(mT)56 b Fh(2)43 b Fn([0)p Fk(;)15 b(T)e Fn([)43 b(and)d(\014x)h Fk(m)p Fn(.)73 b(Changing)42 b(the)f(v)-5 b(ariable)0 2312 y Fk(s)25 b Fn(=)g Fk(s)207 2279 y Ff(0)250 2312 y Fn(+)20 b Fk(mT)43 b Fn(and)29 b(using)h(the)h(prop)s(ert)m(y)f Fk(U)10 b Fn(\()p Fk(t)20 b Fn(+)g Fk(mT)8 b(;)15 b(s)20 b Fn(+)g Fk(mT)13 b Fn(\))25 b(=)g Fk(U)10 b Fn(\()p Fk(t;)15 b(s)p Fn(\),)31 b(w)m(e)g(obtain)1104 2492 y(1)p 1077 2532 V 1077 2616 a(2)p Fk(\031)1202 2430 y Fl(Z)1293 2456 y Fi(t)1318 2432 y Fp(0)1253 2636 y Ff(\0001)1398 2553 y Fk('U)10 b Fn(\()p Fk(t)1597 2516 y Ff(0)1620 2553 y Fk(;)15 b(s)1703 2516 y Ff(0)1727 2553 y Fn(\))p Fk(Q)p Fn(\()p Fk(s)1912 2516 y Ff(0)1935 2553 y Fn(\))1985 2430 y Fl(Z)2036 2636 y Fi(d)2072 2644 y Fr(\013)2136 2553 y Fk(e)2178 2516 y Ff(\000)p Fi(im\022)2359 2553 y Fk(g)s Fn(\()p Fk(\022)s(;)g(s)2569 2516 y Ff(0)2592 2553 y Fn(\))p Fk(d\022)s(ds)2810 2516 y Ff(0)928 2887 y Fn(=)1062 2826 y(1)p 1034 2866 V 1034 2950 a(2)p Fk(\031)1160 2764 y Fl(Z)1251 2790 y Fi(t)1276 2766 y Fp(0)1211 2970 y Fj(0)1318 2887 y Fk('U)10 b Fn(\()p Fk(t)1517 2850 y Ff(0)1541 2887 y Fk(;)15 b(s)1624 2850 y Ff(0)1647 2887 y Fn(\))p Fk(Q)p Fn(\()p Fk(s)1832 2850 y Ff(0)1856 2887 y Fn(\))p Fk(U)1953 2901 y Fj(0)1993 2887 y Fn(\()p Fk(s)2071 2850 y Ff(0)2094 2887 y Fn(\))2144 2764 y Fl(Z)2195 2970 y Fi(d)2231 2978 y Fr(\013)2295 2887 y Fk(e)2337 2850 y Ff(\000)p Fi(im\022)2517 2887 y Fk(g)s Fn(\()p Fk(\022)s(;)g Fn(0\))p Fk(d\022)s(ds)2947 2850 y Ff(0)584 3205 y Fn(+)692 3144 y(1)p 665 3184 V 665 3267 a(2)p Fk(\031)821 3091 y Ff(1)790 3119 y Fl(X)792 3316 y Fi(k)r Fj(=0)937 3081 y Fl(Z)1028 3108 y Ff(\000)p Fi(k)r(T)987 3288 y Ff(\000)p Fi(k)r(T)10 b Ff(\000)p Fi(T)1257 3205 y Fk('U)g Fn(\()p Fk(t)1456 3168 y Ff(0)1480 3205 y Fk(;)15 b(s)1563 3168 y Ff(0)1586 3205 y Fn(\))p Fk(Q)p Fn(\()p Fk(s)1771 3168 y Ff(0)1794 3205 y Fn(\))1844 3081 y Fl(Z)1895 3288 y Fi(d)1931 3296 y Fr(\013)1995 3205 y Fk(e)2037 3168 y Ff(\000)p Fi(im\022)2218 3205 y Fk(g)s Fn(\()p Fk(\022)s(;)g(s)2428 3168 y Ff(0)2451 3205 y Fn(\))p Fk(d\022)s(ds)2669 3168 y Ff(0)2718 3205 y Fn(=)25 b Fk(I)2854 3219 y Fj(1)2893 3205 y Fn(\()p Fk(t)p Fn(\))c(+)f Fk(I)3148 3219 y Fj(2)3187 3205 y Fn(\()p Fk(t)p Fn(\))p Fk(:)150 3445 y Fn(The)31 b(in)m(tegral)j Fk(I)711 3459 y Fj(1)750 3445 y Fn(\()p Fk(t)p Fn(\))e(can)g(b)s(e)f(estimated)i (follo)m(wing)h(the)e(argumen)m(t)g(giv)m(en)h(b)s(elo)m(w)e(and)h(w)m (e)g(will)g(deal)g(with)0 3553 y(the)f(in\014nite)f(sum.)39 b(Changing)31 b(the)f(v)-5 b(ariable)31 b Fk(s)1642 3520 y Ff(0)1690 3553 y Fn(=)25 b Fh(\000)p Fk(T)33 b Fh(\000)20 b Fk(k)s(T)33 b Fn(+)20 b Fk(\030)t Fn(,)30 b(w)m(e)h(get)h(the)e (series)640 3679 y Ff(1)609 3706 y Fl(X)611 3904 y Fi(k)r Fj(=0)756 3669 y Fl(Z)847 3695 y Fi(T)806 3875 y Fj(0)917 3669 y Fl(Z)968 3875 y Fi(d)1004 3883 y Fr(\013)1067 3792 y Fk('U)10 b Fn(\()p Fk(t)1266 3755 y Ff(0)1311 3792 y Fn(+)19 b Fk(T)8 b(;)15 b Fn(0\))p Fk(U)10 b Fn(\()p Fk(k)s(T)j Fn(\))p Fk(U)d Fn(\(0)p Fk(;)15 b(\030)t Fn(\))p Fk(e)2153 3755 y Fi(i)p Fj(\()p Fi(k)r Fj(+1\))p Fi(\022)2403 3792 y Fk(Q)p Fn(\()p Fk(\030)t Fn(\))p Fk(e)2631 3755 y Ff(\000)p Fi(im\022)2813 3792 y Fk(g)s Fn(\()p Fk(\022)s(;)g(s)3023 3755 y Ff(0)3046 3792 y Fn(\))p Fk(d\022)s(d\030)t(:)423 b Fn(\(3.1\))0 4046 y(Here)31 b(w)m(e)g(ha)m(v)m(e)g(used)f(the)g(fact) i(that)1042 4216 y Fk(U)10 b Fn(\()p Fk(t)1182 4179 y Ff(0)1226 4216 y Fn(+)20 b Fk(T)32 b Fn(+)20 b Fk(k)s(T)8 b(;)15 b(\030)t Fn(\))26 b(=)f Fk(U)10 b Fn(\()p Fk(t)1985 4179 y Ff(0)2029 4216 y Fn(+)20 b Fk(T)8 b(;)15 b Fn(0\))p Fk(U)10 b Fn(\()p 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5106 y Ff(0)1697 5143 y Fn(+)20 b Fk(T)8 b(;)15 b Fn(0\)\(1)22 b Fh(\000)e Fk(\037)p Fn(\))25 b(=)g(0)p Fk(:)p eop end %%Page: 9 9 TeXDict begin 9 8 bop 1330 66 a Fj(GLOBAL)31 b(STRICHAR)-6 b(TZ)31 b(ESTIMA)-6 b(TES)1295 b(9)0 266 y Fn(Notice)38 b(that)f(the)f(function)g Fk(\037)g Fn(dep)s(ends)e(on)i Fk(':)h Fn(According)g(to)g(the)f(prop)s(erties)g(\(1.6\),)k(\(2.1\))d (of)g Fk( )3513 280 y Fj(1)3552 266 y Fk(;)j( )s Fn(,)e(it)f(is)0 374 y(clear)31 b(that)g(\(3.1\))i(can)d(b)s(e)g(written)g(in)g(the)h (form)404 491 y Fl(Z)454 698 y Fi(d)490 706 y Fr(\013)554 491 y Fl(Z)645 518 y Fi(T)605 698 y Fj(0)715 615 y Fk('U)10 b Fn(\()p Fk(t)914 578 y Ff(0)958 615 y Fn(+)20 b Fk(T)8 b(;)15 b Fn(0\))p Fk(\037)p Fn(\()p Fk(e)1364 578 y Ff(\000)p Fi(i\022)1484 615 y Fk(I)27 b Fh(\000)20 b Fk(U)10 b Fn(\()p Fk(T)j Fn(\)\))1885 578 y Ff(\000)p Fj(1)1980 615 y Fk( )2039 629 y Fj(1)2079 615 y Fk(U)d Fn(\(0)p Fk(;)15 b(\030)t Fn(\))p Fk(Q)p Fn(\()p Fk(\030)t Fn(\))p Fk(U)2598 629 y Fj(0)2639 615 y Fn(\()p Fk(\030)t Fn(\))p Fk(e)2795 578 y Ff(\000)p Fi(im\022)2976 615 y Fk( )s(g)s Fn(\()p Fk(\022)s(;)g Fn(0\))p Fk(d\030)t(d\022)s(:)219 b Fn(\(3.2\))0 850 y(First)41 b(assume)f(that)h Fk(n)g Fh(\025)h Fn(3)e(is)h(o)s(dd.)69 b(Then)39 b(\()p Fh(R)p Fn(\))i(implies)g(that)g Fk(R)2433 864 y Fi(\037)2481 850 y Fn(\()p Fk(\022)s Fn(\))f(has)g(no)g(p)s(oles)h(with)f Fh(=)p Fk(\022)j Fh(\025)f Fn(0)f(and)0 958 y(w)m(e)c(can)f(c)m(ho)s (ose)h Fk(\016)h(>)d Fn(0)h(so)h(that)f Fk(R)1260 972 y Fi(\037)1308 958 y Fn(\()p Fk(\022)s Fn(\))g(has)g(no)g(p)s(oles)g Fk(\022)i Fn(with)e Fh(=)p Fk(\022)g Fh(\025)f(\000)p Fk(\016)s(T)8 b(;)39 b Fh(\000)p Fk(\031)f(<)c Fh(<)p Fk(\022)j Fh(\024)d Fk(\031)s Fn(.)58 b(Let)37 b Fk(d)3651 973 y Ff(\000)p Fi(\016)r(T)3829 958 y Fn(=)0 1066 y([)p Fh(\000)p Fk(i\016)s(T)d Fh(\000)20 b Fk(\031)s(;)15 b Fh(\000)p Fk(i\016)s(T)34 b Fn(+)20 b Fk(\031)s Fn(])p Fk(:)31 b Fn(Recall)h(that)f Fk(t)25 b Fn(=)g Fk(mT)33 b Fn(+)20 b Fk(t)1817 1033 y Ff(0)1840 1066 y Fn(,)30 b(so)1649 1248 y Fk(e)1691 1211 y Ff(\000)p Fi(m\016)r(T)1923 1248 y Fh(\024)25 b Fk(C)7 b(e)2133 1211 y Ff(\000)p Fi(\016)r(t)0 1424 y Fn(with)30 b Fk(C)i(>)25 b Fn(0)31 b(indep)s(enden)m(t)e(on)h Fk(m)g Fn(and)g Fk(t)p Fn(.)40 b(On)30 b(the)g(other)h(hand,)652 1669 y Fk( )s(g)s Fn(\()p Fk(\022)s(;)15 b Fn(0\))27 b(=)e Fk( )1192 1555 y Ff(1)1162 1583 y Fl(X)1163 1780 y Fi(k)r Fj(=0)1309 1669 y Fk(U)1371 1683 y Fj(0)1410 1669 y Fn(\()p Fk(k)s(T)13 b Fn(\))p Fk(e)1638 1632 y Fi(ik)r(\022)1741 1669 y Fk(f)34 b Fn(=)25 b Fk(e)1958 1632 y Ff(\000)p Fi(i\022)2076 1669 y Fk( )s Fn(\()p Fk(e)2215 1632 y Ff(\000)p Fi(i\022)2334 1669 y Fk(I)j Fh(\000)20 b Fk(U)2555 1683 y Fj(0)2594 1669 y Fn(\()p Fk(T)13 b Fn(\)\))2765 1632 y Ff(\000)p Fj(1)2860 1669 y Fk(f)5 b(;)35 b Fh(=)p Fk(\022)27 b(>)e Fn(0)0 1934 y(and)33 b(w)m(e)i(conclude)f(that)h Fk( )s(g)s Fn(\()p Fk(\022)s(;)15 b Fn(0\))36 b(admits)e(an)g(analytic)h(con)m (tin)m(uation)h(in)e Fg(C)p Fn(.)51 b(W)-8 b(e)35 b(shift)f(the)g(con)m (tour)h(of)f(the)0 2042 y(in)m(tegration)e(from)e Fk(d)724 2056 y Fi(\013)804 2042 y Fn(to)h Fk(d)962 2057 y Ff(\000)p Fi(\016)r(T)1137 2042 y Fn(and)f(w)m(e)g(obtain)1405 2223 y Fh(k)p Fk(I)1490 2237 y Fj(2)1530 2223 y Fn(\()p Fk(t)p Fn(\))p Fh(k)c(\024)f Fk(C)1865 2237 y Fj(1)1904 2223 y Fk(e)1946 2185 y Ff(\000)p Fi(\016)r(t)2065 2223 y Fh(k)p Fk(f)10 b Fh(k)p Fk(;)36 b(t)25 b Fh(\025)g Fn(0)p Fk(;)0 2399 y Fn(where)30 b Fk(C)328 2413 y Fj(1)393 2399 y Fk(>)25 b Fn(0)31 b(dep)s(ends)e(on)h Fk(')h Fn(and)f Fk(R)q Fn(.)41 b(T)-8 b(o)31 b(estimate)h Fk(I)1984 2413 y Fj(1)2024 2399 y Fn(\()p Fk(t)p Fn(\),)f(w)m(e)g(shift)f(again)i(the) f(con)m(tour)g(of)g(in)m(tegration)h(to)0 2507 y Fk(d)47 2522 y Ff(\000)p Fi(\016)r(T)221 2507 y Fn(and)e(w)m(e)h(obtain)g(the)f (same)h(estimate)h(as)f(that)g(for)f Fk(I)2051 2521 y Fj(2)2090 2507 y Fn(\()p Fk(t)p Fn(\).)41 b(Com)m(bining)31 b(these)f(estimates,)j(w)m(e)e(get)1439 2689 y Fh(k)p Fk('U)10 b Fn(\()p Fk(t;)15 b Fn(0\))p Fk(f)10 b Fh(k)27 b(\024)e Fk(C)2091 2703 y Fj(2)2130 2689 y Fk(e)2172 2651 y Ff(\000)p Fi(\016)r(t)2291 2689 y Fh(k)p Fk(f)10 b Fh(k)p Fk(:)1252 b Fn(\(3.3\))0 2865 y(Next)31 b(let)h(0)25 b Fh(\024)g Fk(s)g Fh(\024)g Fk(t)30 b Fn(and)g(let)h Fk(s)20 b Fh(\000)g Fk(j)5 b(T)39 b Fh(2)24 b Fn([0)p Fk(;)15 b(T)e Fn([)p Fk(;)37 b(j)31 b Fh(2)25 b Fg(N)p Fk(:)30 b Fn(Then)1043 3040 y Fh(k)p Fk('U)10 b Fn(\()p Fk(t;)15 b(s)p Fn(\))p Fk(f)10 b Fh(k)27 b Fn(=)e Fh(k)p Fk('U)10 b Fn(\()p Fk(t)21 b Fh(\000)f Fk(j)5 b(T)j(;)15 b Fn(0\))p Fk(U)10 b Fn(\(0)p Fk(;)15 b(s)22 b Fh(\000)e Fk(j)5 b(T)13 b Fn(\))p Fk(f)d Fh(k)935 3186 y(\024)25 b Fk(C)1096 3200 y Fj(3)1136 3186 y Fk(e)1178 3148 y Ff(\000)p Fi(\016)r Fj(\()p Fi(t)p Ff(\000)p Fi(j)t(T)10 b Fj(\))1489 3186 y Fh(k)p Fk(U)g Fn(\(0)p Fk(;)15 b(s)22 b Fh(\000)e Fk(j)5 b(T)13 b Fn(\))p Fk(f)d Fh(k)25 b(\024)g Fk(C)2311 3200 y Fj(4)2351 3186 y Fk(e)2393 3148 y Ff(\000)p Fi(\016)r Fj(\()p Fi(t)p Ff(\000)p Fi(s)p Fj(\))2654 3186 y Fh(k)p Fk(f)10 b Fh(k)914 b Fn(\(3.4\))0 3361 y(with)29 b(a)g(constan)m(t)i Fk(C)711 3375 y Fj(4)779 3361 y Fn(dep)s(ending)d(on)g Fk(')i Fn(and)e Fk(R)19 b Fn(+)e Fk(T)c Fn(.)40 b(Here)30 b(w)m(e)f(ha)m(v)m(e)h(used)f(the)g (fact)h(that)f Fk(U)10 b Fn(\(0)p Fk(;)15 b(s)k Fh(\000)e Fk(j)5 b(T)13 b Fn(\))p Fk(f)39 b Fn(has)0 3469 y(a)31 b(compact)g(supp)s(ort)e(indep)s(enden)m(t)g(of)i Fk(s)p Fn(.)150 3685 y(P)m(assing)38 b(to)h(the)f(case)g Fk(n)f Fn(ev)m(en,)k(w)m(e)d(will)g(estimate)h(the)f(in)m(tegral)i(\(3.2\).)64 b(Cho)s(ose)37 b(again)i Fk(\016)i(>)c Fn(0)h(so)g(that)0 3793 y Fk(R)69 3807 y Fi(\037)117 3793 y Fn(\()p Fk(\022)s Fn(\))30 b(has)g(no)h(p)s(oles)f Fk(\022)i Fn(lying)f(in)1300 3969 y Fh(f)p Fk(\022)d Fn(:)d Fh(=)p Fk(\022)i Fh(\025)e(\000)p Fk(\016)s(T)8 b(;)36 b Fh(\000)p Fk(\031)28 b Fh(\024)d(\006<)p Fk(\022)i(<)e Fn(0)p Fh(g)p Fk(:)0 4144 y Fn(Next)42 b(c)m(ho)s(ose)g Fk(\016)48 b Fh(\025)43 b Fk(\017)771 4158 y Fj(0)853 4144 y Fk(>)h Fn(0)d(so)h(that)g Fk(R)1454 4158 y Fi(\037)1502 4144 y Fn(\()p Fk(\025)p Fn(\))f(is)h(b)s(ounded)d (for)i Fh(j)p Fk(\022)s Fh(j)i(\024)g Fk(\017)2591 4158 y Fj(0)2672 4144 y Fn(and)d(consider)h(the)h(con)m(tour)g Fk(\015)48 b Fn(=)0 4252 y(\000)57 4266 y Fj(1)116 4252 y Fh([)20 b Fk(!)j Fh([)d Fn(\000)415 4266 y Fj(2)454 4252 y Fk(;)31 b Fn(where)863 4428 y(\000)920 4442 y Fj(1)984 4428 y Fn(=)25 b([)p Fh(\000)p Fk(i\016)s(T)34 b Fh(\000)20 b Fk(\031)s(;)15 b Fh(\000)p Fk(i\016)s(T)34 b Fh(\000)20 b Fk(\027)6 b Fn(])p Fk(;)36 b Fn(\000)2040 4442 y Fj(2)2104 4428 y Fn(=)25 b([)p Fh(\000)p Fk(i\016)s(T)34 b Fn(+)20 b Fk(\027)q(;)15 b Fh(\000)p Fk(i\016)s(T)34 b Fn(+)20 b Fk(\031)s Fn(])0 4604 y(and)30 b(0)c Fk(<)g(\027)31 b(<)25 b(\017)554 4618 y Fj(0)624 4604 y Fn(is)31 b(su\016cien)m(tly)g (small.)42 b(The)30 b(con)m(tour)i Fk(!)h Fn(is)e(a)g(curv)m(e)g (connecting)h Fh(\000)p Fk(i\016)s(T)i Fh(\000)20 b Fk(\027)36 b Fn(and)30 b Fh(\000)p Fk(i\016)s(T)k Fn(+)20 b Fk(\027)6 b Fn(,)0 4712 y(symmetric)36 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Fn(can)d(b)s(e)f(handled)0 5143 y(follo)m(wing)34 b(Lemma)f(7)g(in)g(Chapter)f(IX,)h([18)q(].)48 b(In)32 b(fact)i(w)m(e)f(m)m(ust)g(estimate)i(only)d(the)h(in)m(tegral)i(o)m(v) m(er)f(a)f(part)g(of)p eop end %%Page: 10 10 TeXDict begin 10 9 bop 0 66 a Fj(10)1679 b(V.)23 b(PETK)n(O)n(V)0 266 y Fn(the)32 b(circle)i Fh(j)p Fk(\022)s Fh(j)28 b Fn(=)g Fk(\017)658 280 y Fj(0)697 266 y Fk(:)k Fn(Since)h(\()p Fk(I)28 b Fh(\000)22 b Fk(K)7 b Fn(\()p Fk(e)1351 233 y Ff(\000)p Fi(i\022)1469 266 y Fn(\)\))1539 233 y Ff(\000)p Fj(1)1666 266 y Fn(is)32 b(b)s(ounded)e(for)i Fh(j)p Fk(\022)s Fh(j)c(\024)g Fk(\017)2533 280 y Fj(0)2573 266 y Fk(;)k Fn(the)g(leading)h(term)f(of)h(the)f(singular)0 376 y(part)e(of)h(\()p Fk(I)c Fh(\000)20 b Fk(K)7 b Fn(\()p Fk(e)655 343 y Ff(\000)p Fi(i\022)774 376 y Fn(\)\))844 343 y Ff(\000)p Fj(1)969 376 y Fn(is)30 b(giv)m(en)i(b)m(y)1106 614 y Fk(A)1174 628 y Fj(0)1233 614 y Fn(+)1379 500 y Fi(l)1324 528 y Fl(X)1327 725 y Fi(k)r Fj(=)p Fi(j)1471 614 y Fn(ln)1547 576 y 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Fk(t)30 b Fn(w)m(e)h(get)963 2635 y Fh(k)p Fk('U)10 b Fn(\()p Fk(t;)15 b(s)p Fn(\))p Fk(f)10 b Fh(k)1455 2648 y Fj(_)1425 2665 y Ff(H)1485 2674 y Fs(1)1521 2665 y Fj(\()p Fe(R)1596 2646 y Fr(n)1639 2665 y Fj(\))1696 2635 y Fh(\024)25 b Fk(C)7 b Fn(\()p Fk(n;)35 b(';)15 b(R)q Fn(\))p Fk(p)p Fn(\()p Fk(t)21 b Fh(\000)f Fk(s)p Fn(\))p Fh(k)p Fk(f)10 b Fh(k)2695 2648 y Fj(_)2667 2665 y Ff(H)2727 2674 y Fs(1)2762 2665 y Fj(\()p Fe(R)2837 2646 y Fr(n)2880 2665 y Fj(\))2911 2635 y Fk(;)777 b Fn(\(3.6\))0 2791 y(where)30 b(for)g Fk(t)25 b Fh(\025)g Fk(t)589 2805 y Fj(0)654 2791 y Fk(>)g Fn(1)30 b(w)m(e)h(ha)m(v)m(e)1275 3010 y Fk(p)p Fn(\()p Fk(t)p Fn(\))26 b(=)1546 2855 y Fl(\()1619 2949 y Fk(e)1661 2916 y Ff(\000)p Fi(\016)r(t)1780 2949 y Fk(;)35 b(n)25 b Fh(\025)g Fn(3)p Fk(;)36 b Fn(o)s(dd)o Fk(;)1619 3078 y(t)1652 3045 y Ff(\000)p Fj(1)1761 3078 y Fn(ln)1837 3041 y Ff(\000)p Fj(2)1947 3078 y Fk(t;)f(n)25 b Fh(\025)g Fn(2)p Fk(;)36 b Fn(ev)m(en)q Fk(:)3713 3010 y Fn(\(3.7\))1020 3285 y(4.)46 b Fk(L)1198 3252 y Fj(2)1237 3285 y Fm(-integrability)34 b(of)g(the)f(local)g(ener)n(gy)150 3447 y Fn(W)-8 b(e)32 b(start)e(with)h(the)f(follo)m(wing)0 3609 y Fw(Prop)s(osition)37 b(1.)42 b Fd(Assume)33 b(the)g(c)-5 b(ondition)35 b Fn(\()p Fh(R)p Fn(\))e Fd(ful\014l)5 b(le)-5 b(d)34 b(and)g Fn(0)26 b Fh(\024)g Fk(\015)31 b Fh(\024)26 b Fn(min)o Fh(f)p Fn(1)p Fk(;)15 b Fn(\()p Fk(n)22 b Fh(\000)e Fn(1\))p Fk(=)p Fn(2)p Fh(g)p Fk(:)36 b Fd(L)-5 b(et)33 b Fn(\()p Fk(f)3594 3623 y Fj(0)3633 3609 y Fk(;)j(f)3739 3623 y Fj(1)3778 3609 y Fn(\))26 b Fh(2)36 3700 y Fn(_)0 3723 y Fh(H)77 3737 y Fi(\015)121 3723 y Fn(\()p Fg(R)222 3690 y Fi(n)269 3723 y Fn(\))43 b Fd(and)h(let)f Fk(F)57 b Fh(2)44 b Fk(L)954 3690 y Fj(2)954 3746 y Fi(t)993 3723 y Fn(\()p Fg(R)p Fn(;)1196 3700 y(_)1162 3723 y Fk(H)1245 3679 y Fi(\015)1238 3735 y(x)1290 3723 y Fn(\()p Fg(R)1391 3690 y Fi(n)1438 3723 y Fn(\)\))f Fd(b)-5 b(e)43 b(supp)-5 b(orte)g(d)46 b(in)d Fh(f)p Fn(\()p Fk(t;)15 b(x)p Fn(\))45 b(:)72 b Fh(j)p Fk(x)p Fh(j)44 b(\024)g Fk(R)q Fh(g)p Fd(.)73 b(Then)43 b(for)g(every)g(\014xe)-5 b(d)0 3831 y Fk(')26 b Fh(2)f Fk(C)243 3798 y Ff(1)236 3856 y Fj(0)317 3831 y Fn(\()p Fg(R)418 3798 y Fi(n)465 3831 y Fn(\))33 b Fd(the)g(solution)g Fk(u)p Fn(\()p Fk(t;)15 b(x)p Fn(\))34 b Fd(of)f Fn(\(1)p Fk(:)p Fn(1\))h Fd(with)g Fk(\034)h Fn(=)25 b(0)33 b Fd(satis\014es)g(the)g(estimate) 1576 3905 y Fl(Z)1666 3931 y Ff(1)1626 4111 y(\0001)1771 4028 y Fh(k)p Fn(\()p Fk('u)p Fn(\()p Fk(t;)15 b(x)p Fn(\))p Fk(;)g('@)2304 4042 y Fi(t)2337 4028 y Fk(u)p Fn(\()p Fk(t;)g(x)p Fn(\)\))p Fh(k)2664 3991 y Fj(2)2694 4050 y(_)2664 4067 y Ff(H)2724 4075 y Fr(\015)2765 4067 y Fj(\()p Fe(R)2840 4048 y Fr(n)2883 4067 y Fj(\))2915 4028 y Fk(dt)905 4274 y Fh(\024)25 b Fk(C)1073 4173 y Fl(\020)1127 4274 y Fh(k)p Fk(f)1217 4288 y Fj(0)1256 4274 y Fh(k)1327 4287 y Fj(_)1301 4304 y Fi(H)1364 4285 y Fr(\015)1404 4304 y Fj(\()p Fe(R)1479 4285 y Fr(n)1522 4304 y Fj(\))1574 4274 y Fn(+)20 b Fh(k)p Fk(f)1755 4288 y Fj(1)1794 4274 y Fh(k)1866 4287 y Fj(_)1839 4304 y Fi(H)1902 4285 y Fr(\015)s Fp(\000)p Fs(1)2021 4304 y Fj(\()p Fe(R)2096 4285 y Fr(n)2139 4304 y Fj(\))2191 4274 y Fn(+)g Fh(k)p Fk(F)13 b Fh(k)2443 4304 y Fi(L)2491 4281 y Fs(2)2491 4324 y Fr(t)2526 4304 y Fj(\()p Fe(R)p Fj(;)2647 4287 y(_)2621 4304 y Fi(H)2684 4271 y Fr(\015)2679 4315 y(x)2724 4304 y Fj(\()p Fe(R)2799 4285 y Fr(n)2842 4304 y Fj(\)\))2901 4173 y Fl(\021)2956 4196 y Fj(2)3713 4274 y Fn(\(4.1\))0 4448 y Fd(with)33 b(a)g(c)-5 b(onstant)35 b Fk(C)d Fn(=)25 b Fk(C)7 b Fn(\()p Fk(n;)15 b(\015)5 b(;)15 b(';)g(R)q Fn(\))26 b Fk(>)f Fn(0)33 b Fd(dep)-5 b(ending)34 b(only)f(on)g Fk(n;)j(\015)5 b(;)35 b(')e Fd(and)h Fk(R)q Fd(.)150 4617 y Fw(Pro)s(of.)41 b Fn(First)31 b(notice)h(that)f(for)f(the)g(free)h(w)m(a)m(v)m(e)h(equation)f(and)f Fk(f)35 b Fn(=)25 b(\()p Fk(f)2656 4631 y Fj(0)2695 4617 y Fk(;)15 b(f)2780 4631 y Fj(1)2819 4617 y Fn(\))26 b Fh(2)3002 4594 y Fn(_)2966 4617 y Fh(H)3043 4631 y Fi(\015)3087 4617 y Fn(\()p 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1605 y Ff(H)2472 1613 y Fr(\015)2514 1605 y Ff(!)2612 1589 y Fj(_)2585 1605 y Ff(H)2645 1613 y Fr(\015)2721 1575 y Fn(whic)m(h)33 b(can)h(b)s(e)e(deduced)h(from)g(a)0 1700 y(deca)m(y)f(of)e Fh(k)p Fk('U)10 b Fn(\()p Fk(t;)15 b(s)p Fn(\))p Fk(\037)p Fh(k)850 1713 y Fj(_)820 1730 y Ff(H)880 1739 y Fs(0)917 1730 y Ff(!)1015 1713 y Fj(_)988 1730 y Ff(H)1048 1739 y Fs(0)1116 1700 y Fn(and)30 b(in)m(terp)s (olation.)150 1916 y(W)-8 b(e)32 b(need)e(also)h(a)g(result)f (concerning)h(the)f(non-homogeneous)h(free)g(w)m(a)m(v)m(e)h(equation.) 0 2092 y Fw(Prop)s(osition)k(2.)41 b Fd(Assume)32 b Fn(1)25 b Fh(\024)33 b Fn(~)-53 b Fk(p)25 b(<)g Fn(2)p Fk(:)33 b Fd(L)-5 b(et)33 b Fk(f)h Fn(=)25 b(\()p Fk(f)1874 2106 y Fj(0)1913 2092 y Fk(;)15 b(f)1998 2106 y Fj(1)2038 2092 y Fn(\))25 b Fh(2)2220 2069 y Fn(_)2184 2092 y Fh(H)2261 2106 y Fi(\015)2305 2092 y Fn(\()p Fg(R)2406 2059 y Fi(n)2453 2092 y Fn(\))p Fk(;)35 b(F)j Fh(2)25 b Fk(L)2798 2048 y Fj(~)-41 b Fi(p)2792 2116 y(t)2832 2092 y Fn(\()p Fg(R)p 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Fj(0)1727 1449 y Fh(2)25 b Fg(R)1879 1416 y Fj(+)1970 1449 y Fd(and)34 b Fk(g)29 b Fh(2)c Fk(L)2367 1416 y Fi(p)2406 1449 y Fn(\()p Fg(R)2507 1416 y Fj(+)2566 1449 y Fn(;)36 b Fk(X)7 b Fn(\))p Fd(.)42 b(Then)33 b(we)g(have)912 1538 y Fl(\015)912 1593 y(\015)912 1647 y(\015)962 1519 y(Z)1013 1725 y Fj(0)p Ff(\024)p Fi(s)25 b Fn(0)33 b Fd(dep)-5 b(ends)34 b(only)g(on)f Fk(p;)15 b(q)s(:)150 2010 y Fn(In)39 b([8])h(the)f(ab)s (o)m(v)m(e)i(result)e(is)h(form)m(ulated)g(with)f Fg(R)g Fn(instead)g(of)h Fg(R)2494 1977 y Fj(+)2592 2010 y Fn(and)e Fk(s;)k(t;)f(t)3019 2024 y Fj(0)3099 2010 y Fh(2)e Fg(R)p Fn(,)j(but,)f(as)f(it)g(w)m(as)0 2117 y(men)m(tioned)31 b(in)f([8],)h(the)f(same)h(pro)s(of)e(w)m(orks)h(for)g(other)h(in)m (terv)-5 b(als)31 b(and)f(in)f(particular)i(for)f Fg(R)3226 2084 y Fj(+)3284 2117 y Fk(:)h Fn(By)f(h)m(yp)s(othesis)8 2225 y(~)-53 b Fk(p)25 b(<)g Fn(2,)31 b(so)g(taking)g(in)m(to)g(accoun) m(t)h(\(4.5\),)h(w)m(e)d(deduce)g(from)g(Lemma)h(2)g(the)f(estimate)696 2377 y 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(of)f(of)g(Theorem)g(1)h(is)f(complete.)150 2404 y(T)-8 b(o)37 b(obtain)g(homogeneous)g(Stric)m(hartz)g(estimates,)j(w)m(e)d (need)f(to)h(apply)f(Prop)s(osition)h(1)g(com)m(bined)f(with)0 2512 y(the)31 b(estimate)h(\(4.2\).)42 b(Moreo)m(v)m(er,)33 b Fk(\015)i Fn(is)c(related)g(only)g(to)g Fk(n;)15 b(p;)g(q)s(:)0 2675 y Fw(Theorem)28 b(3.)36 b Fd(L)-5 b(et)27 b(the)g(c)-5 b(ondition)29 b Fn(\()p Fh(R)p Fn(\))e Fd(b)-5 b(e)27 b(ful\014l)5 b(le)-5 b(d.)40 b(Supp)-5 b(ose)29 b(that)f Fn(2)d Fh(\024)g Fk(p;)15 b(q)29 b Fh(\024)c Fn(+)p Fh(1)p Fk(;)d Fn(0)k Fh(\024)f Fk(\015)30 b Fh(\024)25 b Fn(\()p Fk(n)7 b Fh(\000)g Fn(1\))p Fk(=)p Fn(2)p Fk(;)25 b(p)g(>)0 2783 y Fn(2)38 b Fd(ar)-5 b(e)39 b(such)f(that)h(the)f(solution)h Fk(u)1203 2797 y Fj(0)1243 2783 y Fn(\()p Fk(t;)15 b(x)p Fn(\))38 b Fd(of)g(the)h(pr)-5 b(oblem)39 b Fn(\(1)p Fk(:)p Fn(1\))h Fd(with)e Fk(V)55 b Fn(=)35 b(0)p Fk(;)40 b(F)47 b Fn(=)35 b(0)p Fk(;)40 b(\034)k Fn(=)35 b(0)j Fd(satis\014es)h(the)0 2891 y(estimate)944 3006 y Fh(k)p Fk(u)1041 3020 y Fj(0)1081 3006 y Fh(k)1126 3028 y Fi(L)1174 2996 y Fr(p)1174 3048 y(t)1211 3028 y Fj(\()p Fe(R)p Fj(;)15 b Fi(L)1369 2996 y Fr(q)1369 3039 y(x)1408 3028 y Fj(\()p Fe(R)1483 3009 y Fr(n)1526 3028 y Fj(\)\))1610 3006 y Fh(\024)25 b Fk(C)1778 2905 y Fl(\020)1832 3006 y Fh(k)p Fk(f)1922 3020 y Fj(0)1962 3006 y Fh(k)2033 3019 y Fj(_)2007 3036 y Fi(H)2070 3017 y Fr(\015)2110 3036 y Fj(\()p Fe(R)2185 3017 y Fr(n)2228 3036 y Fj(\))2280 3006 y Fn(+)20 b Fh(k)p Fk(f)2461 3020 y Fj(1)2500 3006 y Fh(k)2571 3019 y Fj(_)2545 3036 y Fi(H)2608 3017 y Fr(\015)s Fp(\000)p Fs(1)2727 3036 y Fj(\()p Fe(R)2802 3017 y Fr(n)2845 3036 y Fj(\))2876 2905 y Fl(\021)2931 3006 y Fk(:)0 3162 y Fd(Then)33 b(the)g(solution)h Fk(u)p Fn(\()p Fk(t;)15 b(x)p Fn(\))33 b Fd(of)g(the)g(pr)-5 b(oblem)35 b Fn(\(1)p Fk(:)p Fn(1\))f Fd(with)f Fk(F)39 b Fn(=)25 b(0)p Fk(;)36 b(\034)f Fn(=)25 b(0)33 b Fd(satis\014es)g(the) g(fol)5 b(lowing)34 b(estimate)947 3337 y Fh(k)p Fk(u)p Fh(k)1089 3360 y Fi(L)1137 3327 y Fr(p)1137 3380 y(t)1174 3360 y Fj(\()p Fe(R)p Fj(;)16 b Fi(L)1333 3327 y Fr(q)1333 3371 y(x)1372 3360 y Fj(\()p Fe(R)1447 3341 y Fr(n)1490 3360 y Fj(\)\))1574 3337 y Fh(\024)25 b Fk(C)1735 3351 y Fj(1)1775 3236 y Fl(\020)1829 3337 y Fh(k)p Fk(f)1919 3351 y Fj(0)1958 3337 y Fh(k)2029 3350 y Fj(_)2003 3367 y Fi(H)2066 3348 y Fr(\015)2107 3367 y Fj(\()p Fe(R)2182 3348 y Fr(n)2225 3367 y Fj(\))2276 3337 y Fn(+)20 b Fh(k)p Fk(f)2457 3351 y Fj(1)2497 3337 y Fh(k)2568 3350 y Fj(_)2542 3367 y Fi(H)2605 3348 y Fr(\015)s Fp(\000)p Fs(1)2724 3367 y Fj(\()p Fe(R)2799 3348 y Fr(n)2841 3367 y Fj(\))2873 3236 y Fl(\021)2927 3337 y Fk(:)1732 3537 y Fu(References)38 3682 y Fv([1])40 b(A.)28 b(Bac)n(helot)h(and)e(V.)h(P)n(etk)n(o)n(v,)g Fc(Existenc)l(e)j(des)f(op)n(\023)-37 b(er)l(ateurs)33 b(d'ondes)d(p)l(our)h(les)e(syst)n(\022)-37 b(emes)32 b(hyp)l(erb)l(oliques)f(ave)l(c)g(un)f(p)l(otentiel)158 3773 y(p)n(\023)-37 b(erio)l(dique)28 b(en)g(temps)p Fv(,)f(Ann.)e(Inst.)h(H.)f(P)n(oincare)n(\023)-36 b(e)27 b(\(Ph)n(ys.)e(Th)n(\023)-36 b(eorique\),)27 b Fb(47)f Fv(\(1987\),)h(383-428.)38 3865 y([2])40 b(J.-F.)26 b(Bon)n(y)f(and)h (V.)f(P)n(etk)n(o)n(v,)g Fc(R)l(esonanc)l(es)30 b(for)d(non-tr)l (apping)j(time-p)l(erio)l(dic)f(p)l(erturb)l(ations)p Fv(,)g(J.)d(Ph)n(ys.)f(A:)h(Math.)g(Gen.)g Fb(37)158 3956 y Fv(\(2004\),)h(9439-9449.)38 4047 y([3])40 b(N.)29 b(Burq,)g Fc(Glob)l(al)i(Strichartz)h(estimates)h(for)d(non-tr)l (apping)j(ge)l(ometries:)42 b(A)n(b)l(out)32 b(an)e(article)i(by)e(H.)g (Smith)h(and)g(C.)e(So)l(gge)p Fv(,)158 4139 y(Comm)n(un.)d(PDE,)g Fb(28)g Fv(\(2003\),)h(1675-1683.)38 4230 y([4])40 b(M.)26 b(Christ)g(and)g(A.)f(Kiselev,)i Fc(Maximal)g(functions)h(asso)l(ciate) l(d)i(to)e(\014ltr)l(ations)p Fv(,)f(J.)g(F)-6 b(unct.)25 b(Anal.)g Fb(179)i Fv(\(2001\),)g(409-425.)38 4321 y([5])40 b(J.)23 b(Co)r(op)r(er)h(and)e(W.)h(Strauss,)h Fc(Sc)l(attering)j(of)d (waves)i(by)f(p)l(erio)l(dic)l(al)t(ly)g(moving)g(b)l(o)l(dies)p Fv(,)g(J.)e(F)-6 b(unct.)22 b(Anal.)h Fb(47)g Fv(\(1982\),)i(180-229.) 38 4413 y([6])40 b(A.)34 b(Galtaba)n(y)n(ar,)j(A.)d(Y)-6 b(ensen)33 b(and)g(K.)h(Y)-6 b(a)t(jima,)37 b Fc(L)l(o)l(c)l(al)f (time-de)l(c)l(ay)h(of)e(solutions)h(to)g(Schr\177)-39 b(odinger)37 b(e)l(quations)f(with)g(time-)158 4504 y(p)l(erio)l(dic)28 b(p)l(otentials)p Fv(,)g(Journal)e(of)h(Statistical)g(Ph)n(ysics,)f Fb(114)g Fv(\(2004\),)h(231-281.)38 4595 y([7])40 b(J.)31 b(Ginibre)f(and)g(G.)h(V)-6 b(elo,)31 b Fc(Gener)l(alize)l(d)j (Strichartz)g(ine)l(qualities)e(for)f(the)i(wave)f(e)l(quation)p Fv(,)h(J.)e(F)-6 b(unct.)29 b(Anal.)i Fb(133)f Fv(\(1995\),)158 4687 y(50-68.)38 4778 y([8])40 b(A.)31 b(Hassell,)j(T.)e(T)-6 b(ao)32 b(and)f(J.)h(W)-6 b(unsc)n(h,)31 b Fc(Sharp)j(Strichartz)h (estimates)f(on)f(non-tr)l(apping)i(asymptotic)l(al)t(ly)e(c)l(onic)g (manifols)p Fv(,)158 4869 y(Preprin)n(t)25 b(2004.)38 4961 y([9])40 b(P)-6 b(.)26 b(D.)f(Lax)h(and)f(R.)h(S.)f(Phillips,)i Fc(Sc)l(attering)j(The)l(ory)p Fv(,)d(2nd)e(Edition,)i(Academic)f (Press,)h(New)f(Y)-6 b(ork,)25 b(1989.)0 5052 y([10])40 b(H.)24 b(Lindblad)f(and)h(C.)h(D.)f(Sogge,)h Fc(On)h(existenc)l(e)i (and)e(sc)l(attering)j(with)d(minimal)e(r)l(e)l(gularity)k(for)e (semiline)l(ar)g(wave)g(e)l(quation)p Fv(,)158 5143 y(J.)g(F)-6 b(unct.)25 b(Anal.)h Fb(130)g Fv(\(1995\),)i(357-426.)p eop end %%Page: 16 16 TeXDict begin 16 15 bop 0 66 a Fj(16)1679 b(V.)23 b(PETK)n(O)n(V)0 266 y Fv([11])40 b(M.)26 b(Keel)g(and)f(T.)i(T)-6 b(ao,)27 b Fc(Endp)l(oint)h(Strichartz)i(Estimates)p Fv(,)d(Amer.)f(J.)g(Math.)h Fb(120)f Fv(\(1998\),)h(995-980.)0 357 y([12])40 b(V.)25 b(P)n(etk)n(o)n(v,)h Fc(Sc)l(attering)j(The)l(ory)g(for)f(Hyp)l(erb)l (olic)g(Op)l(er)l(ators)p Fv(,)g(North)e(Holland,)g(Amsterdam,)g(1989.) 0 448 y([13])40 b(M.)29 b(Reissig)i(and)d(K.)h(Y)-6 b(agdjian,)31 b Fc(One)g(applic)l(ation)g(of)f(Flo)l(quet's)h(the)l(ory)i(to)e Ft(L)2514 456 y Fr(p)2570 448 y Fq(\000)19 b Ft(L)2701 456 y Fr(q)2766 448 y Fc(estimates)32 b(for)e(hyp)l(erb)l(olic)i(e)l (quations)158 540 y(with)27 b(very)i(fast)f(oscil)t(lations)p Fv(,)e(Math.)h(Meth.)f(Appl.)f(Sci.)h Fb(22)g Fv(\(1999\),)h(937-951.)0 631 y([14])40 b(M.)j(Reissig)h(and)e(K.)h(Y)-6 b(agdjian,)47 b Fc(A)n(b)l(out)e(the)f(in\015uenc)l(e)g(of)e(oscil)t(lations)i(on)f (Strichartz-typ)l(e)k(estimates)p Fv(,)i(Rend.)42 b(Sem.)158 722 y(Mat.Univ.)26 b(P)n(olitec.)h(T)-6 b(orino,)27 b Fb(58)f Fv(\(2000\),)h(375-388.)0 814 y([15])40 b(M.)27 b(Reissig)h(and)e(K.)g(Y)-6 b(agdjian,)28 b Ft(L)1189 822 y Fr(p)1243 814 y Fq(\000)17 b Ft(L)1372 822 y Fr(q)1435 814 y Fc(estimates)30 b(for)e(the)i(solutions)f(of)f(strictly)h(hyp)l (erb)l(olic)h(e)l(quations)g(of)e(se)l(c)l(ond)i(or)l(der)158 905 y(with)d(incr)l(e)l(asing)i(in)e(time)h(c)l(o)l(e\016cients)p Fv(,)f(Math.)g(Nac)n(hr.)e Fb(214)h Fv(\(2000\),)i(71-104.)0 996 y([16])40 b(H.)21 b(F.)h(Smith)f(and)h(C.)g(Sogge,)i Fc(Glob)l(al)g(Strichartz)i(estimates)f(for)f(non-tr)l(apping)i(p)l (erturb)l(ations)h(of)d(the)g(L)l(aplacian)p Fv(,)g(Comm)n(un.)158 1088 y(PDE,)i Fb(25)g Fv(\(2000\),)h(2171-2183.)0 1179 y([17])40 b(T.)30 b(T)-6 b(ao,)31 b Fc(Spheric)l(al)t(ly)h(aver)l(age)g (endp)l(oint)g(Strichartz)h(estimates)g(for)e(the)g(two-dimensional)h (Schr\177)-39 b(odinger)33 b(e)l(quation)p Fv(,)f(Com-)158 1270 y(m)n(un.)25 b(PDE,)h Fb(25)g Fv(\(2000\),)h(1471-1485.)0 1362 y([18])40 b(B.)23 b(R.)f(V)-6 b(ain)n(b)r(erg,)23 b Fc(Asymptotic)j(metho)l(ds)g(in)e(e)l(quations)i(of)e(mathematic)l (al)i(physics)p Fv(,)e(Gordon)f(and)f(Breac)n(h,)i(New)f(Y)-6 b(ork,)22 b(1989.)0 1453 y([19])40 b(B.)26 b(V)-6 b(ain)n(b)r(erg,)25 b Fc(Sc)l(attering)k(by)e(p)l(erio)l(dic)l(al)t(ly)h(moving)f (obstacles)p Fv(,)g(\(Russian\),)e(Mat.)h(Sb)r(ornik,)f Fb(182)h Fv(\(1991\),)h(911-927;)g(English)158 1544 y(translation:)36 b(Math)25 b(USSR)g(Sb)r(ornik,)g Fb(73)h Fv(\(1992\),)h(289-304.)0 1636 y([20])40 b(B.)c(V)-6 b(ain)n(b)r(erg,)38 b Fc(On)e(the)i(lo)l(c)l (al)e(ener)l(gy)i(of)e(solutions)i(of)e(exterior)i(mixe)l(d)f(pr)l (oblems)g(that)h(ar)l(e)f(p)l(erio)l(dic)g(with)g(r)l(esp)l(e)l(ct)i (to)e Ft(t)p Fv(,)158 1727 y(\(Russian\),)25 b(T)-6 b(rudy)23 b(Mosco)n(w)k(Math.)e(So)r(c.)g Fb(54)g Fv(\(1992\):)35 b(English)25 b(translation:)35 b(T)-6 b(rans.)26 b(Mosco)n(w)g(Math.)f (So)r(c.)g(1993,)i(191-216.)0 1818 y([21])40 b(G.)26 b(V)-6 b(o)r(dev,)25 b Fc(On)i(the)i(uniform)e(de)l(c)l(ay)i(of)e(lo)l (c)l(al)g(ener)l(gy)p Fv(,)h(Serdik)l(a)d(Math.)h(J.)h Fb(25)f Fv(\(1999\),)h(191-206.)150 1989 y Fu(D)212 1983 y(\023)212 1989 y(ep)-5 b(ar)g(tement)28 b(de)i(Ma)-5 b(th)1003 1983 y(\023)1003 1989 y(ema)g(tiques)29 b(Appliqu)1706 1983 y(\023)1706 1989 y(ees,)g(Universit)2241 1983 y(\023)2241 1989 y(e)f(Bordea)n(ux)i(I,)g(351,)h(Cours)f(de)f(la)i(Lib)3589 1983 y(\023)3589 1989 y(era)-5 b(tion,)0 2081 y(33405)30 b(T)-7 b(alence,)28 b(France)150 2172 y Fc(E-mail)f(addr)l(ess)6 b Fv(:)37 b Fa(petkov@math.u-bordeaux1.fr)p eop end %%Trailer userdict 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