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\hfuzz 1pt
 
\TITLE NORMAL FORMS FOR PARABOLIC PARTIAL DIFFERENTIAL EQUATIONS
 
\AUTHOR J.-P. Eckmann \FROM
D\'epartement de Physique Th\'eorique
Universit\'e de Gen\`eve
CH-1211 Gen\`eve 4
Switzerland
\AUTHOR H. Epstein \FROM
IHES
35, Route de Chartres
F-91440 Bures-sur-Yvette
France
\AUTHOR C. E. Wayne \FROM
Department of Mathematics
Pennsylvania State University
University Park, PA 16802
USA
\ENDTITLE
\ABSTRACT
We begin a study of normal form theorems for parabolic
partial differential equations.  We show that despite the
presence of resonances one can construct
a partial normal form for perturbations
of the Ginzburg-Landau equation.  The normal form transformation is
expressed in terms of singular integral operators, whose behavior
can be controlled in the appropriate function spaces.
\ENDABSTRACT
 
 
\SECTION Introduction
 
In hydrodynamics, as in many other phenomena described by partial
differential
equations, simplified equations appear naturally as descriptions of
bifurcations. These
simplified equations usually describe ``amplitudes'' or other envelope
functions
on time and space scales which are related to the bifurcation parameter. An
example where this reduction can be proved with mathematical rigor is provided
by the bifurcation near the instability threshold of the Swift-Hohenberg
equation [CE1, CE2]:
This equation is
$$
\partial_t u \,=\,  \bigl (\alpha -(1+\partial_x^2)^2\bigr )u-u^3~,
\EQ(i1)
$$
where $u=u(x,t)$, $u:\real\times\real^+\to\real$.
If one rescales the solutions as
$$
u(x,t) \,=\,  \epsilon {\rm Re~} v(\epsilon x,\epsilon ^2t) e^{ix}~,
\EQ(i2)
$$
with $\epsilon =\sqrt{\alpha /3}$, then the function $v$ satisfies
the equation
$$
\partial_t v \,=\,  (4\partial_x^2+3 -4i\epsilon \partial_x^3-\epsilon
^2\partial_x^4)v -3v|v|^2-e^{2ix/\epsilon }v^3~,
\EQ(i3)
$$
when $\alpha >0$. One can then show that for times of order $\OO(1)$ in
\equ(i3), or, equivalently, for times of order $\OO(\epsilon ^{-2})$ in
\equ(i1), the evolution given by \equ(i3) is close to that of
$$
\partial_t w \,=\,  (4\partial_x^2+3 -\epsilon
^2\partial_x^4)w -3w|w|^2~,
\EQ(i4)
$$
when $w(x,0)=v(x,0)$.
Thus, \equ(i4) is some sort of {\em normal form} for this type of bifurcation.
We may thus ask in which sense the solutions of equations such as \equ(i4) are
dynamically equivalent. Is the term $\epsilon ^2\partial_x^4$ negligible? Do
higher order nonlinear terms matter in \equ(i1)?
 
The aim of the present paper is to discuss the normal forms of some parabolic
PDE's. We consider the equation
$$
\partial_t v \,=\,  \partial_x^2 v+ \mu v -v^3 -\epsilon R(v)~,
\EQ(i5)
$$
where $R$ is a polynomial whose terms are all of degree 4 or higher.
We ask whether there is a coordinate transformation (in function space),
$H: v\mapsto H(v)$, depending on $\epsilon $ which {\em eliminates} $R$.
Furthermore, we require $H$ to exist {\em uniformly} in $\mu$ when $\mu$ is
greater than or equal to zero and not
too large. We shall show that the lowest order monomial of degree $n\ge4$
in
$R$ can indeed be
eliminated by a coordinate transformation, leading to a new remainder term
$Q$,
whose lowest order term is of degree $n'>n$. We would like to iterate the
procedure, eliminating the lowest order term of $Q$, but this raises new
difficulties,
since $Q$ contains terms which are not pure powers of $v$ (as in $R$), but
convoluted kernels. We have only preliminary results in the direction of
eliminating further terms.
 
Before stating our result in detail, it is perhaps useful to
relate it to the existing literature on normal forms.
Normal forms for diffeomorphisms
and flows have been studied extensively, see [PD] for a review of the
literature. Our result can viewed as a first step towards an analogue of the
Sternberg Linearization Theorem for PDE's of the form \equ(i5).
In {\em finite} dimension,
the Sternberg Linearization Theorem
asserts that $\CC^\infty$ vector fields $X:\real^s\to\real^s$ with
$X(0)=0$ can be linearized by a $\CC^\infty $ local diffeomorphism if $DX(0)$
satisfies the ``eigenvalue condition'' (nonresonance condition) [N,p.38] :
The eigenvalues $\lambda _i$, $i=1,\dots,s$ of $DX(0)$ satisfy
$$
\lambda _i \,\ne\, m_1\lambda _1 +m_2\lambda _2+\cdots+ m_s\lambda _s~,
\EQ(i6)
$$
for all choices of positive integers $m_i$ satisfying
$2\le\sum m_i$. In the infinite dimensional case,
there is continuous spectrum, and more importantly,
the
analogue of the condition \equ(i6) is violated, since the continuous spectrum
of
the Laplacian is $\lambda _k=-k^2$, $k\in\real$ and it
is easy to find values of $k_1$, $k_2$, $k_3$ for which
$\lambda _{k_1}=\lambda _{k_2}+\lambda _{k_3}$.
 
In the case of ordinary differential equations, violation of the
non-resonance condition \equ(i6) produces zero denominators in terms in the
sum which defines the normal form transformation.  (And hence, renders
the transformation meaningless.)  However, due to the presence of
continuous spectrum, the normal form transformation is defined
in terms of integrals rather than sums.  Under certain conditions, we
can integrate right through the resonances.  This can be done
because the singularity is relatively weak
and
the
phase space volume over which we integrate is sufficiently large if the term
we
want to eliminate is of order 4 or higher.
Note that there are weaker sorts of normal form theorems than the Sternberg
theorem.  For instance, the Grobman-Hartmann theorem requires only
that the linear vector field be hyperbolic, at the cost of
being able to conclude only that the function conjugating the vector field
to the normal form is only continuous.  However, even such a result
(naively) fails here since for $\mu \ge 0$, zero is in the spectrum
of the linearized version of \equ(i5).
 
Another fact which is often encountered in finite
dimensional normal form theorems
is that while it may be impossible to reduce a system to a linear
normal form, there may nonetheless be a useful nonlinear normal
form.  The most common example of this phenomenon is a hamiltonian
vector field near a hyperbolic fixed point.  Relationships among
the eigenvalues which are inherent in the hamiltonian nature
of the problem mean that \equ(i6) is inevitably violated.
Nonetheless one can still (often) transform the system to
the Birkhoff normal form.  We encounter a similar situation here.
Our normal form is not linear, but contains a cubic term.  Technically, the
reason for this is that the integrals referred to in the previous paragraph
diverge if one tries to eliminate terms of order three.
On a more fundamental level, these terms cannot be eliminated because
it is known that if $\mu = 0$ in \equ(i5), the asymptotic behavior of
the solutions of the linear and non-linear equations is different,
and hence they cannot be conjugate.
 
We conclude this introduction by reviewing some related work.
An interesting paper dealing
with continuous spectrum is [S], which deals with Klein-Gordon equations.
In that case, one finds essentially $\lambda _k=\sqrt{k^2+m^2}$, with $m>0$
and
such
eigenvalues always satisfy the condition \equ(i6).
Two further related papers are [Se] and [D]. In [D], the discrete Laplacian is
considered in \equ(i5), and then $\mu$ is required to be sufficiently large as
a
function of the discretization parameter. In [Se],
the Laplacian is not discrete but it is multiplied by
a non-real coefficient which, again, prevents the appearance of
resonances.  This is similar to earlier
work [Ni] on the nonlinear Schr\"odinger equation.
 
After a first version of this paper had been completed, the related paper
of McKean and Shatah [MS] came to our attention.  They also study normal
forms of partial differential equations.  In the result closest to ours,
they show that if in \equ(i5) one sets $\mu = 0$, and if the cubic term
is not present, then a nonlinearity $\epsilon R(u) = \epsilon u^p$
can be completely eliminated if $p \ge 4$, {\it i.e.} under
these hypotheses one can conjugate the equation to a linear normal
form.  In contrast to our use of singular integrals to construct
the normal form, they rely on an infinite dimensional analogue
of the scattering method proof of the Sternberg theorem.  In our
example, this method seems unfortunately inapplicable because we
do not understand sufficiently well the asymptotic behavior
of the solutions of the normal form equation.  Furthermore, in the case
$\mu > 0$, one would need a stable manifold theorem for partial differential
equations in order to apply this method, and that result is also
unavailable.
 
\SECTION The Conjugation
 
We next describe the derivation of an equation for
conjugation, in momentum space. We define the operator $P$ by
$$
(Pv)(k) \,=\,  k v(k)~.
$$
We want to transform the problem (as expressed in momentum space)
$$
\partial_t v \,=\,   - (P^2-\mu)v - v^{*3} +U_n(v)~,
$$
by a transformation $\TT$ given by
$$
\TT v \,=\,  v+H_n(v)~.
$$
Here, $U_n$ is of degree $n\ge4$ and the effect of the transformation
should be to replace it with terms of degree $\ge5$. The notation $v^{*3}$
stands for threefold
convolution.
 
Our aim is to determine $H_n$. We now assume that $U_n$ is homogeneous of
degree $n\ge4$, and we
require that $H_n$ to be homogeneous of the same degree.
The transformation $H_n$ found in this way will be adequate for any polynomial
whose lowest order homogeneous term is equal to $U_n$.
Let $w=\TT v$. Then
$$
\eqalign{
\partial_t&w\,=\,\partial_t v +DH_n(v)\partial_t v\cr
\,&=\,-(P^2-\mu)v-v^{*3}+U_n(v)-DH_n(v) \cdot (P^2-\mu) v -DH_n(v)
v^{*3}+DH_n(v) U_n(v).\cr
}
\EQ(m1)
$$
Hence
$$
\eqalign{
\partial_tw+(P^2-\mu)w+w^{*3}\,&=\,
U_n(v)+(P^2-\mu)H_n(v)-DH_n(v)(P^2-\mu)v\cr
~&+~3 v^{*2}*H_n(v)-DH_n(v) \cdot v^{*3}\cr
~&+DH_n(v)U_n(v)\cr
~&+3v* \bigl (H_n(v)\bigr )^{*2}\cr
~&+H_n(v)^{*3}~.\cr
}
\EQ(m2)
$$
The terms on the r.h.s.~in \equ(m2) have been arranged by degrees and
the five lines correspond to degrees $n$, $n+2$, $2n-1$, $2n+1$, and $3n$,
respectively. We can eliminate the terms of degree $n$ in Eq.\equ(m2) by
setting
$$
U_n(v)+(P^2-\mu) H_n(v) - DH_n(v) (P^2-\mu) v  \,=\,  0~.
\EQ(m3)
$$
 
This is our main equation, and we now rewrite it in more explicit form.
Since $U_n$ is typically $v^{*n}$, we will restrict our attention to
operators of the form
$$
U_n(v)(k) \,=\,  \int\delta
(k-\sum_{i=1}^n{p_i})u(k,p_1,\dots,p_n)\prod_{i=1}^n
v(p_i) dp_i~,
\EQ(m4)
$$
and try to construct $H_n$ in the same form, i.e.,
$$
H_n(v)(k) \,=\,  \int\delta
(k-\sum_{i=1}^n{p_i})h(k,p_1,\dots,p_n)\prod_{i=1}^n
v(p_i) dp_i~.
\EQ(m6)
$$
Despite the functional notation, the kernels $u$, $h$, etc.,
should be thought of as tempered distributions (of some relatively
mild type) on the manifold defined by $k = \sum p_i$.
In terms of these kernels, Eq.\equ(m3) is equivalent to
$$
u(k,p_1,\dots,p_n) \,=\,   \bigl( \sum_{i=1}^n p_i^2 - k^2-(n-1)\mu \bigr)
h(k,p_1,\dots,p_n)~.
\EQ(nm7)$$
Conversely, if $h$ is a solution of this division problem
which defines an operator $v \mapsto H_n(v)$ on the space of
functions $v$ of interest to us, then the main equation \equ(m3)
will hold. It is always possible to define, for $\veps \not= 0$,
$$
h_\veps (k,p_1,\dots,p_n) =
{1 \over \sum_{j=1}^n p_j^2 -k^2 -(n-1)\mu -i\veps}\, u(k,p_1,\dots,p_n)~,
\EQ(nm8)$$
as a tempered distribution on the manifold $k = \sum p_i$,
as well as the function
$$
H_{n,\veps}(k)\,=\,  \int{\delta
(k-\sum_{j=1}^n{p_j})\over \sum_{j=1}^n p_j^2- k^2 -(n-1)\mu -i\veps} \,
u(k,p_1,\dots,p_n)\prod_{j=1}^n v(p_j) dp_j~,
\EQ(nm9) $$
since this just amounts to multiplying $u$ by a $\CC^\infty$
function and integrating. If the limit of $h_\veps$ as $\veps \downarrow 0$
exists as a tempered
distribution on the manifold defined by $k = \sum p_i$, it is a
solution of the division problem \equ(nm7), and it only
remains to see whether the limit $H_n=\lim_{\veps \downarrow 0}H_{n,\veps}$
defines a continuous
operator on the right space.
 
There are now two problems we want to address:
\item{i)} On which space of functions can the map $v\mapsto H_n(v)$
be defined?
\item{ii)} Is the map $v\mapsto v+H_n(v)$ invertible?
 
\SECTION Statement of Results
 
We want to study $H_n$ for the case $u\equiv1$. We change variables of
integration in the Eq.\equ(nm9) to $p_i=k/n+q_i$. Then,
$$
\eqalign{
H_n(v) \,&=\, \lim_{\epsilon \downarrow 0} H_{n,\epsilon}(v)~,\cr
H_{n,\epsilon}(v)(k) \, &=\, \int_0^\infty  d\rho
{2\rho\over \rho^2- k^2(1-1/n)-(n-1)\mu  -i\epsilon }\,W_n(k,\rho,v)~,\cr}
$$
where
$$
W_n(k,\r,v)\,=\,\int \delta(\r^2 - \ssum_{i=1}^n q_i^2)
\delta(\ssum_{i=1}^n q_i)
\prod_{i=1}^n v(\textstyle{k \over n}+q_i) dq_i~.
\EQ(1)$$
It is also convenient to define, for $\rho>0$ and real $k$,
the corresponding linear functional
$$
V_n(k,\r,\phi)\,=\,\int \delta(\r^2 - \ssum_{i=1}^n
q_i^2) \delta(\ssum_{i=1}^n q_i)\,
\vhi(\textstyle{k \over n}+q_1,\dots,\textstyle{k \over n}+q_n)\,
\prod_{i=1}^n dq_i~.
\EQ(2)$$
Thus
$$
W_n(k,\r,v)\,=\, V_n(k,\r, v\otimes\dots\otimes v)~.
$$
With these notations, we have
$$
H_n(v)(k) \,=\,  F_n(k,\sqrt{(n-1)(k^2/n+\mu) + i0},v)~,
\EQ(121)$$
where
$$
F_n(k,\z,v)\,=\,
\int_0^\infty {2\r \over \r^2 -\z^2} W_n(k,\r,v) d\r~.
\EQ(3)$$
Throughout, we consider the norms
$$
\eqalign{
|v|_\beta  \,&=\,\sup_{p \in\real } (1+p^2)^\b |v(p)|
~,\cr
\|v\|_\beta  \,&=\,\max\{|v|_\beta,|v'|_\beta\}   ~.\cr
}
\EQ(norm)
$$
 
We call $\VV_\beta$ and $\VV'_\beta$ the corresponding Banach
spaces of complex functions.
Note that functions in $\VV'_\beta$ need not be continuously differentiable.
In particular, for $\mu=0$, the function $k \mapsto H_n(v)(k)$ defined by
\equ(121)  has, in general, a discontinuous derivative at $k=0$, even
when $v$ is $\CC^\infty $. In Appendix B we show, by an explicit 
calculation, that this happens even for the case of a Gaussian $v$.
For functions of $n$ real variables, we define analogously
$$
\eqalign{
|\vhi|_{n,\beta}  \,&=\,\sup_{p_1,\dots,p_n \in \bR}
|\vhi(p_1,\dots,p_n)|
\prod_{i=1}^n (1+p_i^2)^\b ~,\cr
\|\vhi\|_{n,\beta}  \,&=\,
\max_{0 \le \alpha_i \le 1}\
\sup_{p_1,\dots,p_n \in \bR} \
|D^\alpha \,\vhi(p_1,\dots,p_n)|
\prod_{i=1}^n (1+p_i^2)^\b ~.\cr}
\EQ(nnorm)$$
We denote $\VV_{n,\b}$ and $\VV'_{n,\b}$ the corresponding 
Banach spaces. 
\CLAIM Theorem(main) Suppose $v\in\VV'_\beta $
for some $\b > 1/2$.
Then for all $n\ge 4$,
the function $k\mapsto\bigl ( H_n v\bigr )(k) =
F_n(k,\sqrt{(n-1)(k^2/n+\mu) + i0},v)$ is continuous on $\bR$.
If $\mu>0$, this function has a H\"older continuous derivative on $\bR$.
If $\mu=0$, its restrictions to $[0,\infty)$ and to $(-\infty,0]$ have
H\"older continuous derivatives. In these two cases,
there is a constant $B_n$
such that $$
\|H_n(v)\|_\beta \,\le\, B_n  \|v\|_\beta ^n~.
\EQ(6)$$
 
\REMARK If $\mu>0$, the linear part of the r.h.s.~of Eq.\equ(i5) is unstable,
for $\mu=0$ it is marginally stable, and, for $\mu<0$ it is globally stable.
In this last case, the function $u=0$ is a stable fixed point in $L^2\cap L^
\infty $. Our theorem covers the unstable and marginally stable cases in a
uniform setting.

The operator $H_n$, as defined above does not map real functions
into real functions. But if $v$ is real, the operator defined by
$v \mapsto {\rm Re\,}H_n(v)$ is also a solution to our problem.
The proof of \clm(main) will be based on inductive bounds on the
functions $W_n$ and
$V_n$ in Section 4. These bounds will then be combined with estimates
on the integral \equ(3) in Section 5.
 
 
\CLAIM Corollary(mainc) Suppose $v\in\VV'_\beta $
for some $\b > 1/2$ and $\mu\ge0$.
Then for all $n\ge 4$, the map $v\mapsto v+H_n(v)$ is bounded and
invertible on a small ball in $\VV'_\b$, centered at the origin.
 
\PROOF It follows at once from \clm(main) that, on a
sufficiently small ball centered at the origin in $\VV'_\b$,
the map $v \mapsto H_n(v)$ is a contraction.
 
 
\LIKEREMARK {Remark}
In this paper, the function $\z \mapsto \log(\z)$
is always understood as
being defined in $\bC \setminus \bR_-$ and real for $\z >0$.
Similarly $\sqrt{\z} = \exp(\log(\z)/2)$ is positive on $\bR_+$.
 
\SECTION Phase Space Bounds
 
We assume throughout $\beta >1/2$. The bounds on $W_n$ and $V_n$ are
summarized in
\CLAIM Proposition(Wbds)
For every $n\ge 3$, there is a constant $C_n$ such that, for any
$v \in {\cal V}_\b$,
$$
|W_n(k,\r,v)| \le
{C_n\r^{n-3} \over (1+\r)^{n-3}
(1+k^2+\r^2)^{\b+1/2}} |v|_\b^n~.
\EQ(22)
$$
If $\vhi \in\VV_{n,\b}$, then
$$
|V_n(k,\r,\vhi )| \le
{C_n\r^{n-3} \over (1+\r)^{n-3}
(1+k^2+\r^2)^{\b+1/2}} |\vhi |_\b~.
\EQ(22a)
$$
 
\LIKEREMARK{Proof of \clm(Wbds)}
Throughout, the letters $C$, $C_n$ denote constants
which can vary between equations, and which may depend on $\beta$,
but not on $v$.
>From Eq.\equ(2), we have for $\r >0$, $n>1$, by a change
of variables,
$$
\eqalignno{
W_n(k,\r,v)&=\r^{n-3}\int \delta(1 - \ssum_{i=1}^n z_i^2)
\,\delta(\ssum_{i=1}^n z_i)  \prod_{i=1}^n
v(\textstyle {k \over n}+\r z_i)dz_i ~,
\NR(10)V_n(k,\r,\vhi )&=\r^{n-3}\int \delta(1 - \ssum_{i=1}^n
z_i^2)
\,\delta(\ssum_{i=1}^n z_i)\vhi (\textstyle{k \over n}+\r z_1,\dots,
{\textstyle{k \over n}}+\r z_n) \displaystyle\prod_{i=1}^n
dz_i
.
\NR(10a)}
$$
In particular,
$$
W_n(k,\r,1)\,=\,\r^{n-3} L_n~,
$$
with
$$
L_n\,=\,\int \delta(1 - \ssum_{i=1}^n z_i^2)\, \delta(\ssum_{i=1}^n z_i)
\prod_{i=1}^n  dz_i~.
\EQ(13)$$
Clearly, $L_n$ is nothing but the volume of the $n-2$ dimensional unit sphere. 
Thus, if $v$ is bounded, then
$$
|W_n(k,\r,v)| \le \r^{n-3} L_n \|v\|_{L^\infty}^n~.
\EQ(14)$$
This bound is valid for all $k$. However, if $|k|>2n\r$, then in the integrand
of \equ(10), the argument of every $v$ has modulus at least $|k|/2n$, and
therefore $v\in\VV_\b$ implies
$$
|W_n(k,\r,v)| \,\le \, \r^{n-3}\, L_n (1+k^2/4n^2)^{-n\b}\,|v|_\b ^n~.
\EQ(15)
$$
Note also that if $\vhi  \in \VV_{n,\b}$, $n \ge 2$,
$$
|V_n(k,\r,\vhi )| \,\le\,
W_n(k,\r,\,u)
|\vhi |_\beta  ~,
\ \ \ u(p)= (1+p^2)^{-\b} ~,
\EQ(151)$$
so that Eq.\equ(22a) follows from Eq.\equ(22).

In view of the bounds \equ(14) and \equ(15),
we can distinguish the two cases $\r \le1$ and $\r >1$.
\LIKEREMARK{Case 1: $\rho\le1$}Combining \equ(14) and \equ(15) yields
Eq.\equ(22) for $\rho\le1$.
The proof of Eq.\equ(22a) is then obvious in the case $\rho\le1$.
\LIKEREMARK{Case 2: $\rho>1$}In this case, we proceed by induction on $n$,
starting from $n=2$. From Eq.\equ(2), we have $$
V_2(k,\r,\vhi )\,=\,
\textstyle{\sqrt 2 \over 4\r} \vhi ({k \over 2} + {\r \over \sqrt 2},
{k \over 2} - {\r \over \sqrt 2})
+ {\sqrt 2 \over 4\r} \vhi ({k \over 2} - {\r \over \sqrt 2},
{k \over 2} + {\r \over \sqrt 2})~.
\EQ(8)$$
Hence,
$$
|V_2(k,\r,\vhi )| \le
{2^{2\b - 1/2}
\over \r (1+k^2 + \r^2)^\b} |\vhi |_\b ~.
\EQ(9)$$
 
To simplify the notation, we formulate our further calculations for functions
$\vhi $ of the form
$\vhi = v_1\otimes \dots\otimes  v_{n}$. The extension to general $\vhi $ is
immediate.  Starting from \equ(2) for $n+1$ instead of $n$, and
writing $q_j\,=\,y_j -q/n$ for $j \le n$, and $q_{n+1} =q$, we find:
$$
\eqalign{
&V_{n+1}(k,\r,v_1\otimes \dots\otimes  v_{n+1})\cr \,&=\,\int_{-\nnn  \r}
^{\nnn  \r}~dq\,\textstyle
V_n\left ( {nk \over n+1} -q,\sqrt{\r^2 - {n+1 \over n}q^2},~
v_1\otimes \dots\otimes  v_n\right )
v_{n+1}\left ( {k \over n+1} +q \right )~,\cr}
\EQ(11)$$
and, by a further change of variables,
$$
\eqalign{
&V_{n+1}(k,\r,v_1\otimes \dots\otimes  v_{n+1})\cr
&=\rho \nnn  \int_{-1}^1dt\,
\textstyle V_n\left ( {nk \over n+1} - \nnn  \r t,
\r \sqrt{1-t^2},v_1\otimes \dots\otimes  v_n\right )
v_{n+1}\left ( {k \over n+1} + \nnn  \r t \right ).
\cr }
\EQ(12)$$
 
Before using Eqs.\equ(11) and \equ(12) to obtain iterative bounds on
$W_n$, we note a simplifying effect of the symmetry of Eq.\equ(1). The
integral
of Eq.\equ(10) is also equal to $n$ times the integral over the
region where $z_j \le z_n$. In this region, because of the constraint
$\sum_{i=0}^n z_i=0$, $z_n$ must be strictly positive. Suppose $z_n\le x$ and
that there are $p$ of the $z_j$ taking values $>0$: these add up to at most
$px$. The other variables add up to the opposite amount, so that
the sum of their squares is at most $(px)^2$. Hence we have
$1\,=\,\sum z_i^2 \le (p+1)px^2 \le n(n-1)x^2$. Thus, in this region,
$z_n^2 \ge {1\over n(n-1)}$. Therefore,
$$
\eqalign{
|&W_{n+1}(k,\r,v)|\cr\,&\le\, \r^{n-2}(n+1)\int_{z_{n+1}\ge 1/\sqrt{n(n+1)}}
\delta(1 - \ssum_{i=1}^{n+1} z_i^2)
\delta(\ssum_{i=1}^{n+1} z_i)\prod_{i=1}^{n+1} |v|({\textstyle {k \over
n+1}+\r
z_i}) dz_i~.\cr} \EQ(17)$$
Going through the same changes of variables as before, we obtain:
$$
\eqalign{
|&W_{n+1}(k,\r,v)| \cr
\,&\le\,\r\sqrt{n (n+1)} \int_{1/n}^1dt\,
\textstyle W_n\left ( {nk \over n+1} - \nnn  \r t,
\r \sqrt{1-t^2},|v| \right )
\,|v|\left ( {k \over n+1} + \nnn  \r t \right )~.
\cr }
\EQ(18)$$
In view of \equ(151),
it suffices to prove, for $\rho>1$,
$$
W_n(k,\r,u) \le {C_n\over (1+k^2+\r^2)^{\b+1/2}}~,
\EQ(20)$$
where $u(p)= (1+p^2)^{-\b}$.
The corresponding result for $V_n$ is then obvious. Note also that
$W_n(k,\r,v)\,=\,W_n(-k,\r,\widetilde v)$,
with $\widetilde v(p)\,=\,v(-p)$. It
suffices therefore to consider only $k \ge 0$.
 
We perform now an inductive step from $n\ge3$ to $n+1$. The step from
$n=2$ to $n=3$ will be dealt with later.
Suppose that \equ(22) holds for some
$n\ge 3$. To prove
that it holds for $n+1$, we use Eq.\equ(18), substituting the postulated bound
for $|W_n|$, and set $k\,=\,(n+1)a$, $\r\,=\,b\sqrt{(n+1)/n}$.
Some elementary  estimates on the resulting integral, and variants
of these estimates we will need later are stated in the following lemma.
\CLAIM Lemma(iter)
Let $\beta>\HALF$ and $\gamma \ge \b+1/2$.
For any $n\ge 2$ there is a constant
$M_n$ such that for all $a > 0$, $b >0$,
$$
\eqalignno{
I_n(a,b) \,&=\, \int_{1/n}^1 dt\,{b  \over \hphantom{\sqrt{1-t^2}}
[1+ (na -bt)^2 + b^2(1-
t^2)]^\gamma [1 + (a+bt)^2]^\b}\cr
&\le {M_n\over (1+ a^2 +b^2)^{\b + 1/2}}~,\NR(21)
J_n(a,b) \,&=\, \int_{1/n}^1dt\, {1 \over \sqrt{1-t^2}
[1+ (na -bt)^2 + b^2(1- t^2)]^\b [1 + (a+bt)^2]^\b} \cr
\,&\le\,{M_n\over (1+ a^2 +b^2)^{\b + 1/2}}~,\NR(21a)
K_n(a,b) \,&=\, \int_{-1}^1 dt \,{b \over \sqrt{1-t^2}
[1+ (na -bt)^2 + b^2(1- t^2)]^\gamma [1 + (a+bt)^2]^\b} \cr
\,&\le\,{M_n\over (1+ a^2 +b^2)^{\b}}~.\NR(21b)
}
$$
 
\PROOF The proofs are given in Appendix A.
 
It is now clear that Eq.\equ(21) proves the induction step $n\ge3\to n+1$.
We next consider the step from $n=2$ to $n=3$. Using the bounds Eq.\equ(9) and
\equ(18), we get the desired estimate from Eq.\equ(21a).
 
\beginsection{Bounds on First Derivatives}
 
In addition to the bounds on $W_n$ and $V_n$ of \clm(Wbds),
we shall also need  
bounds on their first and second derivatives.
$W_n$ and $V_n$.
Since
$$
\partial_k W_n(k,\r,v)\,=\,
V_n(k,\r,v\otimes \dots\otimes v\otimes  v')~,
\EQ(23)$$
we have immediately  from \clm(Wbds) and Eq.\equ(9):
 
\CLAIM Corollary(dkbds)
For all $n\ge 3$, there is a constant $C_n$ such that, whenever
$v$ and $v'$ are both in ${\cal V}_\b$,
$$
|\partial_k W_n(k,\r,v)| \le
{C_n\r^{n-3} \over (1+\r)^{n-3}
(1+k^2+\r^2)^{\b+1/2}} |v|_\b^{n-1} |v'|_\b ~.
\EQ(24)$$
For $n=2$,
$$
|\partial_k W_2(k,\r,v)| \le
{C_n\over
\r (1+k^2+\r^2)^{\b}} |v|_\b|v'|_\b ~.
\EQ(24a)$$
 
We consider next the $\rho$-derivative. For all $n\ge 3$, we get from
\equ(10),
$$
\eqalign{
\partial_\r& W_n(k,\r,v)\,=\, \textstyle{n-3 \over \r} W_n(k,\r,v) \quad  \cr
&+n\r^{n-3}\int  \delta(1 -\ssum_{i=1}^n z_i^2)
\delta(\ssum_{i=1}^n z_i)\prod_{i=1}^{n-1}
v(\textstyle{k \over n}+\r z_i) dz_i \cdot
z_n\cdot v'(\textstyle{k \over n}+\r z_n)dz_n ~. \cr}
\EQ(25)$$
The second term is bounded in modulus by
$nV_n(|v|\otimes \dots\otimes |v| \otimes |v'|)$.
Therefore, using Eq.\equ(22a), we find:
 
\CLAIM Proposition(drobds)
For all $n\ge 4$ there is a constant $K'_n$ such that, whenever
$v$ and $v'$ are both in ${\cal V}_\b$,
$$
|\partial_\r W_n(k,\r,v)| \,\le\,
{K'_n\r^{n-4} \over
(1+\r)^{n-4} (1+k^2+\r^2)^{\b+1/2}} |v|_\b^{n-1} \,|v'|_\b~.
\EQ(26)$$
For $n=3$,
$$
|\partial_\r W_3(k,\r,v)| \,\le\,
{3C_3 \over
(1+k^2+\r^2)^{\b+1/2}} |v|_\b^2 \,|v'|_\b ~.
\EQ(27)$$
 
\beginsection{Bounds on Second Derivatives}
 
We shall assume throughout that $v$, $v'\in\VV_\b$. We start by recasting
Eq.\equ(25) in the following ``inductive'' form, valid for
$n\ge 2$:
$$
\eqalign{
\partial_\r& W_{n+1}(k,\r,v)\,=\,
\textstyle {n-2 \over \r} W_{n+1}(k,\r,v)  \cr
&+(n+1) \int_{-\nnn  \r}
^{\nnn  \r} \,dq\,\,
W_n\left ( \textstyle {nk \over n+1} -q,{\textstyle \sqrt{\r^2 - {n+1 \over
n}q^2}}, ~v \right)
{q \over \r}~v' \left (\textstyle  {k \over n+1} +q \right ) ~.\cr}
\EQ(28)$$
 
\beginsection{-- Bound on $\partial^2_\r W_n$}
 
We now take $n\ge 3$ and differentiate in $\r$. This yields
several terms which we call $T_1+ \dots  +T_6$.
 
\noindent 1) We consider the first term in \equ(28)
reexpressed with the help of \equ(10) for $n+1$:
$$
{ {n-2\over \rho}}
W_{n+1}(k,\r,v)\,=\,(n-2)\r^{n-3}\int \delta(1 - \ssum_{i=1}^{n+1} z_i^2)
\,\delta(\ssum_{i=1}^{n+1} z_i)\, \prod_{i=1}^{n+1}
v({\textstyle{ k \over n+1}+\r z_i})
 dz_i ~.
$$
Differentiating with respect to $\rho$, this
gives two terms, $T_1$ and $T_2$:
$$
\eqalignno{
T_1 \,&=\,  (n-2)(n-3) \r^{-2} W_{n+1}(k,\r,v)
~,
\NR(29)
T_2\,&=\,(n-2)(n+1)\r^{n-3} \int \delta(1 -\ssum_{i=1}^{n+1} z_i^2)
\delta(\ssum_{i=1}^{n+1} z_i) \cr
&\times\prod_{i=1}^n
v(\textstyle {k \over n+1}+\r z_i) dz_i \cdot
z_{n+1}\cdot v'(\textstyle {k \over {n+1}}+\r z_{n+1})dz_{n+1} ~.
\NR(30)}$$
The first term is bounded using Eq.\equ(22), and yields
$$
|T_1|\,\le\,{(n-2)(n-3) C_{n+1} \r^{n+1-5}\over (1+\r)^{n+1-3} (1+k^2
+\r^2)^{\beta+1/2}}\, |v|_\beta ^{n+1}~. $$
Since $|z_{n+1}|\le1$, the term $T_2$  can be bounded using Eq.\equ(22a),
yielding
$$
|T_2|\,\le\,{(n-2)(n+1) C_{n+1} \r^{n+1-4}\over (1+\r)^{n+1-3} (1+k^2
+\r^2)^{\beta+1/2}}\, |v|_\beta ^{n}\,|v'|_\beta~. $$
Since $T_1=0$ for $n=3$, we find
$$
|T_1|+|T_2| \,\le\,{C \over (1 +k^2 + \r^2)^{\b+1/2}} \,\|v\|_\beta ^{n+1}~.
$$
 
\noindent 2) We consider now the integral in \equ(28) and differentiate with
respect to $\rho$.
This gives two boundary terms, $T_3$, $T_4$, and two terms $T_5$, $T_6$ from
differentiating
the factor $\rho^{-1}$ and the square root in $W_n$, respectively.
The boundary terms are
$$\eqalign{
T_3\,&=\,\hphantom{-}n W_n\left (\textstyle  {nk \over n+1} - \nnn \r,
~0 ,v \right) ~v'\left ( \textstyle {k \over n+1} +\nnn \r
\right )~,\cr
T_4\,&=\,-n W_n\left ( \textstyle {nk \over n+1} + \nnn \r,
~0 ,v \right) ~v'\left ( \textstyle {k \over n+1} -\nnn \r
\right )~.\cr}
\EQ(31)$$
Using again Eq.\equ(22) we see that they vanish for $n > 3$, and when
$n=3$, the first of these two terms decreases slower at
infinity, and their total contribution is bounded by
$$
|T_3|+|T_4| \,\le\,{C \over (1+k^2+\r^2)^\b}\,\|v\|_\beta^{n+1}~.
$$
The term $T_5$ is bounded by 
$(1/\r)(n+1)V_{n+1}(k,\r,|v\otimes \dots \otimes v \otimes v'|)$,
which by Eq.\equ(22)
leads to
$$
|T_5|\,\le\,{C \over (1+k^2+\r^2)^{\b+1/2}}\,\|v\|_\beta^{n+1}~.
$$
The term $T_6$ is equal to
$$
\eqalign{
T_6\,=\,&(n+1) \int_{-\nnn  \r}
^{\nnn  \r} \,dq\,\,
\partial_\r W_n\left ( \textstyle {nk \over n+1} -q,~
\sqrt{\r^2 - {n+1 \over n}q^2}, ~v \right)\cr
&\times{q \over \sqrt{\r^2 - {n+1 \over n}q^2}} ~
v' \left ( \textstyle {k \over n+1} +q \right ) ~.\cr}
\EQ(32)$$
After changing variables $q\,=\,\rho t\sqrt{n\over (n+1)}$, and using
$|t|\le1$, this term can be estimated by Eq.\equ(21b) and yields
$$
|T_6|\,\le\,{C \over (1+k^2+\r^2)^{\b}}\,\|v\|_\beta^{n+1}~.
$$
Combining the bounds, and replacing $n+1$ by $n$, we get the
\CLAIM Proposition(d2r)
For every $n\ge 4$, there is a constant $C_n$ such that
$$
|\partial^2_\r W_n(k,\r,v)| \,\le\, {C_n \over (1 +k^2
+\r^2)^\b}\,\|v\|_\beta^n~. \EQ(33)$$
 
\beginsection{-- Bound on $\partial_k \partial_\r W_n$}
 
In analogy with \clm(d2r), we have
\CLAIM Proposition(dkdr)
For every $n\ge 4$, there is a constant $C_n$ such that
$$
|\partial_k\partial_\r W_n(k,\r,v)|\, \le\, {C_n \over (1 +k^2 +\r^2)^\b}
\,\,\|v\|_\beta^n~.
\EQ(35)$$
 
\PROOF We go back to Eq.\equ(28) for $n\ge 3$,
and differentiate in $k$. This gives a sum of three
terms
$$
\eqalign{
\partial_k&\partial_\r W_{n+1}(k,\r,v) \cr
\,&=\,\textstyle {n-2 \over \r} \partial_kW_{n+1}(k,\r,v)\cr
&+n \int_{-\nnn  \r}
^{\nnn  \r} \,dq\,\,
\partial_k W_n\left ( \textstyle {nk \over n+1} -q ,{\textstyle \sqrt{\r^2 -
{n+1 \over n}q^2}}, ~v \right)
{q \over \r}~v' \left (\textstyle  {k \over n+1} +q \right )\cr
&+\hphantom{n} \int_{-\nnn  \r}
^{\nnn  \r} \,dq\,\,
\hphantom{\partial_k} W_n\left ( \textstyle {nk \over n+1} -q ,{\textstyle
\sqrt{\r^2 - {n+1 \over n}q^2}}, ~v \right)
{q \over \r}~v'' \left (\textstyle  {k \over n+1} +q \right )
\cr\,&=\,S_1+S_2+S_3
~.\cr}
$$
The terms $S_1$ and $S_2$ are readily bounded by
$$
|S_1|+|S_2|\,\le\,{C\over (1+k^2+\r^2)^\b}  \|v\|_\beta ^{n+1}~,
$$
using \clm(dkbds) for $S_1$, then \equ(23), \equ(11), and \equ(22a)
for $S_2$.
We now deal with $S_3$.
Here, the $v''$ must be interpreted in the weak sense, i.e., it
has to be integrated by parts with respect to the variable $q$.
(Alternatively, the same expression can be obtained without ever
mentioning $v''$,
e.g., by changing to the integration variable $q+k/(n+1)$,
differentiating in $k$, then changing back to $q$.)
This yields:
$$
S_3\,=\,S_{31}+\dots +S_{35}~.
\EQ(34)$$
The term $S_{31}$ arises, when integrating by parts, from differentiating
the first argument of the $W_n$ factor. It is equal to $S_2/n$.
The boundary terms give $S_{32}$ and $S_{33}$:
$$
\eqalign{
S_{32} \,&=\,
\nnn W_n\left (\textstyle  {nk \over n+1} - \nnn \r,
0,v \right) ~v'\left (\textstyle  {k \over n+1} +\nnn \r
\right )~,\cr
S_{33} \,&=\,
\nnn W_n\left ( \textstyle {nk \over n+1} + \nnn \r,
0,v \right) ~v'\left (\textstyle  {k \over n+1} -\nnn \r
\right ) ~.\cr}
$$
These terms are handled like $T_3$ and $T_4$. They vanish for $n>3$, and for
$n=3$ they are bounded by
$$
|S_{32}|+|S_{33}|\,\le\,C(1+k^2+\r^2)^{-\b}\,\|v\|_\beta ^{n+1}~.
$$
Upon integrating by parts, there is a term coming from $\partial_q q/\r$
which is equal to
$$
S_{34}\,=\,(1/\r) V_{n+1}(k,\r,v\otimes \dots\otimes v\otimes v')~.
$$
This term is bounded by $|S_{34}|\le C(1+k^2+\r^2)^{-\b-1/2}\|v\|_\beta
^{n+1}$.
Finally there is a term where the derivative acts on the second argument of
$W_n$:
$$
\eqalign{
S_{35} \,&=\,{\textstyle {n+1 \over n}} \int_{-\nnn  \r}
^{\nnn  \r} \,dq\,\,
\partial_\r W_n\left (\textstyle  {nk \over n+1} -q,~
\sqrt{\r^2 - {n+1 \over n}q^2}, ~v \right)\cr
&\times {q \over \r}{q \over \sqrt{\r^2 - {n+1 \over n}q^2}} ~
v' \left ( \textstyle {k \over n+1} +q \right ) dq~.\cr}
$$
It is estimated exactly in the same way as the term $T_6$ in Eq.\equ(32),
and is bounded by $C(1+k^2+\r^2)^{-\b}\|v\|_\beta ^{n+1}$.
The proof of \clm(dkdr) is complete.
 
\beginsection{-- Bound on $\partial^2_k W_n$}
 
Finally, we bound the second derivative with respect to $k$:
\CLAIM Proposition(dkdk)
For any $n\ge 4$, there is a constant $C_n$ such that
$$
|\partial^2_k W_n(k,\r,v)| \le {C_n \over (1 +k^2 +\r^2)^\b}\,\|v\|_\beta
^{n}~. \EQ(40)$$
 
\PROOF For $n \ge 3$, a formula for $\partial^2_k W_{n+1}$
is obtained in the same way as
in the case of $\partial_k\partial_\r W_{n+1}$ (using formal
integration by parts). This gives
$$
\partial^2_k W_{n+1}(k,\r,v)\,=\,R_1+\dots +R_4~,
\EQ(36)$$
with
$$
R_1\,=\,V_{n+1}(k,\r,v\otimes \dots\otimes v\otimes v'\otimes v')~,
\EQ(37)$$
$$
\eqalign{
R_2\,=\,&{1 \over n}
\int_{-\nnn  \r}
^{\nnn  \r}
\partial_\r W_n\left (\textstyle  {nk \over n+1} -q,~
\textstyle \sqrt{\r^2 - {n+1 \over n}q^2}, ~v \right)\cr
&\times {q \over \sqrt{\r^2 - {n+1 \over n}q^2}} ~
v' \left (\textstyle  {k \over n+1} +q \right ) dq\cr}
\EQ(38)$$
$$
\eqalign{
&R_3\,=\,\textstyle {1 \over n+1}
W_n\left ( \textstyle {nk \over n+1} - \nnn \r,
0,v \right) ~v'\left (\textstyle  {k \over n+1} +\nnn \r
\right ),\cr
&R_4\,=\,\textstyle {-1 \over n+1}
W_n\left (\textstyle  {nk \over n+1} + \nnn \r,
0,v \right) ~v'\left ( \textstyle {k \over n+1} -\nnn \r
\right )~.\cr}
\EQ(39)$$
The details are now essentially the same as in the case of
$\partial_k\partial_\rho$ and are left to the reader.
 
\SECTION Properties of $F_n$
 
In this section, $v$ is fixed and such that $|v|_\b \le 1$ and
$|v'|_\b \le 1$. Recall that $F_n$ has been defined as:
$$
\eqalign{
F_n (k,\z,v) \,&=\,
\int_0^\infty {2\r \over \r^2 -\z^2} W_n(k,\r,v) d\r\cr
&=\,\int_0^\infty \left (
{1 \over \r - \z} \,+\, {1 \over \r + \z} \right )
\ W_n(k,\r,v) d\r~.\cr}
\EQ(41)$$
We take $n \ge 4$, so that the bounds established in the previous
section give:
$$
\eqalignno{
|W_n(k,\r,v)|+|\partial_k W_n(k,\r,v)| \,&\le\,
{C\r \over (1+\r) (1+k^2+\r^2)^{\b+1/2}}~,
\NR(43)
|\partial_\r W_n(k,\r,v)| \,&\le\,
{C \over (1+k^2+\r^2)^{\b+1/2}}~,
\NR(44)
|\partial^m W_n(k,\r,v)| \,&\le\,
{C \over (1+k^2+\r^2)^\b} \
{\rm for\ } |m| \,=\,  2 ~.
\NR(45)
}
$$
Because of these bounds, $\z \mapsto F_n(k,\z,v)$,
as defined by Eq.\equ(41),
is even and holomorphic in $\z$ for ${\rm Im\ }\z \not= 0$, and has
(not necessarily even)
H\"older continuous boundary values on $\bR$ from the upper and
lower half-planes. We note the following identities:
$$
\partial_k F_n(k,\z,v) \,=\,
\int_0^\infty {2\r d\r \over \r^2 - \z^2}
\partial_k W_n(k,\r,v)~,
\EQ(46)$$
$$
\partial_\z F_n(k,\z,v) \,=\,
\int_0^\infty {2\z d\r \over \r^2 - \z^2}
\partial_\r W_n(k,\r,v)~.
\EQ(47)
$$ 

We now propose to find bounds on $|F_n(k,\z,v)|$,
$|\partial_k F_n(k,\z,v)|$, and $|\partial_\z F_n(k,\z,v)|$
when $\z \,=\,  \xi + i\eta$, for real $\xi$ and small, positive $\eta$.
It suffices to deal with the case $k \ge 0$.
We give all details for the case of $\partial_\z F_n(k,\z,v)$
and give indications on the changes needed for the two other
cases.
 
We rewrite Eq.\equ(47) as
$$
\partial_\z F_n(k,\z,v) \,=\,
\psi(\z) \partial_\r W_n(k,0,v) + R_+ (k,\z) + R_- (k,\z)~,
\EQ(48)$$
$$
\psi(\z) \,=\,
\int_{-\infty}^\infty
{d\r \over (\r -\z)(1+\r^2)^2}  \,=\,
-{\pi (\z + 2i)\over 2(\z+i)^2} ~,
\EQ(49)$$
$$
R_{\pm}(k,\z) \,=\,
\pm  \int_{-\infty}^\infty
{\vhi (k,\r) d\r \over \r \mp \z}~,
\EQ(50)$$
$$
\vhi (k,\r)  \,=\,  \theta (\r) \left (\partial_\r W_n(k,\r,v) -
\partial_\r W_n(k,0,v)(1+\r^2)^{-2}\right )~.
\EQ(51)$$
(Here we make the inessential assumption that
$\b \le 3/2$. Otherwise the definition of $\psi$ and $\vhi$
would be changed by replacing $(1+\r^2)^{-2}$ with
$(1+\r^2)^{-p}$ with $p \ge \b+1/2$.)
Each of the functions $\z \mapsto R_{\pm}(k,\z)$  is a Cauchy
transform of the Lipschitz function $\r \mapsto \vhi (k,\r)$,
and its boundary value as ${\rm Im\ } \z \downarrow 0$
is therefore H\"older continuous (of any exponent $<1$) in the variable
$\z$.
 
We now fix $\z \,=\,  \xi + i\eta$, $\eta > 0$ and $0 < h < 1/2$.
We have by Eq.\equ(44):
$$
|\vhi (k,\r)| \le {C \over (1+k^2+\r^2)^{\b+1/2}}~.
\EQ(52)$$
For all $\alpha \in [0,1)$, combining Eq.\equ(44) and \equ(45), we have:
$$
|\vhi (k+h,\r) - \vhi (k,\r)| \le
{Ch^{1-\alpha} \over
(1+k^2+\r^2)^{\b + \alpha/2}}~.
\EQ(53)$$
Finally, if $|\r -\xi| < 1/2$, we have for all $\theta \in [0,1]$, by
Eq.\equ(44) and \equ(45):
$$
|\vhi (k,\r) - \vhi (k,\xi)|  \le
{C|\r-\xi|^{1-\theta} \over
(1+k^2+\r^2)^{\b + \theta/2}}~,
\EQ(54)$$
and, by interpolation,
$$
|\vhi (k+h,\r) - \vhi (k,\r)
-\vhi (k+h,\xi) + \vhi (k,\xi)| \ \le
{Ch^{(1-\alpha)(1-\theta)} |\r -\xi|^{(1-\alpha)\theta} \over
(1+k^2+\r^2)^{\b + \alpha/2}}~.
\EQ(55)$$
Analogous estimates can be obtained for the case of the
function $F_n(k,\z,v)$, with $\vhi$ replaced by $\theta(\rho)W_n(k,\r,v)$ and
for $\partial_k F_n(k,\z,v)$,
with $\vhi$ replaced by $ \theta(\rho)\partial_k W_n(k,\r,v)$.
 
\beginsection{Bounds on $|R_\pm(k,\z)|$}
 
We choose $a \in (0,\HALF)$ and split the integral \equ(50)
representing $R_+(k,\z)$
$$
\int_{-\infty}^\infty
{\vhi (k,\r) d\r \over \r -\z}
$$
into three contributions, denoted $K_1$, $K_2$, and $K_3$,
corresponding to the ranges of integration
$(-\infty ,\xi -a]$, $[\xi -a,\xi +a]$, and
$[\xi+a,\infty)$, respectively.
Then, using the bound \equ(52),
$$
|K_1| \le {C \over (1+k^2)^{\b-1/2}}
\int_{-\infty}^{ \xi -a}
{d\r \over (\xi - \r)(1+k^2 +\r^2)}~.
$$
The last integral is equal to
$$
{1 \over 2\pi i}
\int_{\cal C}d\r\, {\log (\r - \xi +a)
\over (\xi - \r)(1+k^2 +\r^2)} ~,
$$
where the contour ${\cal C}$ follows the boundary of ${\bf C} \setminus (\xi
-a
+{\bf R}_-)$ in the positive direction.
It can thus be computed by residues, yielding
$$
{\log a \over (1+k^2+\xi^2)}
+ 2{\rm Re\ }
{\log (i\sqrt{1+k^2} -\xi +a) \over
(\xi - i\sqrt{1+k^2})(2i\sqrt{1+k^2})}
$$
The term $K_3$ is treated in a similar manner.
Therefore:
$$
|K_1|+|K_3| \le {C(1+|\log a| +\log (1+|k|) +\log (1+|\xi|)\,) \over
(1+k^2)^\b (1+k^2 +\xi^2)^{1/2}}~.
\EQ(56)$$
We next split $K_2$ as $K_{21} +K_{22}$:
$$
K_{21} \,=\,  \int_{\xi -a}^{\xi +a}
{\vhi(k,\r) - \vhi(k,\xi) \over
\r - \xi}~.
$$
This is bounded by
$$
\eqalign{
|K_{21}| &\le {C \over (1 +k^2 +\xi^2)^{\b + \theta/2}}
\int_{\xi -a}^{\xi +a} |\r - \xi|^{-\theta} d\r \cr &\le
{2C a^{1-\theta} \over
(1-\theta)(1 +k^2 +\xi^2)^{\b + \theta/2}}~. \cr}
\EQ(57)$$
The term $K_{22}$ is equal to
$$
K_{22} \,=\,  \vhi(k,\xi) \int_{\xi-a}^{\xi +a}
{d\r \over \r - \z} \,=\,
\vhi(k,\xi) \log {a -i\eta \over -a -i\eta}~,
$$
so that
$$|K_{22}| \le {C \over (1+ k^2 +\xi^2)^{\b +1/2}}~.
\EQ(58)$$
The quantity $|R_-(k,\z)|$ is estimated in the same way.
Combining these estimates, we get
$$
|R_{\pm}(k,\xi +i\eta)| \le
{C_\veps \over (1+k^2)^{\b} (1+k^2+\xi^2)^{1/2-\veps}}~,
\EQ(59)$$
for all sufficiently small $\veps >0$.
 
\beginsection{H\"older continuity of $k \mapsto R_+(k,\z)$}
 
We apply the same treatment as above to
$$
R_+(k+h,\z) - R_+(k,\z) \,=\,
\int_{-\infty}^{\infty}
{[\vhi(k+h,\r) - \vhi(k,\r)]d\r \over \r-\z}~,
$$
using now the bounds \equ(53) and \equ(55) instead of
\equ(52) and \equ(54). We have to choose here
$\alpha \ge 2 - 2\b$. All the estimates are completely
analogous to those for $R_+(k,\z)$ and we find that for
every $\alpha \in (2-2\b,1)$, $\kappa \in (0,1-\alpha)$
and sufficiently small $\veps >0$, there is a constant $C$
such that
$$
|R_+(k+h,\z)- R_+(k,\z)| \le
{Ch^{\kappa} \over (1+k^2)^{\b - (1-\alpha)/2}
(1+k^2+\xi^2)^{1/2-\veps}} ~~.
\EQ(60)$$
The same holds for $R_-$. It is clear that the same
kind of estimates work for
$R_{\pm}(k,\z +h) - R_{\pm}(k,\z)$, with the same
result. Therefore,
 
\CLAIM Theorem(fbd) Let $n \ge 4$.
The function $(k,\z) \mapsto F_n(k,\z,v)$ extends to a $\CC^1$
function in $\bR \times \{\z\,:\ {\rm Im\ }\z \ge 0\}$. 
For each sufficiently small $\veps >0$, there is a
constant $C_n(\veps)$ such that for all real $k$ and $\xi$, 
and every bi-index $m$ with $|m| \le 1$, 
$$
|D^m\, F_n(k,\xi+i0,v)| \ \le
{C_n(\veps) \over (1+k^2)^{\b} (1+k^2+\xi^2)^{1/2 -\veps}} ~~.
\EQ(61)$$
Moreover, if $0 < \kappa < 2\b -1$,
there is a $C'_n(\veps,\kappa)$ such that for $|h_1|,|h_2| \le 1/2$,
$$
\eqalign{
|D^m\,&F_n(k+h_1,\xi+h_2 + i0,v) - D^m\,F_n(k,\xi+ i0,v)|\cr
\,&\le\,
{C'_n(\veps,\kappa) ( |h_1|^\kappa + |h_2|^\kappa) \over
(1+k^2)^{\b - \kappa/2}(1+k^2+\xi^2)^{1/2 -\veps}}~.\cr
}
\EQ(62)
$$

\noindent To prove \clm(main), it suffices to apply \clm(fbd) with $\zeta$$=$$
\sqrt{(n-1)(k^2/n+\mu)+i0}$. If $\mu=0$, then this means
$\zeta=|k|\sqrt{1-1/n}+i0$, while for $\mu>0$, the square root depends
differentiably on $k$. If $\mu<0$, the square root is not differentiable at
$k=\pm\sqrt{n|\mu|}$.
\SECTIONNONR Appendix A
 
Here, we give the proof of \clm(iter). We begin by proving Eq.\equ(21).
We distinguish two cases:
 
\noindent 1) $na \ge 2b$: in that case
$$
I_n(a,b) \le {b \over
[1 + (na/2)^2 ]^{\gamma} [1 + a^2]^\b} \quad \le\quad
{C \over (1+a^2 +b^2)^{\b +1/2}}~.
$$
2) $na < 2b$: then
$$
\eqalign{
I_n(a,b) &\le {1 \over (1 + a^2 +b^2/n^2)^\b}
\int_0^1 {b dt \over
[1 + b^2(1-t)]^\gamma}\cr
&= {1 \over (1 + a^2 +b^2/n^2)^\b}
\int_0^1 {b dy \over
[1 + b^2y]^\gamma}~. \cr}
$$
The last integral is bounded by $1/b(\gamma -1)$ and also
bounded by 1 if $b \le 1$, hence in the present case,
$$
I_n(a,b) \le {2\gamma \over
(\gamma -1)(1 + a^2 +b^2/n^2)^\b (1+b)} \,\le\,
{C \over (1+a^2 +b^2)^{\b +1/2}}~.
$$
This proves Eq.\equ(21).
 
We next consider Eq.\equ(21a).
We again distinguish two cases:
 
\noindent 1) $na \ge 2b$: then
$$
J_n(a,b) \le {1 \over
[1 + (na/2)^2 ]^\b [1 + a^2]^\b}
\int_0^1 {1 \over \sqrt{1-t^2}}  \,\le\,
{C \over (1+ a^2 + b^2)^{2\b}} ~.
$$
2) $na < 2b$: then
$$
\noindent\eqalign{
J_n(a,b) \le & {1 \over (1 +a^2 + b^2/n^2)^\b}
\int_{1/n}^1 {nt dt \over
[1 + b^2(1-t^2)]^\b \sqrt{1-t^2}} \cr
\le & ~{n\over (1 +a^2 + b^2/n^2)^\b} \int_0^1 {dy \over
(1+ b^2 y^2)^\b}~. \cr}
$$
The last integral is bounded by 1 and also by
$$
{1 \over b} \int_0^1 {2^\b b dy \over (1 + by)^{2\b}} \quad \le\quad
{2^\b \over b(2\b -1)}~,
$$
and the assertion follows.
 
Finally, we prove Eq.\equ(21b).
Note that if $b \le 1$, the integral is bounded.
If $b \le a/2$ the integral is bounded by
$Cb/(1+a^2)^{\b+\gamma}$. Therefore we can restrict our
attention to the case when $b \ge 1$ and $a < 2b$.
The part of the integral from $t\,=\,1/n$ to $t=1$ can be estimated by
\equ(21a);
that part is bounded by
$$
{Cb \over (1+ a^2 +b^2)^{\b + 1/2}} \,\le\,
{C \over (1+ a^2 +b^2)^{\b}}~.
$$
We next consider the range of integration $0 \le t \le 1/n$.
This contribution is bounded by
$$
\int_0^{1/n}dt\, {Cb \over
(1+b^2)^\gamma [1+ (a+bt)^2]^\b}\,\le\,
{C \over (1+b^2)^\gamma }
\int_{-\infty}^{\infty} {dx \over (1+x^2)^\b} \,\le\,
{C \over (1+ a^2 +b^2)^\gamma }~.
$$
Finally, the contribution from $-1 \le t \le 0$ can be bounded by
$$
{Cb  \over (1+ a^2 +b^2)^\gamma } \int_0^1 {dt \over \sqrt{1-t^2}}\,\le\,
{C \over (1+ a^2 +b^2)^{\b}}~.
$$
The proof is complete.
 
\SECTIONNONR Appendix B
 
In this section, we briefly discuss the Gaussian example, i.e.,
the case when
$$
v(p) \,=\,   g(p) \equiv \exp (-Ap^2)~,
\EQ(B1)$$
with some fixed $A > 0$.
Then, for any integer $n \ge 2$,
$$
W_n(k,\r,g)  \,=\,   e^{-A(\r^2 + k^2/n)} W_n(k,\r,1) \,=\,  
L_n\,e^{-A(\r^2 + k^2/n)}\,\theta(\r)\, \r^{n-3} ~.
\EQ(B2)$$
We now consider, for any integer $n \ge 3$,
$$
F_n(k,\zeta,g)  \,=\, 
L_n\,e^{-Ak^2/n}\, \int_0^{\infty}d\r\, 
{2\r^{n-2}\,e^{-A\r^2}\over
\r^2 - \zeta^2}~.
\EQ(B3)$$
If $n$ is odd and $n\ge 3$, we get
$$
F_n(k,\zeta,g)  \,=\,   -L_n
e^{-Ak^2/n}\left (
\log(-\zeta^2)e^{-A\zeta^2}\zeta^{n-3}+
E_n(\zeta^2)\right )~.
\EQ(B5)$$
If $n$ is even and $\ge 4$, we  find:
$$
F_n(k,\zeta,g)  \,=\,   L_n
e^{-Ak^2/n}\left (
i\pi {\rm sign}({\rm Im~} \zeta) e^{-A\zeta^2}\zeta^{n-3}+
E_n(\zeta^2)\right )~.
\EQ(B9)$$
Here $E_n$ is an entire function such that $E_n(z) = E_n(z^*)^*$.

If we choose $n=4$ and $\zeta=\sqrt{(n-1)(k^2/n+\mu)+i0}$,
then there are three cases:
\item{a)}$\mu>0$: Then $\zeta $ is a holomorphic function of $k$ in a
neighborhood of $\bR$. Therefore, $k\mapsto F_n(k,\zeta,g)$ is $\CC^1$ on
$\bR$ with a H\"older continuous derivative.
\item{b)}$\mu=0$: Then $\zeta =\sqrt{1-1/n}|k|+i0$.
Therefore, $k\mapsto F_n(k,\zeta,g)$ has
a H\"older continuous derivative on $\bR \setminus 0$.
\item{c)}$\mu<0$: Then the derivative of $\zeta$ is infinite at
$k=\pm\sqrt{-n\mu}$.
\LIKEREMARK{Acknowledgements}This work was made possible by the
hospitality of the IHES and the University of Geneva. It was partially
supported by the NSF DMS-9002059 and the Fonds National Suisse.
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