Table Of ContentNORTH-HOLLAND MATHEMAICS STUDIES 111
Notas de Matematica (101)
Editor: Leopoldo Nachbin
Centro Brasileiro de Pesquisas Fisicas
Rio de Janeiro
and University of Rochester
NORTH-HOLLAND -AMSTERDAM NEW YORK *OXFORD
RECENT PROGRESS IN
FOURIER ANALYSIS
Proceedings of the Seminar on FburierAnalysis held in
El Escorial, Spain, June 30 -July 5/ 1983
Edited by
1. PERAL and J.-L. RUB10 de FRANCIA
UniversidadAutonomad e Madrid
Madrid
Spain
1985
NORTH-HOLLAND -AMSTERDAM NEW YORK *OXFORD
Elsevier Science PublishersB .V., 1985
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Library of Congress Cataloging In PmbUeatlon Data
Seminar 011 Fourier Analysis (1983 : Escorial)
Recent progress in Fourier analysis.
(North-Bolland mathem&ticas tudies ; lll) (Notas de
matcmatica ; 101)
w i s h o r French.
1. Fourier analysis-Congreasea. I. .P eral, Ireneo.
11. Rubio de Rancia, J.-L., 1949- 111. Title.
IV. Series. V. Sariaa: Rotas de matdtics (Ria de
Janeiro, Brazil) ; no. 101.
u.K86 no. lola 510 s C515'.24333 85-4531
tqA403.5 3
ISEN 0-444-87745-2 (Elae-der Science Pub. )
PRINTED IN THE NETHERLANDS
RECENT PROGRESS IN FOURIER ANALYSIS
The following contributions were presented at the Seminar on Fourier
Analysis which was held in El Escorial from 30 June to 5 July 1983.
This meeting was sponsored by the Asociacidn Matemdtica Espaiiola
with financial support from the Comisidn Asesora de Investigacidn
Cientifica y Te'cnica (project 4192).
A decisive factor with respect to the organization toas the financial
help, together with the facilities, provided by the Vicerrectorado
de Investigacidn of the Universidad Autdnoma de Madrid.
The articles we present give a good idea of how work in the area has
evolved and of the scientific, character of the meeting. The friend%
and cordial atmosphere meant that the organization, far from being a
chore, became a pleasurable experience. For this toe owe our
sincerest thanks to all participants.
Special thanks must also go to the invited speakers for their magni-
ficent collaboration, and to Caroline, without whose presence we
hate to think what could have happened!
We should also like to express our gratitude to our colleagues in
the Divisidn de Materndticas in the Universidad Autdnoma de Madrid,
for their help in correcting proofs, and to Soledad, for typing the
.
man u sc r i pt
The Editors
V
Recent Pr0gre.m in Fourier Analysis
1. Perd and J.-L. Rubio de Francin (Editors)
0 Elsevier Science Publishers B.V. (North-Holland), 1985
FUNCTIONS OF LP-BOUNDED PSEUDO-DIFFERENTIAL
OPE RAT0R S
Josefina Alvarez Alonso
Universidad de Buenos Aires
The aim of this paper is to construct a functional calculus over
an algebra of LP-bounded pseudo-differential operators acting on
functions defined on a compact manifold without boundary.
The operators we consider here depend on amplitudes or symbols
with a finite number of derivatives, without any hypothesis of homo-
geneity. The manifolds where the operators act are also of class CM
for a suitable M. In this way is it possible to control the number
of derivatives of f that we need in order to give meaning to f(A),
when A is a self-adjoint operator in that algebra.
Indeed, this program was carried out in [l] and [2] when p = 2.
In [l] an algebra of pseudo-differential operators acting on func-
tions defined in Rn is constructed. The main tool to do that is the
sharp L2 estimates obtained by R. Coifman and Y. Meyer in [3].
Then, functions%f those operators are defined by means of the H.
Weyl formula (see [4] , for example). Since it seems not to be
possible to obtain directly a polynomial estimate for the exponential
exp(-2nitA) in terms of t, a roundabout argument is employed by
introducing an adapted version of the characteristic operators de-
fined by A. P. CalderBn in [5].
All this machinery is extended in [Z] to non-infinitely differen-
tiable compact manifolds without boundary.
In order to get the Lp version of these results the first thing
to do is to obtain the analogous of the algebra constructed in [l].
The main point is to observe that amplitudes in a subclass of Sy,l
give rise to operators on which the classical theory of CalderBn and
Zygmund works (see [6]). Unfortunately as far as I know, it is an
open question to get in the euclidean case a non trivial estimate for
the exponential exp(-ZnitA). However, when the operators act on
3
4 J. Alvarez Alonso
functions defined in compact manifolds, a suitable estimate can be
obtained and so, a non-infinitely differentiable functional calculus
runs.
,...,
Given 0 6 < 1, k = 1,2 let
[
k/l-d if this is an integer
We will consider operators K acting on S in the following way
I
Kf = e - 2a ixS pj(x,S)2(S)dS + Rf
j=O
where
i) The function belongs to the class Sj; that is to say,
pj
is a continuous function defined on Wn x Rn; it has continuous
pj
derivatives in the variable 5 up to the order n+N+2-j and each
function Di pj has continuous derivatives in x,c up to the order
2[n/2] +N+k+2-j , satisfying
I DS DB& Y Pj (X,S) I
-'
X,S SeU PR (1+l5 1) . (1 -6)+la(6 -j B+yl < m
a, B, Y
ii) For 1 < po 5 2 fixed, R is a linear and continuous opera-
tor from Lp into itself for po 5 p 5 p;. Moreover, R and the
adjoint R* are continuous from Lp into ,!L where Li denotes
the Sobolev space of order k and p; is the conjugate exponent
of PO'
Let klk be the class of the operators K.
Now, let X be a differentiable compact manifold of dimension n
and class CM, M = 2[n/2]+n+2N+k+S, without boundary; X has a
measure which in terms of any local coordinate system
..., ...
x = (x,, xn) can be express as G(x)dxl dx,, where G > 0
is a function of class cM-l.
We will introduce the following notation.
Let U1, U2 be open bounded subsets of X or Rn let
$ : U1 + U2 be a diffeomorphism of class CM; if f is a function
defined on the ambient space of U2, $*(f) will denote the function
Functions of JI.D.0. 5
defined on the ambient space of U1 which coincides with foe on
U1 and vanishes outside U1. On the other hand, if A is an ope-
rator acting on functions defined on the ambient space of by
U ,,
$*(A) we denote the operator acting on functions defined on the
ambient space of U, as
$*(A)(f) = $*[A($-'*(f))]
Now, we are ready to define classes of operators on X.
Given 1 < po 5 2, R belongs to Rk(X) if R is a linear
continuous operator from Lp(X) into itself for po 5 p 5 p; and
R, R* map continuously LP(x) into L{(x) for po 5 p 5 pi.
Rk(X) is a self-adjoint Banach algebra with the norm
Now, given 1 < po 5 2, Mk(X) is the class of linear continuous
operators A fron Lp(X) into itself for po 5 p 5 p:, which
satisfy the following two conditions
M
i) Given $1,$2 e C,(X) with disjoint supports, the operator
$1 A $2 belongs to Rk(X). Here $1, $, stand for the operators
of multiplication by the function $1, $,, respectively.
ii) Let U C X be an open subset and let $ : U + Ul be a
u,
diffeomorphism of class CM extendable to a neighborhood of
where U I C Rn. There exists an operator Al e Mk such that if
$1,@2 e c;(u),
Mk(X) is a self-adjoint algebra and Rk(X) is a two-sided ideal of
M~(x); moreover, operators in M ~ ( x ) are continuous from L:(x)
into itself for po 5 p 5 p;, 0 5 m 2 k.
It is possible to endow Mk(X) with a complete norm. In order to
avoid technical details, we will not precise the definition. With
this norm Rk(X) is continuously included in Mk(X) and Mk(X) is
continuously included in L(L:(x)), the space of linear and
continuous operators from L~(x) into itself, for po 5 p 5 p;,
O z m z k .
6 J. Alvarez Alonso
THEOREM 1. po, k E d n be suck that l/po - k/n - 1/2.
Given a self-adjoint operator A e Mk(X) and a function f i n
the Soholev space Lf, wkem s > Z!J + 5/2, !J = 2[n/2]+n+k+
+ N(N+3)/2+4, the Bockner in$egral
J -m
belongs to Rk(X) and coincides with f(A) calculated by means of
the spectral formula in L(L~(x)).
Remarks :
a) It is possible to impose on f additional conditions under
which the operator f(A) belongs to Rk(X).
b) When po = 2 the above theorem remains true with s > !J + 3/2.
c) The Weyl's formula also allows to define functions of a tuple
of non-commuting self-adjoint operators.
We will include here the proof of the theorem 1 in a particular
but significant case.
Suppose that 6 = 0, k = 1; it follows that N = 1. It is
clear that theorem 1 can be deduced from a suitable estimate for
lexp(-ZnitA) I in terms of t e R.
M1 (XI
In order to get this estimate, some notations and results will be
needed. We fix in X coordinate neighborhoods U., diffeomorphisms
$j : Uj + $.(U.) of class CM, whkre M = 2[n/2j+n+8, functions
1 3
8. E C!(Uj), ej 2 0 and a finite partition of unity {nj} of
cIl ass CM , such that supp(ej)c Ui whenever supp(ej) n supp(ei)#
# 0; ei = 1 in a neighborhood of supp(nj) if j = i or if
n +
SUPP(~~I SUPP(~~) 0.
Now, we define an space of symbols for operators in M,(X). More
exactly, for each j we consider the restriction to g.(U.) of a
3 1
function p(J) e So. We define a norm of such a restriction as
where the supremum is taken over x e gj(Uj), 5 E Rn, [El 5 n+3,
Functions of Jl.D.0. 7
la+^( 5 ~[n/~]+4, j.
We note N,(X) this space. With the pointwise multiplication
(p(j)). (q(J)) = (p(j)q(j))
as a product, N1(X) becomes a commutative Banach algebra.
LEMMA. Let H = (p'j)) be an element in €ll(X); we suppose that
d p i s a real function. Then, if t e R,
lexp(-ZnitH)I 5 C~(I + IHl)(l + [tl)lp
where c = C(X) > 0, p = Z[n/2]+n+7.
Proof
Since N,(X) is a Banach algebra, the exponential exp(-2nitH)
is well defined; moreover it is equal to exp(-2nitp(j))
j'
According to the norm that the space N1(X) has, the conclusion
follows.
Now, we will introduce the space M,(X) in the following way
An element K of IM1(X) is an operator R in R,(X) and a vector
(p(j)) in €l,(X) subject to the condition that if Uin Uj # 0
and = ~$1~014 ;, then
Such an element K will be denoted as {(p(j)) ,R).
We define a norm in IMl(X) as follows
IKI IM, = I(,(j))l lRIRl
+
Given K eIM,(X) we define an operator A(K) in the following way
f
A(K) = njI$i(Aj)Bj + R
where
8 J. Alvarez Alonso
It can be proved that A(K) belongs to Ml(X). Moreover the linear
-
map
m1( XI -h E(1 (XI
K A(K)
is into and continuous. Furthermore, if A E E41(X) is self-adjoint,
A = A(K) for some K = i(p(j)),R}, with p (j) real for all j.
It is possible to define a product in W ,(X) in such a way that
IM1(X) becomes a Banach algebra and the map A above is a conti-
nuous homomorphism of algebras.
Finally, let us consider the maps
fi is a continuous homomorphism of algebras and the linear map
Ql
is a right continuous inverse of R.
THEOREM 2. Suppose that l/po - l/n 5 1/2.
-Let H = {(p(j)),R} be an element of Wl(X) such that A(H)
is a self-adjoint operator and the functions p (J' are real for all
j. Then, if t e R,
Proof
According to the notations above, we set A = A(H), K = Q(H) =
= (P (j)) E B1(X).
We assert that
-2aitH - (PitK
e Ql 1
it an element of the form {(O) ,R(t)l.
In fact, since n is a continuous homomorphism of algebras and
n1 is a right inverse of n, we have
-2aitH n,(e-2nitKl~ -2"" - nQl(e -2nitK) =
n re
