Christ M., Kenig C.E., Sadosky C. Harmonic Analysis and Partial Differential Equations: Essays in Honor of Alberto P. Calderon
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Chapter 19: Developments in Inverse Problems
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Remarks Addressed to the Conference
Alberto P. Calderdn
Dear Friends:
Thank you for attending this conference. I have turned seventy-five and
I am starting to realize that the cardinality of this number is larger than I
felt it was.
Turning now to the purpose of this conference, I believe that a few math-
ematical comments of mine before the end of the conference would not be
totally out of place. Well, here they are.
They concern the relationship
between singular integral operators and pseudodifferential operators. The
latter are a very refined mathematical tool whose full power is yet to be
exploited. Singular integral whose kernels are infinitely differentiable off the
diagonal are a very special case of pseudodifferential operators. However,
singular integral operators with kernels which are not infinitely differentiable
as above are not pseudodifferential. Similarly, linear partial differential op-
erators are pseudodifferential operators only if the coefficients are infinitely
differentiable. Such differential operators with nonsmooth (i.e., noninfinitely
differentiable) coefficients have important applications. They appear in prob-
lems in physics and engineering, and they can be studied by using singular
integral operators with nonsmooth kernels. It is' not difficult to construct
an algebra of singular integral operators K in such a way that differential
operators of a given order m with coefficients with specified regularity be
contained in the module, over this algebra, of operators of the form KAm,
where K is an operator in the algebra and A is the operator defined by
(Af)-(x) = T(x) co(x), where - denotes Fourier transform and V(x) is a pos-
itive infinitely differentiable function such that V(x) = lxi for Jxj > 1. The
study of these algebras is not yet complete. There are still open questions
about best possible assumptions.
Thank you.
347
Contributors
Alexandra Bellow
Department of Mathematics
Northwestern University
Evanston, IL 60208
Earl Berkson
Department of Mathematics
University of Illinois, Urbana
Urbana, IL 61801
Jean Bourgain
School of Mathematics
Institute for Advanced Study
Princeton, NJ 08540
Donald L. Burkholder
Department of Mathematics
University of Illinois, Urbana
Urbana, IL 61801
Luis A. Caffarelli
Department of Mathematics
University of Texas, Austin
Austin, TX 78712
Alberto P. Calderdn (1920-1998)
Department of Mathematics
University of Chicago
Chicago, IL 60637
Sun- Yung A. Chang
Department of Mathematics
Princeton University
Princeton, NJ 08544
349
350
Michael Christ
Department of Mathematics
University of California, Berkeley
Berkeley, CA 94720
A. B. Cruzeiro
Grupo de Fisica-Matematica
Universidade de Lisboa
1649-003 Lisboa
Portugal
Ingrid Daubechies
Program in Applied and Computational Mathematics
and Department of Mathematics
Princeton University
Princeton, NJ 08544
Guy David
Departement de Mathematiques, Batiment 425
University de Paris-Sud
91405 Orsay Cedex
France
Charles Fefferman
Department of Mathematics
Princeton University
Princeton, NJ 08544
Robert A. Fefferman
Department of Mathematics
University of Chicago
Chicago, IL 60637
David Jerison
Department of Mathematics
Massachussetts Institute of Technology
Cambridge, MA 02139
Carlos E. Kenig
Department of Mathematics
University of Chicago
Chicago, IL 60637
Contributors
Contributors
Gilles Lebeau
Centre de Mathematiques
Ecole Polytechnique
F-91128 Plaiseau Cedex
France
Stephane Maes
IBM T. J. Watson Research Center
Human Language Technologies
Acoustic Processing Department
Yorktown Heights, NY 10598
P. Malliavin
10 rue Saint Louis en l'Isle
75004 Paris
France
Yves Meyer
CMLA-CNRS URA. 1611
ENS de Cachan
61 Avenue du President Wilson
94235 Cachan Cedex
France
Louis Nirenberg
Courant Institute
New York University
New York, NY 10012
Maciej Paluszyriski
Institute of Mathematics
University of Wroclaw
Poland
Cora Sadosky
Department of Mathematics
Howard University
Washington, DC 20059
Robert Seeley
Department of Mathematics
University of Massachussetts, Boston
Boston, MA 02125
351
352
Stephen Semmes
Department of Mathematics
Rice University
Houston, TX 77005
Elias M. Stein
Department of Mathematics
Princeton University
Princeton, NJ 08544
Gunther Uhlmann
Department of Mathematics
University of Washington
Seattle, WA 98195
Guido Weiss
Department of Mathematics
Washington University
St. Louis, MO 63130
Contributors
Index
abstract dynamical system, 30
affine invariance, 117
Ahlfors regular measure, 185
algebraic functions, 199
allocation problems, 117
almost-orthogonal decomposition, 13
amenable group, 56
analytic capacity, 183
anisotropic conductors, 334
Ap condition, 49
aperiodic, strictly, 34
area integral, 154
asymptotic expansions, 275
auditory system, 164
Bernstein theorem, 202; abstract,
204
Besov spaces, 243
Beurling-Ahlfors transform, 101
boundary behavior of harmonic
functions, 3
boundary value problems, 277
bounded angular ratio, 33
Brenier's factorization, 121
calculus, symbolic, 7
Calder6n decomposition formula,
164
Calder6n projection, 278
Calder6n-Zygmund theory:
Calder6n-Zygmund decomposi-
tion, 6, 42, 204; Calder6n-
Zygmund operators, 1, 5, 28;
Calder6n-Zygmund rotation
method, 8, 17, 153; multiparame-
ter theory, 18, 207; variable coef-
ficient singular integrals, 7
canonical transformations, 200
Carleman inequalities, 223
Carleson-Ilunt theorem, 66
Cauchy integral, 13
Cauchy operator, 185
Cauchy problem, uniqueness of, 8
cochlear filtering, 165
commutators, 12, 154
complex interpolation theorem, 34
complex methods, 3
complex variables, several, 17
conjugate functions, 1
constant, best possible, 111;
independent of dimension, 153
continuum mechanics, 121
convolution powers, 31
corona construction, 191
covering lemma techniques, 215
de Leeuw theorem, 52
Denjoy's conjecture, 186
Dirichlet to Neumann map, 129, 296
domains, unbounded, 263
doubling: measure, 286; property,
231
eigenfunctions, sums of, 223
eigenvalue, principal, 266
elastic materials, 323
elliptic theory: elliptic estimates, in
infinite dimensions, 151; elliptic
pseudodifferential operator, 276;
elliptic systems, 124; under mini-
mal regularity, 15
ergodic theory, 27
eta invariant, 279
Eulerian coordinates, 121
extension theorem, 203
extrapolation theorem, 212
extremal problems, 99