Offbeat Integral Geometry on Symmetric Spaces

591 Pages·2013·4.24 MB·English

by  Valery V Volchkov

Valery V. Volchkov • Vitaly V. Volchkov
Offbeat Integral Geometry
on Symmetric Spaces
Valery V. Volchkov
Vitaly V. Volchkov
Department of Mathematics
Donetsk National University
Donetsk
Ukraine
ISBN 978-3-0348-0571-1 ISBN 978-3-0348-0572-8 (eBook)
DOI 10.1007/978-3-0348-0572-8
Springer Basel Heidelberg New York Dordrecht London

Library of Congress Control Number: 2013931100
Mathematics Subject Classification (2010): 33C05, 33C10, 33C15, 33C45, 33C55, 33C80, 35P10, 42A38,
42A55, 42A65, 42A75, 42A85, 42B35, 42C30, 42C15, 43A32, 43A45, 43A85, 43A90, 44A12, 44A15, 44A20,
44A35, 45A05, 46F12, 53C35

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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Part I Analysis on Symmetric Spaces
1 Preliminaries
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Distributions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3 Some transcendental functions . . . . . . . . . . . . . . . . . . . . . 18
1.4 Spherical harmonics. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5 The Gegenbauer polynomials . . . . . . . . . . . . . . . . . . . . . 34
1.6 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 38
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2 The Euclidean Case
2.1 Homeomorphisms with the generalized
transmutation property . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Some completeness results . . . . . . . . . . . . . . . . . . . . . . . 53
2.3 Systems of convolution equations . . . . . . . . . . . . . . . . . . . 57
2.4 Abel type integral equations . . . . . . . . . . . . . . . . . . . . . . 64
2.5 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 78
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3 Symmetric Spaces of the Non-compact Type
3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.2 The mapping 𝔄 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
𝛿
3.3 Uniqueness theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 101
3.4 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 103
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
v
vi Contents
4 Analogies for Compact Two-point Homogeneous Spaces
4.1 Introductory considerations . . . . . . . . . . . . . . . . . . . . . . 111
4.2 The functions Φ . . . . . . . . . . . . . . . . . . . . . . . . 117
𝜆,𝜂,𝑘,𝑚,𝑗
4.3 Generalized spherical transform . . . . . . . . . . . . . . . . . . . . 121
4.4 The mapping 𝔄 . . . . . . . . . . . . . . . . . . . . . . . . . . 127
𝑘,𝑚,𝑗
4.5 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 131
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
5 The Phase Space Associated to the Heisenberg Group
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
5.2 The functions 𝜙 . . . . . . . . . . . . . . . . . . . . . . . . . 138
𝜆,𝜂,𝑝,𝑞,𝑙
5.3 The transform ℱ(𝑝,𝑞) . . . . . . . . . . . . . . . . . . . . . . . . . . 143
𝑙
5.4 The mapping 𝔄 . . . . . . . . . . . . . . . . . . . . . . . . . . 148
(𝑝,𝑞),𝑙
5.5 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 150
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
Part II Offbeat Integral Geometry
1 Functions with Zero Ball Means on Euclidean Space
1.1 Simplest properties of functions with zero integrals over balls . . . 159
1.2 Uniqueness results . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
1.3 Description of functions in the classes 𝑉 (𝐵 ) and 𝑈 (𝐵 ) . . . . . 172
𝑟 𝑅 𝑟 𝑅
1.4 Local two-radii theorems . . . . . . . . . . . . . . . . . . . . . . . . 178
1.5 Functions with zero integrals over balls in a spherical annulus . . . 197
1.6 The Liouville property . . . . . . . . . . . . . . . . . . . . . . . . . 201
1.7 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 223
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
2 Two-radii Theorems in Symmetric Spaces
2.1 Auxiliary constructions . . . . . . . . . . . . . . . . . . . . . . . . . 229
2.2 The Jacobi functions . . . . . . . . . . . . . . . . . . . . . . . . . . 233
2.3 The operator 𝒜 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
2.4 Functions with vanishing averages over geodesic balls . . . . . . . . 241
2.5 A definitive version of the local two-radii theorem . . . . . . . . . . 248
2.6 A local two-radii theorem for weighted ball means . . . . . . . . . . 252
2.7 The compact case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
2.8 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 259
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
Contents vii
3 The Problem of Finding a Function from Its Ball Means
3.1 The Berenstein–Gay–Ygerresult . . . . . . . . . . . . . . . . . . . 265
3.2 The Berenstein–Gay–Ygertheorem generalized . . . . . . . . . . . 272
3.3 The case of a ball and a sphere . . . . . . . . . . . . . . . . . . . . 281
3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
A. One-radius theorem on two-point homogeneous spaces . . . . . . . . . 295
B. Over-determined interpolation problems . . . . . . . . . . . . . . . . . 300
3.5 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 303
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306
4 Sets with the Pompeiu Property
4.1 The Pompeiu problem . . . . . . . . . . . . . . . . . . . . . . . . . 309
4.2 Some examples of Pompeiu sets . . . . . . . . . . . . . . . . . . . . 313
4.3 A characterizationof Pompeiu sets . . . . . . . . . . . . . . . . . . 324
4.4 The local Pompeiu property . . . . . . . . . . . . . . . . . . . . . . 327
4.5 Upper and lower estimates for ℛ(𝐴) . . . . . . . . . . . . . . . . . 332
4.6 The value of ℛ(𝐴) for some subsets of the plane . . . . . . . . . . . 342
4.7 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 353
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
5 Functions with Zero Integrals over Polytopes
5.1 The value ℛ(𝐴) for convex polytopes . . . . . . . . . . . . . . . . . 359
5.2 The value ℛ(𝐴) for rectangular parallelepipeds . . . . . . . . . . . 365
5.3 The class 𝔓(𝐴,𝐵 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
𝑟
5.4 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 385
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391
6 Ellipsoidal Means
6.1 Requisite results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
6.2 Functions with zero averagesover ellipsoids . . . . . . . . . . . . . 397
6.3 The value of ℛ(𝐴) for ellipsoids . . . . . . . . . . . . . . . . . . . . 406
6.4 Reconstruction of a function by means of its integrals
over ellipsoids of revolution . . . . . . . . . . . . . . . . . . . . . . 417
6.5 Mean-value characterizationof pluriharmonic and
separately harmonic functions . . . . . . . . . . . . . . . . . . . . . 422
6.6 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 429
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432
viii Contents
7 The Pompeiu Property on a Sphere
7.1 Auxiliary assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
7.2 The functions 𝜓 and Ψ𝑘,𝑙 . . . . . . . . . . . . . . . . . . . . . . 443
𝜈,𝑘 𝜈
7.3 Basic properties of the class 𝒱 (𝐵 ). . . . . . . . . . . . . . . . . . 447
𝑟 𝑅
7.4 Two-radii theorems for the class 𝒱 (𝐵 ) . . . . . . . . . . . . . . . 452
𝑟 𝑅
7.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 457
A. Conical injectivity sets of the spherical Radon transform . . . . . . . . 457
B. Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 462
7.6 The hemispherical transform . . . . . . . . . . . . . . . . . . . . . . 464
7.7 Measures with the Pompeiu property . . . . . . . . . . . . . . . . . 471
7.8 The Pompeiu property for spherical polygons . . . . . . . . . . . . 478
7.9 Extremal versions of the Pompeiu problem on a sphere . . . . . . . 483
7.10 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 491
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
8 The Pompeiu Transform on Symmetric Spaces and Groups
8.1 Main problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 497
8.2 Pompeiu transforms for distributions with support
on a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
8.3 The Pompeiu problem for groups . . . . . . . . . . . . . . . . . . . 511
8.4 Spherical means on the reduced Heisenberg group and
the Pompeiu problem with a twist. . . . . . . . . . . . . . . . . . . 515
8.5 Pompeiu’s problem on discrete space . . . . . . . . . . . . . . . . . 518
8.6 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 521
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
9 Pompeiu Transforms on Manifolds
9.1 Pompeiu transforms on a complete Riemannian manifold . . . . . . 527
9.2 Radial Pompeiu transforms on a locally symmetric space . . . . . . 531
9.3 Freak theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
9.4 General Pompeiu transforms on locally symmetric spaces . . . . . . 540
9.5 Exercises and further results . . . . . . . . . . . . . . . . . . . . . . 553
Bibliographical notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 557
Author Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579
Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585
Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 589
Preface
A typical problem in offbeat integral geometry is as follows. Let ℝ𝑛 be the 𝑛-
dimensional real Euclidean space, M(𝑛) the group of Euclidean motions of ℝ𝑛,
and𝐴aboundedsubsetofℝ𝑛ofpositiveLebesguemeasure.Considerthefollowing
problem: describe the class of locally integrable functions 𝑓 such that
∫
𝑓(𝑥)𝑑𝑥=0 (∗)
𝑔𝐴
for each 𝑔 ∈ M(𝑛). This problem has various generalizations and modifications.
Forinstance,inplaceof(∗)onecaninvestigatesolutionsofasystemofconvolution
equations with fixed distributions.
The first studies in this area were carried out in 1929, by the Rumanian
mathematician D. Pompeiu, who investigated the question on the existence of
non-trivial functions satisfying (∗) for some 𝐴. D. Pompeiu erroneously assumed
thatif𝐴isaballthenequation(∗)hasonlythetrivialsolution.Lateron,F.John
showed that a function 𝑓 ∈𝐶∞ with zero integrals over all balls of fixed radius 𝑟
is uniquely defined by its values in the ball of radius 𝑟.
Afterthat,F.John,J.Delsarte,L.H¨ormander,L.Zalcman,C.A.Berenstein,
andotherauthorsdiscovereddeepconnectionsbetweenthese questionsandmany
areas of contemporary mathematics and its applications.
In recent years, local versions of the above problem have become a point of
attention, in which a function 𝑓 is defined in a bounded domain 𝒪 and equal-
ity (∗) holds for 𝑔 ∈M(𝑛): 𝑔𝐴⊂𝒪. The transition from the global to the local
case makes the problem considerably more complicated, which is related to the
breakdown of the structure of a group action on the solution set of equation (∗).
Among first results in this direction we point out H¨ormander’s approximation
theorem for solutions of a convolution equation on convex domains and the local
two-radii theorem by C.A. Berenstein and R. Gay. Until recently research in this
areawas carriedoutmostly using the technique ofthe Fourier transformandcor-
responding methods of complex analysis. A remarkable result by the first author
at the end of the last century was the development of a universal method for the
completesolutionofmanyproblemsofthis kind,whichallowedone,inparticular,
toremovevirtually allsuperfluous assumptionsimposedbyhis predecessors.This
method is based on the representation of solutions of a broad class of convolu-
tion equations by series in special functions. The results obtained by this method
ix