Operator Theory, Pseudo-Differential Equations, and Mathematical Physics: The Vladimir Rabinovich Anniversary Volume

425 Pages·2013·5.76 MB·English

by  Alexandra Antoniouk

Operator Theory
Advances and Applications
228
Yuri I. Karlovich
Luigi Rodino
Bernd Silbermann
Ilya M. Spitkovsky
Editors
Operator Theory,
Pseudo-Differential
Equations, and
Mathematical
Physics
The Vladimir Rabinovich Anniversary
Volume

Operator Theory: Advances and Applications
Volume 228

Founded in 1979 by Israel Gohberg

Editors:
Joseph A. Ball (Blacksburg, VA, USA)
Harry Dym (Rehovot, Israel)
Marinus A. Kaashoek (Amsterdam, The Netherlands)
Heinz Langer (Vienna, Austria)
Christiane Tretter (Bern, Switzerland)

Associate Editors: Honorary and Advisory Editorial Board:
Vadim Adamyan (Odessa, Ukraine) Lewis A. Coburn (Buffalo, NY, USA)
Albrecht Böttcher (Chemnitz, Germany) Ciprian Foias (College Station, TX, USA)
B. Malcolm Brown (Cardiff, UK) J.William Helton (San Diego, CA, USA)
Raul Curto (Iowa, IA, USA) Thomas Kailath (Stanford, CA, USA)
Fritz Gesztesy (Columbia, MO, USA) Peter Lancaster (Calgary, Canada)
Pavel Kurasov (Lund, Sweden) Peter D. Lax (New York, NY, USA)
Leonid E. Lerer (Haifa, Israel) Donald Sarason (Berkeley, CA, USA)
Vern Paulsen (Houston, TX, USA) Bernd Silbermann (Chemnitz, Germany)
Mihai Putinar (Santa Barbara, CA, USA) Harold Widom (Santa Cruz, CA, USA)
Leiba Rodman (Williamsburg, VA, USA)
Ilya M. Spitkovsky (Williamsburg, VA, USA)
Yuri I. Karlovich
Luigi Rodino
Bernd Silbermann
Ilya M. Spitkovsky
Editors
Operator Theory,
Pseudo-Differential Equations,
and Mathematical Physics
The Vladimir Rabinovich Anniversary Volume
Editors
Yuri I. Karlovich Luigi Rodino
Facultad de Ciencias Dipartimento di Matematica
Universidad Autónoma del Estado de Morelos Università di Torino
Cuernavaca, Morelos Torino
Mexico Italy

Bernd Silbermann Ilya M. Spitkovsky
Fakultät für Mathematik Department of Mathematics
Technische Universität Chemnitz College of William and Mary
Chemnitz Williamsburg, VA
Germany USA

ISBN 978-3-0348-0536-0 ISBN 978-3-0348-0537-7 (eBook)
DOI 10.1007/978-3-0348-0537-7
Springer Basel Heidelberg New York Dordrecht London
Library of Congress Control Number: 2012951622

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Contents
Preface .................................................................. vii
S. Samko
Vladimir Rabinovich: a Mathematician, Colleague and Friend ....... ix
A. Antoniouk and N. Tarkhanov
The Dirichlet Problem for the Heat Equation in Domains
with Cuspidal Points on the Boundary .............................. 1
U. Battisti, T. Gramchev, L. Rodino and S. Pilipovi´c
Globally Bisingular Elliptic Operators ............................... 21
A. B¨ottcher and A. Per¨al¨a
The Index Formula of Douglas for Block Toeplitz Operators
on the Bergman Space of the Ball ................................... 39
G. Burlak and V. Rabinovich
The Green Function and Optical Field Enhancement
in a Multilayered Microsphere with Metamaterial ................... 57
O. Chkadua, R. Duduchava and D. Kapanadze
Potential Methods for Anisotropic Pseudo-Maxwell Equations
in Screen Type Problems ........................................... 73
S. Dobrokhotov, D. Minenkov, V. Nazaikinskii and B. Tirozzi
Functions of Noncommuting Operators in an
Asymptotic Problem for a 2D Wave Equation with
Variable Velocity and Localized Right-hand Side .................... 95
S. Grudsky and A. Rybkin
On Toeplitz and Hankel Operators with Oscillatory Symbols
Containing Blaschke Products and Applications
to the KdV Equation ............................................... 127
N. Habal and B.-W. Schulze
Mellin Quantisation in Corner Operators ............................ 151
vi Contents
A.Yu. Karlovich and I.M. Spitkovsky
Pseudodifferential Operators on Variable Lebesgue Spaces ........... 173
Yu.I. Karlovich and I. Loreto Hern´andez
On Convolution Type Operators with Piecewise Slowly
Oscillating Data .................................................... 185
V.V. Kravchenko and S.M. Torba
Transmutations and Spectral Parameter Power Series
in Eigenvalue Problems ............................................. 209
V.V. Kucherenko
Estimate from Below for the Growth of Solution to the
Navier-Stokes Equation When the Solution Blows Up ............... 239
A. Kufner, L.-E. Persson and N. Samko
Some New Scales of Weight Characterizations
of Hardy-type Inequalities .......................................... 261
G.M. N’Gu´er´ekata and A. Pankov
Almost Periodic Elliptic Equations: Sub- and Super-solutions ....... 275
H. Rafeiro, N. Samko and S. Samko
Morrey-CampanatoSpaces: an Overview ............................ 293
S. Roch
Arveson Dichotomy and Essential Fractality ......................... 325
Y. Safarov
Lower Bounds for the Counting Function of Integral Operators ...... 343
A. Sa´nchez-Nungaray and N. Vasilevski
Toeplitz Operators on the Bergman Spaces with
Pseudodifferential Defining Symbols ................................. 355
M. Seidel and B. Silbermann
Finite Sections of Band-dominated Operators –
Norms, Condition Numbers and Pseudospectra ...................... 375
F.-O. Speck
On the Reduction of Linear Systems Related
to Boundary Value Problems ........................................ 391
Preface
This volume is a collection of papers devoted to the 70th birthday of Professor
Vladimir Rabinovich. The opening article (by Stefan Samko) contains a short
biography of Vladimir Rabinovich, along with some personal recollections and
bibliography of his work. It is followed by twenty research and survey papers in
various branches of Analysis (pseudodifferential operators and partial differential
equations, Toeplitz, Hankel, and convolution type operators, variable Lebesgue
spaces, etc.) close to Professor Rabinovich’s research interests. Many of them are
written by participants of the International workshop “Analysis, Operator The-
ory, and Mathematical Physics” (Ixtapa, Mexico, January 23–27, 2012) having a
longhistoryofscientificcollaborationwithVladimirRabinovich,andarepartially
based on the talks presented there.
The volume will be of great interest to researchersand graduate students in
Differential Equations, Operator Theory, Functional and Harmonic Analysis.
The Editors

OperatorTheory:
Advances andApplications,Vol.228,ix–xxv
⃝c 2013SpringerBasel
Vladimir Rabinovich:
a Mathematician, Colleague and Friend
Stefan Samko
Dedicated to the 70th anniversary of Professor Vladimir Rabinovich
Itwasmypleasuretoaccepttheinvitationtowritethis introductorypapertothe
volumededicatedtothe70thanniversaryofVladimirSamuilovichRabinovich,my
university colleague in the “previous life” in the Soviet Union, collaborator and
friend for more than 40 years.
Vladimir Rabinovich, known to his friends and most of the colleagues as
VolodyaRabinovich,wasborninKiev onSeptember 2, 1940,where his childhood
passed.Inthe beginning ofthe Nazi invasionofUSSR in1941,whenmany people
were evacuated from the Western regions to the interior parts of the country, his
V. Rabinovich at his desk, M´exico City, May 2012.