Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture
Author:   A. V. Sobolev
Publisher:   American Mathematical Society
Volume:   222, 1043
ISBN:  

9780821884874


Pages:   104
Publication Date:   28 May 2013
Format:   Paperback
Availability:   Temporarily unavailable   Availability explained
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Pseudo-Differential Operators with Discontinuous Symbols: Widom's Conjecture


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Overview

Relying on the known two-term quasiclassical asymptotic formula for the trace of the function f(A) of a Wiener-Hopf type operator A in dimension one, in 1982 H. Widom conjectured a multi-dimensional generalisation of that formula for a pseudo-differential operator A with a symbol a(x,ξ) having jump discontinuities in both variables. In 1990 he proved the conjecture for the special case when the jump in any of the two variables occurs on a hyperplane. The present paper provides a proof of Widom's Conjecture under the assumption that the symbol has jumps in both variables on arbitrary smooth bounded surfaces.

Full Product Details

Author:   A. V. Sobolev
Publisher:   American Mathematical Society
Imprint:   American Mathematical Society
Volume:   222, 1043
Weight:   0.300kg
ISBN:  

9780821884874


ISBN 10:   0821884875
Pages:   104
Publication Date:   28 May 2013
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Temporarily unavailable   Availability explained
The supplier advises that this item is temporarily unavailable. It will be ordered for you and placed on backorder. Once it does come back in stock, we will ship it out to you.

Table of Contents

Introduction Main result Estimates for PDO's with smooth symbols Trace-class estimates for operators with non-smooth symbols} Further trace-class estimates for operators with non-smooth symbols A Hilbert-Schmidt class estimate Localisation Model problem in dimension one Partitions of unity, and a reduction to the flat boundary Asymptotics of the trace (9.1) Proof of Theorem 2.9 Closing the asymptotics: Proof of Theorems 2.3 and 2.4 Appendix 1: A lemma by H. Widom Appendix 2: Change of variables Appendix 3: A trace-class formula Appendix 4: Invariance with respect to the affine change of variables Bibliography

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A. V. Sobolev, University College London, UK

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