On integral representations of functions determined by one class of pseudodifferential operators
type: Item
instance of: scholarly article
title: On integral representations of functions determined by one class of pseudodifferential operators
zbMATH Open document ID: 0854.35138
DOI: 10.1007/BF02112543
author: G. A. Shmyrëv
published in: Siberian Mathematical Journal
publication date: 1997-01-05T00:00:00Z
review text: We construct an integral representation for pseudodifferential operators \[ p(D) u= \int_{E_n} e^{ix\xi} p(i\xi) \widehat u(\xi) d\xi, \] with \(p(i\xi)\) a continuous function in \(E_n\) satisfying some conditions. Earlier, S. V. Uspenskiĭ (1972) obtained such a representation for hypoelliptic differential operators and pseudodifferential operators with symbol homogeneous with respect to some vector \(\alpha= (\alpha_1, \alpha_2,\dots, \alpha_n)\). The kernels of the integral representations constructed in (loc. cit.) are regularizations with respect to a parameter of fundamental solutions to the corresponding equations. This approach proved convenient for studying the differential properties and the behavior at infinity of solutions to quasielliptic equations as well as for studying boundary value problems for such equations.
Mathematics Subject Classification ID: 35S05
Mathematics Subject Classification ID: 47G30
zbMATH DE Number: 923570
zbMATH Keywords: integral representation for pseudodifferential operators
MaRDI profile type: Publication
cites work: Orthogonality conditions and behavior at infinity of solutions of a class of quasielliptic equations
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description: scientific article; zbMATH DE number 923570