Inversion of parabolic potentials with \(L_ p\)-densities
type: Item
instance of: scholarly article
title: Inversion of parabolic potentials with \(L_ p\)-densities
zbMATH Open document ID: 0629.35004
DOI: 10.1007/BF01157030
author: V. Nogin
author: Boris Rubin
published in: Mathematical Notes
publication date: 1986-01-01T00:00:00Z
review text: The authors show a general construction of inverse operators (called parabolic hypersingular integrals) for the parabolic potentials \((H^{\alpha}\phi)(x,t)\), (\({\mathcal H}^{\alpha}\phi)(x,t)\), \(\alpha >0\). The latter are defined by \[ F[H^{\alpha}\phi](\xi,\tau)=(| \xi |^ 2+i\tau)^{-\alpha /2}F[\phi](\xi,\tau), \] \[ F[{\mathcal H}^{\alpha}\phi](\xi,\tau)=(1+| \xi |^ 2+i\tau)^{-\alpha /2}F[\phi](\xi,\tau), \] where F[\(\phi\) ] is the Fourier transform of \(\phi\) (x,t). (The authors give references for applications of such potentials.) The construction of the inverse of \(H^{\alpha}\phi\) and \({\mathcal H}^{\alpha}\phi\) with \(L_ p\)-densities gives explicit formulae for positive powers of the differential operators \(-\Delta_ x+\partial /\partial t\) and \(I-\Delta_ x+\partial /\partial t\) in \(L_ p(R^{n+1})\).
Mathematics Subject Classification ID: 35A22
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35K05
zbMATH DE Number: 4023803
zbMATH Keywords: inverse operators
zbMATH Keywords: parabolic hypersingular integrals
zbMATH Keywords: parabolic potentials
zbMATH Keywords: Fourier transform
zbMATH Keywords: explicit formulae
zbMATH Keywords: positive powers of the differential operators
reviewed by: Peter Poláčik
MaRDI profile type: Publication
cites work: Hypersingular Integrals and Parabolic Potentials
cites work: Q3699347
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description: scientific article; zbMATH DE number 4023803