A structure theorem for fundamental solutions of analytic multipliers in Rn

David Scott Winterrose

doi:10.1007/s11868-024-00586-2

A structure theorem for fundamental solutions of analytic multipliers in Rⁿ

David Scott Winterrose^*

^*Corresponding author for this work

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Abstract

Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier Op(p) of order d>0, real-analytic near p^-1(0), has a fundamental solution μ₀. We give an integral representation of μ₀ in terms of the resolutions supplied by Hironaka’s theorem. This μ₀ is weakly approximated in H^t_loc(Rⁿ) for t<d-n/2 by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

Original language	English
Article number	17
Journal	Journal of Pseudo-Differential Operators and Applications
Volume	15
Issue number	17
ISSN	1662-9981
DOIs	https://doi.org/10.1007/s11868-024-00586-2
Publication status	Published - 2024

Keywords

Fundamental solutions
PDE
Pseudo-differential equations

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title = "A structure theorem for fundamental solutions of analytic multipliers in Rn",

abstract = "Using a version of Hironaka{\textquoteright}s resolution of singularities for real-analytic functions, any elliptic multiplier Op(p) of order d>0, real-analytic near p-1(0), has a fundamental solution μ0. We give an integral representation of μ0 in terms of the resolutions supplied by Hironaka{\textquoteright}s theorem. This μ0 is weakly approximated in Htloc(Rn) for t<d-n/2 by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.",

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T1 - A structure theorem for fundamental solutions of analytic multipliers in Rn

AU - Winterrose, David Scott

N1 - Publisher Copyright: © The Author(s) 2024.

PY - 2024

Y1 - 2024

N2 - Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier Op(p) of order d>0, real-analytic near p-1(0), has a fundamental solution μ0. We give an integral representation of μ0 in terms of the resolutions supplied by Hironaka’s theorem. This μ0 is weakly approximated in Htloc(Rn) for t<d-n/2 by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

AB - Using a version of Hironaka’s resolution of singularities for real-analytic functions, any elliptic multiplier Op(p) of order d>0, real-analytic near p-1(0), has a fundamental solution μ0. We give an integral representation of μ0 in terms of the resolutions supplied by Hironaka’s theorem. This μ0 is weakly approximated in Htloc(Rn) for t<d-n/2 by a sequence from a Paley-Wiener space. In special cases of global symmetry, the obtained integral representation can be made fully explicit, and we use this to compute fundamental solutions for two non-polynomial symbols.

KW - Fundamental solutions

KW - PDE

KW - Pseudo-differential equations

U2 - 10.1007/s11868-024-00586-2

DO - 10.1007/s11868-024-00586-2

M3 - Journal article

AN - SCOPUS:85185939053

SN - 1662-9981

VL - 15

JO - Journal of Pseudo-Differential Operators and Applications

JF - Journal of Pseudo-Differential Operators and Applications

IS - 17

M1 - 17

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