Singular integral operators with kernels supported in higher dimensional subvarieties

Ahmad Al-Salman

doi:10.1007/s43037-022-00201-w

Singular integral operators with kernels supported in higher dimensional subvarieties

Ahmad Al-Salman^*

^*Corresponding author for this work

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In this paper, we introduce a class of singular Radon transforms on Rⁿ with kernels supported in a subvariety in Rⁿ× Rⁿ determined by a polynomial mapping from Rⁿ× Rⁿ into Rⁿ. The class of considered operators is related to the composition of homogeneous singular integral operators. We prove that the operators are bounded on L^p provided that the kernels are rough in L(log L) ²(Sⁿ^-¹× Sⁿ^-¹). The condition L(log L) ²(Sⁿ^-¹× Sⁿ^-¹) is observed to be optimal in the sense that the power 2 can not be replaced by a smaller number.

Original language	English
Article number	48
Journal	Banach Journal of Mathematical Analysis
Volume	16
Issue number	3
DOIs	https://doi.org/10.1007/s43037-022-00201-w
Publication status	Published - Jul 2022

Keywords

Fourier transform
L estimates
Maximal functions
Rough kernels
Singular integral operators
Singular Radon transforms

ASJC Scopus subject areas

Analysis
Algebra and Number Theory

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10.1007/s43037-022-00201-w

Cite this

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title = "Singular integral operators with kernels supported in higher dimensional subvarieties",

abstract = "In this paper, we introduce a class of singular Radon transforms on Rn with kernels supported in a subvariety in Rn× Rn determined by a polynomial mapping from Rn× Rn into Rn. The class of considered operators is related to the composition of homogeneous singular integral operators. We prove that the operators are bounded on Lp provided that the kernels are rough in L(log L) 2(Sn-1× Sn-1). The condition L(log L) 2(Sn-1× Sn-1) is observed to be optimal in the sense that the power 2 can not be replaced by a smaller number.",

keywords = "Fourier transform, L estimates, Maximal functions, Rough kernels, Singular integral operators, Singular Radon transforms",

author = "Ahmad Al-Salman",

note = "Publisher Copyright: {\textcopyright} 2022, Tusi Mathematical Research Group (TMRG).",

year = "2022",

month = jul,

doi = "10.1007/s43037-022-00201-w",

language = "English",

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T1 - Singular integral operators with kernels supported in higher dimensional subvarieties

AU - Al-Salman, Ahmad

PY - 2022/7

Y1 - 2022/7

N2 - In this paper, we introduce a class of singular Radon transforms on Rn with kernels supported in a subvariety in Rn× Rn determined by a polynomial mapping from Rn× Rn into Rn. The class of considered operators is related to the composition of homogeneous singular integral operators. We prove that the operators are bounded on Lp provided that the kernels are rough in L(log L) 2(Sn-1× Sn-1). The condition L(log L) 2(Sn-1× Sn-1) is observed to be optimal in the sense that the power 2 can not be replaced by a smaller number.

AB - In this paper, we introduce a class of singular Radon transforms on Rn with kernels supported in a subvariety in Rn× Rn determined by a polynomial mapping from Rn× Rn into Rn. The class of considered operators is related to the composition of homogeneous singular integral operators. We prove that the operators are bounded on Lp provided that the kernels are rough in L(log L) 2(Sn-1× Sn-1). The condition L(log L) 2(Sn-1× Sn-1) is observed to be optimal in the sense that the power 2 can not be replaced by a smaller number.

KW - Fourier transform

KW - L estimates

KW - Maximal functions

KW - Rough kernels

KW - Singular integral operators

KW - Singular Radon transforms

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DO - 10.1007/s43037-022-00201-w

M3 - Article

AN - SCOPUS:85133670634

SN - 1735-8787

VL - 16

JO - Banach Journal of Mathematical Analysis

JF - Banach Journal of Mathematical Analysis

IS - 3

M1 - 48

ER -

Singular integral operators with kernels supported in higher dimensional subvarieties

Abstract

Keywords

ASJC Scopus subject areas

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