Abstract
In this paper, we introduce a class of singular Radon transforms on \({\mathbb {R}}^{n}\) with kernels supported in a subvariety in \({\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\) determined by a polynomial mapping from \({\mathbb {R}}^{n}\times {\mathbb {R}}^{n}\) into \({\mathbb {R}}^{n}\). 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 \(L(\log L)^{2}({\mathbb {S}} ^{n-1}\times {\mathbb {S}}^{n-1})\). The condition \(L(\log L)^{2}({\mathbb {S}} ^{n-1}\times {\mathbb {S}}^{n-1})\) is observed to be optimal in the sense that the power 2 can not be replaced by a smaller number.
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Communicated by Dachun Yang.
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Al-Salman, A. Singular integral operators with kernels supported in higher dimensional subvarieties. Banach J. Math. Anal. 16, 48 (2022). https://doi.org/10.1007/s43037-022-00201-w
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DOI: https://doi.org/10.1007/s43037-022-00201-w
Keywords
- Singular integral operators
- Rough kernels
- \(L^{p}\) estimates
- Singular Radon transforms
- Fourier transform
- Maximal functions