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Pointwise estimates for an anisotropic product type singular integral on the sphere
Abstract
Let Γ={1,…,d}, d≥ 2. For each j∈ Γ, let ℓj≥ 1 be an integer and let αj∈ (1/2,1). We obtain a pointwise estimate for the integral ∫𝕊d−1∏j∈ Γ |xℓjj−ξj|−αj d σ(x), in the region {ξ∈ ℝd: ξj≠ 0, j∈Γ, ∑j∈ F|ξj|2/ℓj≠ 1, F⊆ Γ}. These estimates are used to prove a Stein-Tomas type restriction theorem for the oscillatory integral operator Tf(x)=∫ℝdeimatḣh(x,y)f(y)dy, with the phase h(x,y)=∑j∈Γ aj xjℓj yjmj, where, for each j∈ Γ, aj≠ 0 is a real number and mj>1 is an integer.
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BibTeX
@misc{abdelhakim2026pointwise,
title = {{Pointwise estimates for an anisotropic product type singular integral on the sphere}},
author = {Ahmed A. Abdelhakim},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.11942},
eprint = {2611.11942},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.11942},
}
RIS
TY - UNPB
TI - Pointwise estimates for an anisotropic product type singular integral on the sphere
AU - Ahmed A. Abdelhakim
PY - 2026
DA - 2026/11/09
PB - arXiv
DO - 10.48550/arxiv.2611.11942
UR - https://arxiv.org/abs/2611.11942
AB - Let Γ={1,…,d}, d≥ 2. For each j∈ Γ, let ℓ_j≥ 1 be an integer and let α_j∈ (1/2,1). We obtain a pointwise estimate for the integral ∫_𝕊^d−1∏_j∈ Γ |x^ℓ_j_j−ξ_j|^−α_j d σ(x), in the region {ξ∈ ℝ^d: ξ_j≠ 0, j∈Γ, ∑_j∈ F|ξ_j|^2/ℓ_j≠ 1, F⊆ Γ}. These estimates are used to prove a Stein-Tomas type restriction theorem for the oscillatory integral operator Tf(x)=∫_ℝ^de^imatḣh(x,y)f(y)dy, with the phase h(x,y)=∑_j∈Γ a_j x_j^ℓ_j y_j^m_j, where, for each j∈ Γ, a_j≠ 0 is a real number and m_j>1 is an integer.
ER -