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Sharp constants for weak estimates of the Riesz Potentials
Abstract
In this paper we prove the sharp weak estimates for the Riesz potentials |Is(f)|Ln/(n−s),∞≤ γn,s vn(n−s)/n(Γ(s/2)Γ((n+2−s)/2))/Γ(n/2)|f|L1, when 0<s<min{n,2} and |Isf|Ln/(n−s),∞ ≤ γn,svn(n−s)/n|f|L1, when 2≤ s<n, where γn,s=2−sπ−n/2(Γ((n−s)/2))/Γ(s/2) and vn is the volume of the unit ball. The isoperimetric inequality for the Riesz capacity and the Newtonian isocapacitary inequality play a crucial role in our approach.
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BibTeX
@misc{tang2026sharp,
title = {{Sharp constants for weak estimates of the Riesz Potentials}},
author = {Hanli Tang and Yaojun Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.31043},
eprint = {2608.31043},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.31043},
}
RIS
TY - UNPB
TI - Sharp constants for weak estimates of the Riesz Potentials
AU - Hanli Tang
AU - Yaojun Wang
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2608.31043
UR - https://arxiv.org/abs/2608.31043
AB - In this paper we prove the sharp weak estimates for the Riesz potentials |I_s(f)|_L^n/(n−s),∞≤ γ_n,s v_n^(n−s)/n(Γ(s/2)Γ((n+2−s)/2))/Γ(n/2)|f|_L^1, when 0<s<min{n,2} and |I_sf|_L^n/(n−s),∞ ≤ γ_n,sv_n^(n−s)/n|f|_L^1, when 2≤ s<n, where γ_n,s=2^−sπ^−n/2(Γ((n−s)/2))/Γ(s/2) and v_n is the volume of the unit ball. The isoperimetric inequality for the Riesz capacity and the Newtonian isocapacitary inequality play a crucial role in our approach.
ER -