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Endpoint Lorentz Inequality for Riesz Potentials with Respect to General Measures

  • Pingxu Hu
  • Dachun Yang
  • Wen Yuan
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Abstract

Let n≥2, α∈(0,n), and μ be a locally finite nonnegative Borel measure on ℝn. We prove that, for any f∈ Cc∞(ℝn), |Iαf|Ln/(n−α),1(μ) ≤ Cn,α ∫ℝn |Rf(x)|[Mμ(x)]1−α/n dx, where Iα is the Riesz potential, R is the Riesz transform, Mμ is the Hardy--Littlewood maximal function of μ, and Cn,α is a positive constant depending only on n and α. This answers a question posed by Bortz, Moen, Olivo, Pérez, and Rela, with the case α=1 already known, and extends Spector's optimal unweighted Lorentz inequality to general measures. As applications, we obtain a Lorentz estimate for curl-free vector fields, Sobolev--Lorentz and Hardy inequalities with respect to general measures, and an A1-weighted Lorentz estimate with a sharp exponent of the A1 characteristic.

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Cite

BibTeXRIS
BibTeX
@misc{hu2026endpoint,
  title = {{Endpoint Lorentz Inequality for Riesz Potentials with Respect to General Measures}},
  author = {Pingxu Hu and Dachun Yang and Wen Yuan},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.23154},
  eprint = {2610.23154},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.23154},
}
RIS
TY  - UNPB
TI  - Endpoint Lorentz Inequality for Riesz Potentials with Respect to General Measures
AU  - Pingxu Hu
AU  - Dachun Yang
AU  - Wen Yuan
PY  - 2026
DA  - 2026/10/16
PB  - arXiv
DO  - 10.48550/arxiv.2610.23154
UR  - https://arxiv.org/abs/2610.23154
AB  - Let n≥2, α∈(0,n), and μ be a locally finite nonnegative Borel measure on ℝ^n. We prove that, for any f∈ C_c^∞(ℝ^n), |I_αf|_L^n/(n−α),1(μ) ≤ C_n,α ∫_ℝ^n |Rf(x)|[Mμ(x)]^1−α/n dx, where I_α is the Riesz potential, R is the Riesz transform, Mμ is the Hardy--Littlewood maximal function of μ, and C_n,α is a positive constant depending only on n and α. This answers a question posed by Bortz, Moen, Olivo, Pérez, and Rela, with the case α=1 already known, and extends Spector's optimal unweighted Lorentz inequality to general measures. As applications, we obtain a Lorentz estimate for curl-free vector fields, Sobolev--Lorentz and Hardy inequalities with respect to general measures, and an A_1-weighted Lorentz estimate with a sharp exponent of the A_1 characteristic.
ER  -