Logarithmic fractional Hardy inequalities

Abstract

We establish sharp fractional Hardy inequalities in the critical regimes sp=d and sp=1, with explicit logarithmic weight functions. In the whole-space setting, for p>d≥ 1, sp=d, we prove that ∫_ℝ^d ∫_ℝ^d (|u(x)−u(y)|^p)/(|x−y|^2d) dx dy ≥ ((p−1)/p)^p ∫_{B_R}|u(x)|^p W_p,d,R(x) dx, for u∈ C_c(B_R), where W_p,d,R is a logarithmic-type weight and the constant is sharp. We also show that the corresponding inequality with the simpler weight |x|^−d(log(R/|x|))^−p cannot hold with any positive constant. Furthermore, we derive two-sided estimates and asymptotic formulae for W_p,d,R, including its limiting connection with the classical Hardy inequality. In the half-space ℝ^d₊, we establish an analogous sharp fractional Hardy inequality in the critical case sp=1, with an explicit logarithmic weight depending on the distance to the boundary. We also investigate the asymptotic behaviour of this weight, obtaining estimates that describe its behaviour near the boundary. These results elucidate the role of logarithmic weights in critical fractional Hardy inequalities in both whole-space and half-space settings.

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BibTeX RIS

BibTeX

@misc{kijaczko2026logarithmic,
  title = {{Logarithmic fractional Hardy inequalities}},
  author = {Michał Kijaczko and Vivek Sahu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.38420},
  eprint = {2610.38420},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.38420},
}

RIS

TY  - UNPB
TI  - Logarithmic fractional Hardy inequalities
AU  - Michał Kijaczko
AU  - Vivek Sahu
PY  - 2026
DA  - 2026/10/28
PB  - arXiv
DO  - 10.48550/arxiv.2610.38420
UR  - https://arxiv.org/abs/2610.38420
AB  - We establish sharp fractional Hardy inequalities in the critical regimes sp=d and sp=1, with explicit logarithmic weight functions. In the whole-space setting, for p>d≥ 1, sp=d, we prove that ∫_ℝ^d ∫_ℝ^d (|u(x)−u(y)|^p)/(|x−y|^2d) dx dy ≥ ((p−1)/p)^p ∫_B_R|u(x)|^p W_p,d,R(x) dx, for u∈ C_c(B_R), where W_p,d,R is a logarithmic-type weight and the constant is sharp. We also show that the corresponding inequality with the simpler weight |x|^−d(log(R/|x|))^−p cannot hold with any positive constant. Furthermore, we derive two-sided estimates and asymptotic formulae for W_p,d,R, including its limiting connection with the classical Hardy inequality. In the half-space ℝ^d_+, we establish an analogous sharp fractional Hardy inequality in the critical case sp=1, with an explicit logarithmic weight depending on the distance to the boundary. We also investigate the asymptotic behaviour of this weight, obtaining estimates that describe its behaviour near the boundary. These results elucidate the role of logarithmic weights in critical fractional Hardy inequalities in both whole-space and half-space settings.
ER  -

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