Failure of Pólya-Szegő inequality for weighted nonlocal seminorm and on nonlocal C-K-N inequality

Abstract

We create counterexamples to show that Pólya-Szegő type of inequality fails for the weighted nonlocal seminorm [u]_{W^{s,p, α}(ℝ^d)} := ( ∫_ℝ^d ∫_ℝ^d (|x|^{α₁ p} |y|^{α₂ p} |u(x)−u(y)|^p)/(|x−y|^d+sp) dx dy )^1/p , in all three regimes α₁ +α₂ < 0, α₁ +α₂ = 0, and α₁ +α₂ >0. These counterexamples are such that it completely settles the first two cases for all parameters. We also provide necessary and sufficient conditions on the parameters for $ [u]_{W^{s,p, α}(\mathbb{R}^d)} < \infty$, thus extending an earlier known result of [Serena Dipierro and Enrico Valdinoci, A density property for fractional weighted Sobolev spaces]. We also provide another proof of the fractional C-K-N type inequality, for most of the parameters (Hardy and Sobolev are included), using a familiar technique which is involved in the proof of fractional Sobolev inequality of [Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des sciences mathematiques] together with a symmetrization technique.

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BibTeX RIS

BibTeX

@misc{roy2026failure,
  title = {{Failure of Pólya-Szegő inequality for weighted nonlocal seminorm and on nonlocal C-K-N inequality}},
  author = {Prosenjit Roy and Vivek Sahu},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.08189},
  eprint = {2611.08189},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.08189},
}

RIS

TY  - UNPB
TI  - Failure of Pólya-Szegő inequality for weighted nonlocal seminorm and on nonlocal C-K-N inequality
AU  - Prosenjit Roy
AU  - Vivek Sahu
PY  - 2026
DA  - 2026/11/05
PB  - arXiv
DO  - 10.48550/arxiv.2611.08189
UR  - https://arxiv.org/abs/2611.08189
AB  - We create counterexamples to show that Pólya-Szegő type of inequality fails for the weighted nonlocal seminorm [u]_W^s,p, α(ℝ^d) := ( ∫_ℝ^d ∫_ℝ^d (|x|^α_1 p |y|^α_2 p |u(x)−u(y)|^p)/(|x−y|^d+sp) dx dy )^1/p , in all three regimes α_1 +α_2 < 0, α_1 +α_2 = 0, and α_1 +α_2 >0. These counterexamples are such that it completely settles the first two cases for all parameters. We also provide necessary and sufficient conditions on the parameters for $ [u]_{W^{s,p, α}(\mathbb{R}^d)} < \infty$, thus extending an earlier known result of [Serena Dipierro and Enrico Valdinoci, A density property for fractional weighted Sobolev spaces]. We also provide another proof of the fractional C-K-N type inequality, for most of the parameters (Hardy and Sobolev are included), using a familiar technique which is involved in the proof of fractional Sobolev inequality of [Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des sciences mathematiques] together with a symmetrization technique.
ER  -

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