arXiv subjects

Antoni Szczukiewicz

Publications and source records attributed to Antoni Szczukiewicz.

3 recordsLinked to original sources

arXiv2026

Fractional Hardy--Maz'ya inequality on a half-space

The main purpose of this article is to provide a fractional counterpart of the well-known Maz'ya inequality on the half-space, that is $$ \int_{\mathbb{R}^{d}_{+}}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dy\,dx\ge\mathcal{D}_{d,s,p}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)|^p}{x_d^{sp}}dx+C_{d,s,p,τ}\int_{\mathbb{R}^{d}_{+}}\frac{|u(x)|^p}{x_{d}^{sp-τ}\left(x_{d-1}^2+x_d^2\right)^{τ/2}}dx, $$ where $\mathcal{D}_{d,s,p}$ stands for the sharp constant in the fractional Hardy inequality on a half-space $\mathbb{R}^{d}_{+}$. We also obtain a similar result in the setting of Sobolev--Bregman forms.

math.AP↗

arXiv2026

Critical discrete Hardy inequalities in $L^p$

We establish ground-state representations and critical Hardy inequalities in the discrete $L^p$ setting. In particular, we construct critical Hardy weights for the discrete Dirichlet Laplacian on the half-line and the discrete fractional Laplacian on the integers for all $p\in(1,\infty)$. Our approach uses Sobolev--Bregman forms, for which the ground-state representations are exact identities and the corresponding Hardy weights are expressed through linear operators acting on powers of positive superharmonic functions.

math.CA↗

arXiv2026

Critical fractional Hardy inequalities

We give a roadmap for the study of critical fractional Hardy inequalities for Dirichlet and Sobolev--Bregman forms. We focus on homogeneous forms on the real line, in particular the form of the fractional Laplacian and the Servadei--Valdinoci form. We prove criticality results in both the Dirichlet and Sobolev--Bregman settings.

math.AP↗