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arXiv · 2610.22540

Hardy-Sobolev-Maz'ya inequalities for $\boldsymbol{1<p<2}$

Avas Banerjee·Michał Kijaczko·Vivek Sahu
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Abstract

We establish Hardy--Sobolev--Maz'ya inequalities in the subquadratic range $1<p<2$ for singularities supported on affine subspaces and for arbitrary domains. We first prove a Hardy--Sobolev--Maz'ya inequality for singularities supported on affine subspaces of arbitrary codimension $1\leq k<N$, showing that the deficit in the sharp Hardy inequality controls a suitable weighted Sobolev norm. Combining the same convexity estimate with the one-dimensional approach of Frank and Loss in [9], we also establish Hardy--Sobolev--Maz'ya inequalities on arbitrary domains with the mean distance to the boundary and thereby solve a problem posed in [9]. These results complete the previously unresolved subquadratic range $1<p<2$ and, together with the known results for $p\geq 2$, provide a unified Hardy--Sobolev--Maz'ya theory for all $1<p<N$ associated with plane singularities and general domains.

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Avas Banerjee, Michał Kijaczko, Vivek Sahu. 2026-10-21. Hardy-Sobolev-Maz'ya inequalities for $\boldsymbol{1<p<2}$. https://arxiv.org/abs/2610.22540

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