Stein-Weiss type inequalities with partial variable weight on the upper half space and related weighted inequalities

TL;DR

This paper establishes new Stein-Weiss type inequalities with partial variable weights on the upper half space, proves the existence of extremal functions, and explores their symmetry, explicit forms, and applications to weighted Sobolev inequalities.

Contribution

It introduces a novel class of weighted inequalities on the upper half space, proving extremal functions exist without Riesz rearrangement, and analyzes their symmetry and explicit forms.

Findings

Existence of extremal functions proved via concentration compactness.

Extremal functions exhibit cylindrical symmetry and explicit boundary forms.

Derived weighted Sobolev inequalities for Laplacian and fractional Laplacian.

Abstract

In this paper, we establish a class of Stein-Weiss type inequality with partial variable weight functions on the upper half space using a weighted Hardy type inequality. Overcoming the impact of weighted functions, the existence of extremal functions is proved via the concentration compactness principle, whereas Riesz rearrangement inequality is not available. Moreover, the cylindrical symmetry with respect to $t$-axis and the explicit forms on the boundary of all nonnegative extremal functions are discussed via the method of moving planes and method of moving spheres, as well as, regularity results are obtained by the regularity lift lemma and bootstrap technique. As applications, we obtain some weighted Sobolev inequalities with partial variable weight function for Laplacian and fractional Laplacian.