Some weighted Hardy-type inequalities and applications

TL;DR

This paper establishes a characterization of weighted Hardy-type inequalities involving sums and integrals, extending the understanding of two-weight estimates and their applications to differential operators in weighted Sobolev spaces.

Contribution

It provides a new necessary and sufficient condition for a class of two-weight Hardy inequalities with non-positive functions, linking sum estimates to integral estimates.

Findings

Characterization of two-weight Hardy inequalities for non-positive functions.

Equivalence between sum-based and integral-based estimates under certain conditions.

Application to differential operators in weighted Sobolev spaces.

Abstract

We study the two-weighted estimate \[ \bigg\|\sum_{k=0}^na_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\|\leq c\|f|L_{p,u}(0,\infty)\|,\tag{$*$} \] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<p\leq q\leq\infty$, provided that the weight $u$ satisfies the certain conditions, the estimate $(*)$ holds if and only if the estimate \[ \sum_{k=0}^n\bigg\|a_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\| \leq c\|f|L_{p,u}(0,\infty)\|.\tag{$**$} \] is fulfilled. The necessary and sufficient conditions for $(**)$ to be valid are well-known. The obtained result can be applied to the estimates of differential operators with variable coefficients in some weighted Sobolev spaces.