Some weighted Hardy-type inequalities and applications

Abstract

We study the two-weighted estimate \[ \bigg\|\sum_{k=0}^na_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\|\leq c\|f|L_{p,u}(0,\infty)\|,\tag{$\ast$} \] where the functions $a_k(x)$ are not assumed to be positive. It is shown that for $1<p\le q\le \infty$, provided that the weight $u$ satisfies the certain conditions, the estimate $(\ast )$ holds if and only if the estimate \[ \sum_{k=0}^n\bigg\|a_k(x)\int_0^xt^kf(t)dt|L_{q,v}(0,\infty)\bigg\| \leq c\|f|L_{p,u}(0,\infty)\|.\tag{$\ast \ast$} \] is fulfilled. The necessary and sufficient conditions for $(\ast \ast )$ 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.

15 pages

References in corpus (1)

• Splitting of Volterra Integral Operators with Degenerate Kernels

Cited by in corpus (1)

• On weighted estimates for a class of Volterra integral operators