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