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On some weighted inequalities of the qualitative theory of elliptic equations. (English) Zbl 1069.26016
Let \(1\leq p\leq q\leq r<\infty\). The author proves that for a weight function \(v\) on \(\mathbb R^n\), a locally summable function \(\omega:\mathbb R^n\times\mathbb R^n\to[0,\infty)\) and a non-decreasing continuous function \(\lambda:[0,\infty)\to(0,\infty)\) which are bounded by certain estimate, the Hardy-type inequality \[ \left(\int_{\mathbb R^N}| u| ^rvdx\right)^{1/r}\leq c\left[\int_{\mathbb R^N}\left(\int_{\mathbb R^N} | \,| u(x)|- | u(y)| \,| ^p\lambda(| x-y| )^{-p}\omega(x,y)dy\right)^{q/p}dx\right]^{1/q} \] holds for any function \(u\in C^1_0(\mathbb R^n)\).

MSC:
26D10 Inequalities involving derivatives and differential and integral operators
35J60 Nonlinear elliptic equations
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