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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
Keywords:
Hardy-type inequality
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