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On a variant of the Hardy-Littlewood inequality. (Russian. English summary) Zbl 0566.26004

Let \(1\leq m\leq n\), \(x=(x_ 1,...,x_ n)\in R^ n\), \(x'=(x_ 1,...,x_ m)\), \(1\leq p\leq \infty,\) \(p^{-1}+q^{-1}=1,\) \(\lambda <m/p,\) \(\mu <n/q,\) \(\lambda +\mu >0.\) Let \(| \cdot |\) denote the Euclidean norm. Let \(K(x,t)=| x'|^{-\lambda}| x- t|^{\lambda +\mu -n}| t'|^{-\mu}\) and \[ (Tf)(x)=\int_{R^ n}K(x,t)f(t)dt \] for f in \(L_ p(R^ n).\) The author proves that the operator T is bounded on \(L_ p(R^ n).\)
Reviewer: I.Raşa

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
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