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On scales of equivalent conditions characterizing weighted Stieltjes inequality. (English. Russian summary) Zbl 1261.42005

Dokl. Math. 86, No. 3, 738-739 (2012); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 447, No. 1, 13-14 (2012).
From the introduction: Let \(0<\lambda<\infty\). On the cone \(\mathcal M^+\) of all nonnegative measurable functions on the half line \(\mathbb R_+=[0,\infty)\), the Stieltjes transform is defined by the formula \[ S_\lambda h(x)=\int_0^\infty \frac{h(y)dy}{(x+y)^\lambda}, \quad x \geq 0. \] Let \(u\) and \(v\) be weight functions, i.e., nonnegative locally integrable functions on \(\mathbb R_+\). We consider the problem of characterizing the weighted Stieltjes inequality \[ \left( \int_0^\infty (S_\lambda h(x))^q u(x)dx\right)^{\frac{1}{q}}\leq C \left( \int_0^\infty (h(x))^p v(x)dx\right)^{\frac{1}{p}}, \quad h \in \mathcal M^+, \] where \(p>1\) and \(0<q<\infty\) are fixed parameters and the constant \(C\) is chosen to be as small as possible.
The goal of this work is to find scales of integral conditions equivalent to each other and equivalent to the fulfillment of the weighted Stieltjes inequality.

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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