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Integral operators on cones of monotone functions. (English. Russian original) Zbl 1281.26014

Dokl. Math. 86, No. 1, 562-565 (2012); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 445, No. 6, 618-621 (2012).
Let \(\mathbb{R}^{+}=[0,\infty )\). Let \(\mathcal{M}^{\mathcal{\uparrow }}\) be the set of nondecreasing functions and \(\mathcal{M}^{\mathcal{\downarrow }}\) the set of nonincreasing functions. Let \(u,w\) be weights (i.e., nonnegative locally integrable functions on \(\mathbb{R}^{+})\). In this paper, the authors deal with characterizations of the inequalities \[ \left( \int_{0}^{\infty }(Tf)^{q}w\right) ^{1/q}\leq c\left( \int_{0}^{\infty }f^{p}v\right) ^{1/p} \] on the cones \(\mathcal{M}^{\mathcal{\uparrow }}\) and \(\mathcal{M}^{\mathcal{ \downarrow }}\), where \(T\) is a positive quasilinear operator, \(0<q<\infty \) and \(1\leq p<\infty \). Moreover, if \(T\) satisfies that \(\left\{ f_{n}\right\} \subset \) \(\mathcal{M}^{\mathcal{\uparrow }}\) and \( f_{n}\nearrow f\in \mathcal{M}^{\mathcal{\uparrow }}\Longrightarrow Tf_{n}\nearrow Tf\) (or \(\left\{ f_{n}\right\} \subset \) \(\mathcal{M}^{ \mathcal{\downarrow }}\) and \(f_{n}\nearrow f\in \mathcal{M}^{\mathcal{ \downarrow }}\Longrightarrow Tf_{n}\nearrow Tf\)), the problem is solved in the case \(0<p\leq q<\infty\).

MSC:

26D10 Inequalities involving derivatives and differential and integral operators
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