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Weighted integral inequalities for the Hardy type operator and the fractional maximal operator. (English) Zbl 0794.42015

Summary: Let \(w(x)\), \(u(x)\) and \(v(x)\) be weight functions. In this paper, under appropriate conditions on Young’s functions \(\Phi_ 1\), \(\Phi_ 2\) we characterize the inequality \[ \Phi_ 2^{-1} \left( \int^ \infty_ 0 \Phi_ 2 \bigl( Tf(x) \bigr) w(x)dx \right) \leq \Phi_ 1^{-1} \left( \int^ \infty_ 0\Phi_ 1 \bigl( Cf(x)u(x) \bigr) v(x)dx \right) \] for the Hardy-type operator \(T\) defined by St. Bloom and R. Kerman [Proc. Am. Math. Soc. 113, No. 1, 135-141 (1991; Zbl 0753.42010)] and the inequality \[ \Phi_ 2^{-1} \left( \int_{\mathbb{R}^ n} \Phi_ 2 \bigl( M_{\alpha,\nu} (fv) (x) \bigr) w(x) d \nu(x) \right) \leq \Phi_ 1^{-1} \left( \int_{\mathbb{R}^ n} \Phi_ 1 \bigl( Cf(x) \bigr) v(x)d \nu(x) \right) \] for the fractional maximal operator \(M_{\alpha,\nu}\) defined by the author [J. Lond. Math. Soc., II. Ser. 46, No. 2, 301-318 (1992; Zbl 0758.42012)], as well as the corresponding weak-type inequalities.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
26D15 Inequalities for sums, series and integrals
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