Boundedness of integral operators on decreasing functions. (English) Zbl 1344.47034

Summary: We continue the study of the boundedness of the operator \[ S_af(t)=\int^\infty_0a(s)f(st)\mathrm{d}s \] on the set of decreasing functions in \(L^p(w)\). This operator was first introduced by M. Sh. Braverman [J. Lond. Math. Soc., II. Ser. 47, No. 1, 119–128 (1993; Zbl 0732.47033)] and S. Lai [Trans. Am. Math. Soc. 340, No. 2, 811–836 (1993; Zbl 0819.47044)] and also studied by K. F. Andersen [Can. J. Math. 43, No. 6, 1121–1135 (1991; Zbl 0757.26018)], and although the weighted weak-type estimate \(S_a:L^p_{\operatorname{dec}}(w)\to L^{p,\infty}(w)\) was completely solved, the characterization of the weights \(w\) such that \(S_a:L^p_{\operatorname{dec}}(w)\to L^p(w)\) is bounded is still open for the case in which \(p > 1\). The solution of this problem will have applications in the study of the boundedness on weighted Lorentz spaces of important operators in harmonic analysis.


47G10 Integral operators
42B25 Maximal functions, Littlewood-Paley theory
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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