## 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.

### MSC:

 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.)

### Citations:

Zbl 0732.47033; Zbl 0819.47044; Zbl 0757.26018
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