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

Carro, Mariá J.; Ortiz-Caraballo, Carmen

Proc. R. Soc. Edinb., Sect. A, Math. 145, No. 4, 725-744 (2015).

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.)

Keywords:

weak \(L^p\)-spaces; Hardy-Littlewood maximal operator; Muckenhoupt weights

Citations:

Zbl 0732.47033; Zbl 0819.47044; Zbl 0757.26018