On boundedness of fractional maximal operators between classical Lorentz spaces
Let \(M_\gamma \) be the fractional maximal operator defined at locally integrable functions on \(\mathbb R^n\) by \((M_{\gamma }f)(x)=\sup _{Q\owns x}|Q|^{\frac {\gamma }{n}-1} \int _{Q}|f(y)|dy\), where the supremum is taken over all cubes \(Q\) in \(\mathbb R^n\) with sides parallel to the coordinate axes. The aim of the paper is to give a~characterization of boundedness of \(M_\gamma \) on classical Lorentz spaces. To this end, the author adopts the approach of \textit{M. A. Ariño} and \textit{B. Muckenhoupt} [Trans. Am. Math. Soc. 320, No. 2, 727-735 (1990; Zbl 0716.42016)], who considered the analogous problem for the Hardy-Littlewood maximal operator \(M\) (in other words, for the case \(\gamma =0\)). This approach consists of two main steps: first, a~sharp inequality is proved for the \((Mf)^*\) in terms of a Hardy-type integral operator applied to \(f^*\), where the star denotes the non-increasing rearrangement. The second step consists of characterizing when the Hardy operator is bounded when restricted to the cone of non-increasing functions in a~Lebesgue space. NEWLINENEWLINENEWLINEIn the case of fractional maximal operator, the situation is more complicated. In the first step, the inequality NEWLINE\[NEWLINE(M_\gamma f)^*(t)\leq C \sup _{t<\tau <\infty }\tau ^{\frac {\gamma }{n}-1} \int _{0}^{\tau }f^{*}(s) dsNEWLINE\]NEWLINE is shown for every \(f\) and \(t\) and its sharpness is proved. In the second step, boundedness of the above operator involving supremum restricted to the cone of non-increasing functions in a~Lebesgue space is established. The desired characterization of boundedness of fractional maximal operator on Lorentz spaces then follows. Extensions to more general operators are discussed.NEWLINENEWLINEFor the entire collection see [Zbl 0933.00035].
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