Boundedness properties of maximal operators on Lorentz spaces

Abstract

We study mapping properties of the centered Hardy-Littlewood maximal operator \(M\) acting on Lorentz spaces. Given \(p \in (1,\infty)\) and a metric measure space \(X = (X, \rho, \mu)\) we let \(\Omega^p_{\rm HL}(X) \subset [0,1]^2\) be the set of all pairs \((\frac{1}{q},\frac{1}{r})\) such that \(M\) is bounded from \(L^{p,q}(X)\) to \(L^{p,r}(X)\). Under mild assumptions on \(\mu\), for each fixed \(p\) all possible shapes of \(\Omega^p_{\rm HL}(X)\) are characterized. Namely, we show that the boundary of \(\Omega^p_{\rm HL}(X)\) either is empty or takes the form \(\{ \delta \} \times [0, \lim_{u \rightarrow \delta} F(u)] \cup \{(u, F(u)) \colon u \in (\delta, 1] \}\), where \(\delta \in [0,1]\) and \(F \colon [\delta, 1] \rightarrow [0,1]\) is concave, nondecreasing, and satisfies \(F(u) \leq u\). Conversely, for each such \(F\) we find \(X\) such that \(M\) is bounded from \(L^{p,q}(X)\) to \(L^{p,r}(X)\) if and only if the point \((\frac{1}{q}, \frac{1}{r})\) lies on or under the graph of \(F\), that is, \(\frac{1}{q} \geq \delta\) and \(\frac{1}{r} \leq F\big(\frac{1}{q}\big)\).