Weak-type boundedness of the Hardy-Littlewood maximal operator on weighted Lorentz spaces
The paper deals with the boundedness of the Hardy--Littlewood maximal operator \(M\) on the weighted Lorentz spaces \(\Lambda^p_u(w)\) and \(\Lambda^{p,\infty}_u(w)\) defined by the norms \[ \| f \|_{\Lambda_u^p(w)}^p :=\int_0^\infty pt^{p-1} W(u(\{[f|>t\})) dt, \] \[ \|f\|_{\Lambda_u^{p,\infty}(w)}^p :=\sup_{t>0} t^p W(u(\{|f|>t\})), \] where \(W(t):=\int_0^t w(s)ds\) and \(u(E):=\int_E u(x)dx\). The main result shows that a condition that was previously known to characterise the boundedness of \(M:\Lambda^p_u(w)\to \Lambda^p_u(w)\) [\textit{M. J. Carro} et al., Mem. Am. Math. Soc. 877, 128 p. (2007; Zbl 1126.42005)] is also necessary for \(M:\Lambda^p_u(w)\to \Lambda^{p,\infty}_u(w)\) when \(1<p<\infty\), and therefore these strong and weak type bounds are in fact equivalent. The proof is based on clever estimates of the super level sets of \(M\chi_E\).
scientific article; zbMATH DE number 6673272
scientific article