Weighted modular estimates for a generalized maximal operator on spaces of homogeneous type
The authors prove estimates for maximal operators associated to Young functions on spaces of homogeneous type. The main result generalizes a result of \textit{C. Pérez} and \textit{R. L. Wheeden} [J. Funct. Anal. 181, No. 1, 146--188 (2001; Zbl 0982.42010)] and extends earlier work of the second and the third author [Commentat. Math. Univ. Carol. 44, No. 3, 469--487 (2003; Zbl 1103.46019)]. Suppose that \(a,b\) are nonnegative weight functions on \([0,\infty)\) with \(a(0)=b(0)=0\) and \(b\) nondecreasing with \(b(s)\to\infty\) as \(s\to\infty\). Then \(\psi(t)=\int_0^t b(s)\, ds\) is a Young function (i.e., \(\psi\) is convexly increasing and unbounded on \((0,\infty)\) and \(\psi(0)=0\)). One also defines \(\phi(t)=\int_0^t a(s)\, ds\). Also, one lets \(\eta(t)\) be a doubling Young function (i.e., \(\eta(2t)\leq C\eta(t)\)). If \(f\) is a function on a measure space \((X,\mu)\) and \(w\) is a weight, set \(\|f\|_{\phi,w}=\inf\{\lambda>0:\, \phi(|f|/\lambda)\, w\, d\mu\leq 1\}\), and, if \(E\) is \(\mu\)-measurable, \(\|f\|_{\phi,E}=\inf\{\lambda>0:{1\over \mu(E)}\, \int_E\phi(|f|/\lambda)\, d\mu\leq 1\}\). If \((X,d,\mu)\) is a space of homogeneous type with quasimetric \(d\) one can define the maximal operator \(M_\eta f(x)=\sup_{x\in B}\|f\|_{\eta,B}\) with respect to the family of balls. The main result of this work provides several conditions on a homogeneous space with doubling Young function \(\eta\) and \(a,b,\phi,\psi\) as above such that there is a positive constant \(C\) such that (with \(w=1\)) \[ \|M_\eta f\|_\phi\leq C\|f\|_\psi \] holds for any nonnegative, \(\mu\)-measurable function \(f\) on \(X\). Primarily, this concluding condition is equivalent to the existence of a positive \(C\) such that \[ \int_0^{4t} {a(s)\over s} \eta'(t/s)\, ds\leq C b(Ct) \] for all \(t>0\). Secondarily, it is equivalent to the inequality \[ \|M_\eta f\|_{\phi,w}\leq C\|f\|_{\psi,w} \] for any weight \(w\), and it is also equivalent to \[ \int_X \phi(M_\eta f(x)) w(x)\, d\mu(x)\leq C\int_X \psi (|f(x)|)Mw(x)\, d\mu(x) \] for any \(f\). One requires that \(\mu(X)=\infty\) for some but not all of the equivalences. The tools used to prove this result include a covering lemma, a Calderón-Zygmund decomposition related to Orlicz norms and a reverse modular weak-type inequality.
scientific article; zbMATH DE number 6112448
scientific article