Muckenhoupt-type conditions on weighted Morrey spaces (Q2663273)

The authors deal with the generalized Morrey spaces \(\mathcal M^p(\varphi,w)\), where \(1\le p<\infty\), \(w\) is a nonnegative locally integrable function and \(\varphi\) is a function defined on a set of balls in \(\mathbb R^n\) with values in \((0,\infty)\). These spaces consist of measurable functions in \(\mathbb R^n\) such that \[\|f\|_{\mathcal M^p(\varphi,w)}=\sup_{B}(\frac{1}{\varphi(B)}\int_B |f|^p w)^{1/p}<\infty\] where the supremum is taken over all the euclidean balls in \(\mathbb R^n\). Throughout the paper \(\varphi\) is assumed to be doubling (that is \(\varphi(2B)\le C \varphi(B)\) for some constant \(C>0\) and every balls \(B\)) and satisfy the reverse doubling condition (that is \(\frac{\varphi(B_1)}{\varphi(B_2)}\le C (\frac{|B_1|}{|B_2|})^\delta\) for some \(\delta\), \(C>0\) and every pair of balls \(B_1\subset B_2\)). Several variations of the spaces are also considered. In particular, the weak generalized Morrey spaces \(W\mathcal M^p(\varphi,w)\) defined using \(\sup_B \frac{1}{\varphi(B)^{1/p}}\|f\chi_B\|_{L^{p,\infty}(w)}\) and the weighted local Morrey \(\mathcal L\mathcal M^p(\varphi, w)\) where the supremum is taken only on balls centered at origin. The aim of the paper is to characterize the weights \(w\) such that the Hardy-Littlewood maximal operator \(Mf(x)=\sup_{x\in B} \frac{1}{|B|}\int_B |f|\), is bounded on \(\mathcal M^p(\varphi,w)\). Of course there is a natural condition analogue of Muckenhoupt's \(A_p\) weight which plays an important role. The authors say that \(w\in A(\mathcal M^p(\varphi))\) if \[[w]_{A(\mathcal M^p(\varphi))}= \sup_B \frac{1}{|B|}\|\chi_B\|_{\mathcal M^p(\varphi,w)}\|\chi_B\|_{\mathcal M^p(\varphi,w)^\prime}<\infty\] where \(\mathcal M^p(\varphi,w)^\prime\) stands for the Köthe dual of \(\mathcal M^p(\varphi,w)\) and the supremum run over all the balls in \(\mathbb R^n\). The authors also deal with a more general maximal function, namely for a given collection of balls \(\mathcal B\), the maximal operator associated to \(\mathcal B\) is defined by \(M_\mathcal Bf(x)=\sup_{x\in B, B\in \mathcal B} \frac{1}{|B|}\int_B |f|\) and \(A_\mathcal B(\mathcal M^p(\varphi))\) stand for the class of weights defined as above when the supremum is taken only on balls belonging to \(\mathcal B\). For the collection \(\mathcal B_0\) given by balls centered at the origin, \(M_{\mathcal B_0}\) is denoted by \(M_0\), that is \(M_0f(x)=\sup_{r>|x|}\frac{1}{|B(0,r)|}\int_{B(0,r)} |f|\), and it is shown that \(M_0\) is bounded from \(\mathcal M^p(\varphi,w)\) into \(W\mathcal M^p(\varphi,w)\) iff it is bounded from \(\mathcal M^p(\varphi,w)\) into \(\mathcal M^p(\varphi,w)\) iff \(w\in A_0(\mathcal M^p(\varphi)\). The situation for the Hardy-Littewood maximal operator when restricted to weighted local Morrey spaces is also studied, and the condition \[[w]_{A(\mathcal L\mathcal M^p(\varphi))}= \sup_B \frac{1}{|B|}\|\chi_B\|_{\mathcal L\mathcal M^p(\varphi,w)}\|\chi_B\|_{\mathcal L\mathcal M^p(\varphi,w)'}<\infty\] is necessary and sufficient for the boundedness of the Hardy-Littewood maximal function acting on \(\mathcal L\mathcal M^p(\varphi,w)\) in the cases \(p\in (1,\infty)\). To handle the Hardy-Littewood maximal operator on \(\mathcal M^p(\varphi, w)\) they need to use the condition \(w\in A(\mathcal M^p( \varphi))\) together with a local \(A_p\) condition. This is due to the fact that \(Mf(x) \approx M_0 f(x)+ M_{loc}f(x)\) where \(M_{loc}\) corresponds to the local Hardy-Littlewood maximal operator defined when considering the collection of balls \(B\) such that \(r_B< \kappa |c_B|\) where \(r_B\) and \(c_B\) denote the radius and the center of the ball and \(\kappa\in (0,1)\) (see [\textit{C.-C. Lin} and \textit{K. Stempak}, Math. Ann. 348, No. 4, 797--813 (2010; Zbl 1211.42017)]). These results extend and completes those proved by \textit{S. Nakamura} et al. [Ann. Acad. Sci. Fenn., Math. 45, No. 1, 67--93 (2020; Zbl 1437.42024)]. The boundedness of the Calderón operator on generalized Morrey spaces is also characterized. Finally the paper contains some other interesting results concerning extrapolation of Lebesgue estimates to get boundedness on Morrey spaces.