Boundedness of maximal operators and Sobolev's inequality on Musielak-Orlicz-Morrey spaces (Q1935767)

Let \(L^{\phi,\kappa}(\mathbb{R}^N)\) be the Musielak-Orlicz-Morrey space, with \(\phi\) satisfying suitable conditions. The main result is that the maximal operator \[ Mf(x)= \sup_{r>0}|B(x,r)|^{-1} \int_{B(x,r)} f(y)\,dy \] is bounded from \(L^{\phi,\kappa}(\mathbb{R}^N)\) into itself, i.e., there exists a constant \(C\) such that \(\| Mf\|_{\phi,\kappa}\leq C\| f\|_{\phi,\kappa}\) for all \(f\in L^{\phi,\kappa}(\mathbb{R}^N)\). Furthermore, under suitable conditions for \(\psi\) the following inequality is proved \[ \sup_{x\in\mathbb{R}^N,r> 0} {\kappa(x,r)\over|B(x, r)|} \int_{B(x,r)} \overline\psi\Biggl(y, Jf(y){1\over C}\Biggr)\,dy\leq 1 \] for all \(f\geq 0\) such that \(\| f\|_{\phi,\kappa}\leq 1\). Here, \(J\) is a potential kernel.