Maximal operator with rough kernel in variable Musielak-Morrey-Orlicz type spaces, variable Herz spaces and grand variable Lebesgue spaces
Let us consider the maximal operator \(M_{\Omega}\), defined for a function \(f\) in \(\mathbb{R}^n,\) as follows \[ M_{\Omega} f(x):= \sup_{r>0} \int_{|y|< r} |\Omega(y)| f(x-y) \;dy, \] which is related to a kernel function \(\Omega\) in \(L^1(\mathbb{S}^{n-1})\), homogeneous of degree \(0\). The boundedness of \(M_{\Omega}\) in some non-standard function spaces is considered, such as generalized variable exponent Morrey spaces, generalized Orlicz-Morrey, complementary generalized Morrey or variable exponent Herz spaces. Although the classical rotation method is not applicable in this context, the general technique applied is based on rescaling properties of these spaces. The results obtained are new, even for the counterpart spaces that have \(p\) as a constant exponent.
scientific article; zbMATH DE number 6816337
scientific article