Maximal operator with rough kernel in variable Musielak-Morrey-Orlicz type spaces, variable Herz spaces and grand variable Lebesgue spaces. (English) Zbl 1387.42020
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.
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.
Reviewer: Santiago Boza (Barcelona)
MSC:
| 42B25 | Maximal functions, Littlewood-Paley theory |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
rough operator; maximal operator; generalized Orlicz-Morrey spaces; generalized variable Morrey spaces; variable Herz spacesReferences:
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