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- W2022604324 abstract "Let $Af(x):=frac{1}{|B(0,|x|)|} int_{B(0,|x|)} f(t) dt$ be the $n$-dimensional Hardy averaging operator. It is well known that $A$ is bounded on $Lsp p(Omega)$ with an open set $Omega subset R^n$ whenever $1<pleqinfty$. We improve this result within the framework of generalized Banach function spaces. We in fact find the source'' space $S_X$, which is strictly larger than $X$, and the target'' space $T_X$, which is strictly smaller than $X$, under the assumption that the Hardy-Littlewood maximal operator $M$ is bounded from $X$ into $X$, and prove that $A$ is bounded from $S_X$ into $T_X$. We prove optimality results for the action of $A$ and its associate operator $A'$ on such spaces and present applications of our results to variable Lebesgue spaces $L^{p(cdot)}(Omega)$, as an extension of A.~Nekvinda and L.~Pick [Math.~Nachr. 283 (2010), 262–271; Z.~Anal.~Anwend. 30 (2011), 435–456] in the case when $n=1$ and $Omega$ is a bounded interval." @default.
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- W2022604324 date "2013-01-01" @default.
- W2022604324 modified "2023-09-23" @default.
- W2022604324 title "Hardy Averaging Operator on Generalized Banach Function Spaces and Duality" @default.
- W2022604324 doi "https://doi.org/10.4171/zaa/1483" @default.
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