Hardy averaging operator on generalized Banach function spaces and duality (Q1950976)

Let \(\Omega\subset{\mathbb R}^{n}\) be an open subset. It is well known that the \(n\)-dimensional Hardy operator \( (Af)(x)=\frac{1}{|B(0,|x|)|}\int_{B(0,|x|)}f(t)\,dt \) is bounded on \(L^{p}(\Omega)\), \(1<p\leq\infty\). The authors introduce generalized Banach function spaces of measurable functions on \(\Omega\). For such a space \(X\), there are found the space \(S_{X}\) and the space \(T_{X}\) which are, respectively strictly, larger and strictly smaller than \(X\). Under the assumption that the Hardy-Littlewood operator \(M\) is bounded on \(X\), it is proved that \(A\) is bounded from \(S_{X}\) to \(T_{X}\). Optimality of this result on such spaces is proved. There are given applications of the obtained results to variable exponent Lebesgue spaces, as an extension of [\textit{L. Pick}, Math. Nachr. 283, No. 2, 262--271 (2010; Zbl 1202.47055)].