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Grand Lebesgue spaces with respect to measurable functions. (English) Zbl 1286.46030

Summary: Let \(1<p<\infty\). Given \(\varOmega\subset\mathbb R^n\), a measurable set of finite Lebesgue measure, the norm of the grand Lebesgue spaces \(L^{p)}(\varOmega)\) is given by \[ |f|_{L^{p)}(\varOmega)}=\sup_{0<\varepsilon<p-1} \varepsilon^{\frac{1}{p-\varepsilon}}\left(\frac{1}{|\varOmega|}\int_\varOmega |f|^{p-\varepsilon}dx\right)^{\frac{1}{p-\varepsilon}}. \] In this paper we consider the norm \(|f|_{L^{p),\delta} (\varOmega)}\) obtained replacing \(\varepsilon^{\frac{1}{p-\varepsilon}}\) by a generic nonnegative measurable function \(\delta(\varepsilon)\). We find necessary and sufficient conditions on \(\delta\) in order to get a functional equivalent to a Banach function norm, and we determine the “interesting” class \(\mathcal B_p\) of functions \(\delta\), with the property that every generalized function norm is equivalent to a function norm built with \(\delta\in\mathcal B_p\). We then define the \(L^{p),\delta}(\varOmega)\) spaces, prove some embedding results and conclude with the proof of the generalized Hardy inequality.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B25 Maximal functions, Littlewood-Paley theory
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