Capone, Claudia; Formica, Maria Rosaria; Giova, Raffaella Grand Lebesgue spaces with respect to measurable functions. (English) Zbl 1286.46030 Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 85, 125-131 (2013). 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. Cited in 1 ReviewCited in 53 Documents 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 Keywords:grand Lebesgue spaces; Banach function spaces; rearrangement-invariant spaces; function norm; embedding results; Hardy inequality PDFBibTeX XMLCite \textit{C. Capone} et al., Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 85, 125--131 (2013; Zbl 1286.46030) Full Text: DOI References: [1] Iwaniec, T.; Sbordone, C., On the integrability of the Jacobian under minimal hypothesis, Arch. Ration. Mech. Anal., 119, 129-143 (1992) · Zbl 0766.46016 [2] Boccardo, L., Quelques problemes de Dirichlet avec donnees dans de grands espaces de Sobolev, C. R. Acad. Sci. Paris Sér. I Math., 325, 1269-1272 (1997) · Zbl 0896.35046 [3] Fiorenza, A.; Mercaldo, A.; Rakotoson, J. M., Regularity and comparison results in grand Sobolev spaces for parabolic equations with measure data, Appl. Math. Lett., 14, 979-981 (2001) · Zbl 0983.35067 [4] Fiorenza, A.; Mercaldo, A.; Rakotoson, J. M., Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data, Discrete Contin. Dyn. Syst., 8, 4, 893-906 (2002) · Zbl 1007.35034 [5] Fiorenza, A.; Sbordone, C., Existence and uniqueness results for solutions of nonlinear equations with right hand side in \(L^1\), Studia Math., 127, 3, 223-231 (1998) · Zbl 0891.35039 [6] Greco, L.; Iwaniec, T.; Sbordone, C., Inverting the \(p\)-harmonic operator, Manuscripta Math., 92, 249-258 (1997) · Zbl 0869.35037 [7] Sbordone, C., Nonlinear elliptic equations with right hand side in nonstandard spaces, Atti Sem. Mat. Fis. Univ. Modena, 46, Suppl., 361-368 (1998) · Zbl 0913.35050 [8] Sbordone, C., Grand Sobolev spaces and their applications to variational problems, Matematiche, 51, 2, 335-347 (1996) · Zbl 0915.46030 [9] Capone, C.; Fiorenza, A.; Karadzhov, G. E., Grand Orlicz spaces and global integrability of the Jacobian, Math. Scand., 102, 131-148 (2008) · Zbl 1162.46019 [10] Di Fratta, G.; Fiorenza, A., A direct approach to the duality of grand and small Lebesgue spaces, Nonlinear Anal. TMA, 70, 2582-2592 (2009) · Zbl 1184.46032 [11] Fiorenza, A.; Gupta, B.; Jain, P., The maximal theorem for weighted grand Lebesgue spaces, Studia Math., 188, 2, 123-133 (2008) · Zbl 1161.42011 [12] Koskela, P.; Zhong, X., Minimal assumptions for the integrability of the Jacobian, Ric. Mat., 51, 2, 297-311 (2002) · Zbl 1096.26005 [13] Bennett, C.; Sharpley, R., Interpolation of Operators (1988), Academic Press · Zbl 0647.46057 [14] Capone, C.; Fiorenza, A., On small Lebesgue spaces, J. Funct. Spaces Appl., 3, 1, 73-89 (2005) · Zbl 1078.46017 [15] Fiorenza, A.; Karadzhov, G. E., Grand and small Lebesgue spaces and their analogs, Z. Anal. Anwend., 23, 4, 657-681 (2004) · Zbl 1071.46023 [16] Fiorenza, A.; Rakotoson, J. M., Compactness, interpolation inequalities for small Lebesgue-Sobolev spaces and applications, Calc. Var. Partial Differential Equations, 25, 2, 187-203 (2005) · Zbl 1098.46025 [18] Hewitt, E.; Stromberg, K., Real and Abstract Analysis (1975), Springer-Verlag This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.