Duality and reflexivity in grand Lebesgue spaces
- Autores: Alberto Fiorenza
- Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 51, Fasc. 2, 2000, págs. 131-148
- Idioma: inglés
- Enlaces
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Resumen
The grand $L^p$ space $L^{p)} (\Omega) (1 < p < +\infty)$ introduced by Iwaniec-Sbordone is defined as the \textit{Banach Function Space} of the measurable functions $f$ on $\Omega$ such that $$ \parallel f\parallel_{p)} = \sup_{0 < \varepsilon < p-1}\bigg(\varepsilon\frac{1}{\vert\Omega\vert}\int\limits_{\Omega}\vert f\vert^{p-\varepsilon}dx\bigg)^{1/(p-\varepsilon)} < +\infty.$$ We introduce the \textit{small} $L^{p'}$ \textit{space} denoted by $L^{p')}(\Omega)$ and we prove that the associate space of $L^{p)}(\Omega)$ is $L^{p)'}(\Omega)$. It turns out that $L^{p)'}(\Omega)$ is a \textit{Banach Function Space} whose norm satisfy the Fatou property, and that it is the dual of the closure of $L^\infty(\Omega)$ in $L^{p)}(\Omega)$ . Moreover, we give a characterization of $L^{p)}(\Omega)$ as dual space, and we prove that for any $1 < p < +\infty$ the spaces $L^{p)}(\Omega)$ and $L^{p)'}(\Omega)$ are not reflexive.
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Referencias bibliográficas
- C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988.
- D. Cruz-Uribe, SFO and A. Fiorenza, in preparation.
- M. Carrozza and C. Sbordone, The distance to $L^\infty$ in some function spaces and applications, Differential Integral Equations, 10 (1997),...
- A. Fiorenza and M. Křbec, On the domain and range of the maximal operator, Preprint Academy of Sciences of the Czech Rep. 122, 1997, to appear...
- A. Fiorenza and C. Sbordone, Existence and uniqueness results for solutions of nonlinear equations with right hand side in $L^1$, Studia Math....
- L. Greco, A remark on the equality $\det Df = \Det Df$, Differential Integral Equations 6 (1993), 1089–1100.
- L. Greco, T. Iwaniec and C. Sbordone, Inverting the $p$-harmonic operator, Manuscripta Math. 92 (1997), 249–258.
- T. Iwaniec and C. Sbordone, On the integrability of the Jacobian under minimal hypotheses, Arch. Rational Mech. Anal. 119 (1992), 129–143.
- T. Iwaniec and C. Sbordone, Weak minima of variational integrals, J. Reine Angew. Math. 454 (1994), 143–161.
- T. Iwaniec and C. Sbordone, Riesz transforms and elliptic pde’s with VMO coefficients, J. Analyse Math. 74 (1998), 183–212.
- A. Kufner, O. John and S. Fučik, Function Spaces, Noordhoff International Publishing, Leyden, 1977.
- J. Lang and L. Pick, personal communication.
- C. Sbordone, Grand Sobolev spaces and their applications to variational problems, Le Matematiche LI(2) (1996), 335–347.
- C. Sbordone, Nonlinear elliptic equations with right hand side in nonstandard spaces, Rend. Sem. Mat. Fis. Modena, Supplemento al XLVI (1998),...
- A. C. Zaanen, Integration, North-Holland, 1967.
