New properties of small Lebesgue spaces and their applications.

A. Fiorenza ^{1}”*”

Dipartimento di Costruzioni e Metodi Matematici in Architettura

Universitá di Napoli ”Federico II” via Monteoliveto, 3 I-80134 Napoli, Italy

fiorenza@unina.it

J.M. Rakotoson Laboratoire d’Applications des Mathématiques Teleport 2 Département de Mathématiques Université de Poitiers B.P. 30179 86962 Futuroscope Chasseneuil cedex France rako@mathlabo.univ-poitiers.fr

Keywords : Relative rearrangement, small Lebesgue spaces, monotone rearrangement, elliptic equation, Sobolev embeddings.

Abstract

We prove some new properties of the small Lebesgue spaces introduced by Fiorenza [F]. Combining these properties with the Poincaré-Sobolev inequalities for the relative rearrangement (see [R1]), we derive some new and precises estimates either for small Lebesgue-Sobolev spaces or for quasilinear equations with data in the small Lebesgue spaces.

1 Introduction

In this note, we shall give some new properties of the small Lebesgue spaces introduced by Fiorenza [F] denoted by L^{(p}(Ω) for a bounded set Ω, 1 < p < +∞. This set is smaller than the Lebesgue space L^{p}(Ω) and contains all L^{p+ε}(Ω) for all ε > 0.

Moreover if we denote by

g_{∗∗}(s) = \frac{1}{s} ∫_{0}^{s} |g|_{∗} (t)dt, s ∈ (0, meas(Ω)) = Ω_{∗},

1”*” This work has been partially performed as a part of a National Research Project supported by M.U.R.S.T. and supported by the PROGRAMMA DI SCAMBI INTERNAZIONALI PER LA BREVE MOBILITAI instituted by UNI-VERSITAI DI NAPOLI ”FEDERICO II”