On optimal decomposition in Zygmund spaces

Alberto Fiorenza

Dipartimento di Costruzioni e Metodi Matematici in Architettura, Università degli Studi di Napoli “Federico II” via Monteoliveto 3, 80134 Napoli Italy E-mail: fiorenza@cds.unina.it

and

Miroslav Krbec

Institute of Mathematics, Academy of Sciences of the Czech Republic Žitná 25, 115 67 Prague 1 Czech Republic E-mail: krbecm@matsrv.math.cas.cz

We present an optimal decomposition formula for the norm in the Orlicz space L(log L)^{α}. We apply the decomposition to apriori estimates for elliptic PDEs with right hand side in Zygmund classes.

Key Words: Orlicz spaces, Zygmund classes, extrapolation, duality

1. INTRODUCTION

A lot of attention has been always paid to various expressions for the norm in Orlicz spaces (the Luxemburg norm, the “dual” norm, the Amemyia norm, see, e.g., [24], [28], [25]); indeed, the definition of the norm is one of the very peculiar features of these spaces, which sometimes makes the things rather difficult and—as a rule of thumb—frequently prevents from a straightforward usage of the L^{p}-spaces techniques and approach. There is, however, one special class of Orlicz spaces, where an essential progress has been made in last years, namely the logarithmic Lebesgue spaces.

On the abstract extrapolation theory level this has been done in Jawerth and Milman [22], see also Milman [26]. The identification of the result