Gâteaux differentiability for functionals of type Orlicz-Lorentz.

Levis, F.E.; Cuenya, H.H.

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We prove sharp embeddings of Besov spaces B with the classical smoothness σ and a logarithmic smoothness α into Lorentz-Zygmund spaces. Our results extend those with α = 0, which have been proved by D. E. Edmunds and H. Triebel. On page 88 of their paper (Math. Nachr. 207 (1999), 79-92) they have written: ?Nevertheless a direct proof, avoiding the machinery of function spaces, would be desirable.? In our paper we give such a proof even in a more general context. We cover...

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Let $C V_{p} (F)$ be the left convolution operators on $L^{p} (G)$ with support included in F and $M_{p} (F)$ denote those which are norm limits of convolution by bounded measures in M(F). Conditions on F are given which insure that $C V_{p} (F)$ , $C V_{p} (F) / M_{p} (F)$ and $C V_{p} (F) / W$ are as big as they can be, namely have $l^{\infty}$ as a quotient, where the ergodic space W contains, and at times is very big relative to $M_{p} (F)$ . Other subspaces of $C V_{p} (F)$ are considered. These improve results of Cowling and Fournier, Price and Edwards, Lust-Piquard, and others.

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