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Entropy and approximation numbers of embeddings between weighted Besov spaces

Iwona Piotrowska (2008)

Banach Center Publications

The present paper is devoted to the study of the “quality” of the compactness of the trace operator. More precisely, we characterize the asymptotic behaviour of entropy numbers of the compact map t r Γ : B p , q s ( , w ϰ Γ ) L p ( Γ ) , where Γ is a d-set with 0 < d < n and w ϰ Γ a weight of type w ϰ Γ ( x ) d i s t ( x , Γ ) ϰ near Γ with ϰ > -(n-d). There are parallel results for approximation numbers.

Equivalent quasi-norms and atomic decomposition of weak Triebel-Lizorkin spaces

Wenchang Li, Jingshi Xu (2017)

Czechoslovak Mathematical Journal

Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre's maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given.

Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory

Mark Veraar, Lutz Weis (2015)

Studia Mathematica

We consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form L p ( X ) γ ( X ) L q ( X ) , in terms of the type p and cotype q of the Banach space X. As an application we prove L p -estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

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