Envelopes and Sharp Embeddings of Function Spaces

For the first time in book form, this work presents the new theory of growth and continuity envelopes in function spaces. These concepts originate from the classical result of the Sobolev embedding theorem, ubiquitous in all areas of functional analysis. Self-contained and accessible, the book introduces classical spaces before examining more complex spaces. Including many concrete examples, it first discusses classical spaces, such as Lebesgue and Lorentz, and defines growth and continuity envelopes. The author then examines these functions in subcritical, borderline, and critical cases. The book concludes with several applications that demonstrate the strength of this new theory.

Until now, no book has systematically presented the recently developed concept of envelopes in function spaces. Envelopes are relatively simple tools for the study of classical and more complicated spaces, such as Besov and Triebel-Lizorkin types, in limiting situations. This theory originates from the classical result of the Sobolev embedding theorem, ubiquitous in all areas of functional analysis.

Self-contained and accessible, Envelopes and Sharp Embeddings of Function Spaces provides the first detailed account of the new theory of growth and continuity envelopes in function spaces. The book is well structured into two parts, first providing a comprehensive introduction and then examining more advanced topics. Some of the classical function spaces discussed in the first part include Lebesgue, Lorentz, Lipschitz, and Sobolev. The author defines growth and continuity envelopes and examines their properties. In Part II, the book explores the results for function spaces of Besov and Triebel-Lizorkin types. The author then presents several applications of the results, including Hardy-type inequalities, asymptotic estimates for entropy, and approximation numbers of compact embeddings.

As one of the key researchers in this progressing field, the author offers a coherent presentation of the recent developments in function spaces, providing valuable information for graduate students and researchers in functional analysis.

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