In this book, the author systematically and detailedly presents a new theory of growth and continuity envelopes in function spaces. Envelopes originate from such classical results as the famous Sobolev embedding theorem in [S.L.Sobolev, Am.Math.Soc., Transl.(2) 34, 39–68 (1963; Zbl 0131.11501); transl.from Mat.Sb.(N.Ser.) 4 (46), 471–497 (1963; Zbl 0022.14803)], or, secondly, from the results of H.Brézis and S.Wainger [Commun.Partial Differ.Equations 5, 773–789 (1980; Zbl 0437.35071)] on the almost Lipschitz continuity of functions from a Sobolev space \(H_p^{1+n/p}({\mathbb R^n})\) with \(p\in (1,\infty)\), and they provide relatively simple tools for the study of classical and more complicated spaces, such as Besov and Triebel–Lizorkin types, in limiting cases.
Let \(f^\ast\) be the non-increasing rearrangement of a function \(f\). The author introduces the growth envelope function of a function space \(X\subset L^{\mathrm{loc}}_1\), \[ {\mathcal E}^X_G(t)\equiv\sup_{\| f| X\| \leq 1}f^\ast(t), \quad t\in (0,1).\tag \(*\) \] It turns out that in rearrangement-invariant spaces, there is a connection between \({\mathcal E}^X_G\) and the fundamental function. The pair \({\mathbb E}_G(X)\equiv ({\mathcal E}^X_G, u_G^X)\) is called the growth envelope of \(X\), where \(u_G^X\in (0,\infty]\) is the infimum of all numbers \(\nu\in(0,\infty]\) satisfying that for some \(\varepsilon>0\) and some constant \(C>0\) and all \(f\in X\), \[ \left\{\int^\varepsilon_0\left[\frac {f^\ast(t)} {{\mathcal E}^X_G(t)}\right]^\nu\mu_G(dt)\right\}^{1/\nu}\leq C\| f| X\| \tag \(**\) \] and \(\mu_G\) is the Borel measure associated with \(-\log{\mathcal E}^X_G\). The result reads for Sobolev spaces \[ {\mathbb E}_G(W_p^k)=(t^{-1/p+k/n},p),\quad p\in [1,\infty),\quad k<n/p. \] Let \({\mathcal C}\) be the space of all complex-valued bounded uniformly continuous functions on \({\mathbb R}^n\), equipped with the sup-norm. When \(X\subset{\mathcal C}\), replacing \(f^\ast(t)\) by \(\frac {\omega(f,t)}t\) in (\(\ast\)) and (\(\ast\ast\)), the author then introduces the continuity envelope \({\mathbb E}_{\mathcal C}\), where \(\omega(f,t)\) denotes the modulus of continuity. The above-mentioned result of Brézis and Wainger then appears as \[ {\mathbb E}_{\mathcal C}(H_p^{1+n/p}) =(| \log t| ^{1-1/p},p),\quad p\in (1,\infty). \] The book has two parts. Part I consists of six chapters. Chapter 1 provides a comprehensive introduction, including the background of this new theory. In Chapter 2, the author briefly describes the concept of non-increasing rearrangement and recalls some definitions of the (classical) function spaces under consideration. Chapter 2 ends with a section devoted to Sobolev’s famous embedding theorem. In Chapter 3, the author introduces the growth envelope function and derives some of its properties. This is followed by the introduction of growth envelopes in Chapter 4, which also provides the main results for Lorentz–Zygmund, Sobolev and weighted (Lebesgue) spaces; Chapters 5 and 6 are parallel to Chapters 3 and 4, replacing the growth envelopes by continuity envelopes. To understand Part I, the reader is not assumed to know more function space theory than the basics about Lebesgue (and Lorentz) spaces, Lipschitz spaces and (classical) Sobolev spaces.
Part II consists of five chapters, exploring the results for function spaces of Besov and Triebel–Lizorkin type and presenting several applications of the results, including Hardy-type inequalities, asymptotic estimates for entropy, and approximation numbers of compact embeddings.
The present book is written in a very comprehensive and streamlined way. The author offers a coherent presentation of the recent developments in function spaces. The book is remarkable and will be useful to graduate students and experts in the fields of harmonic analysis, PDE, fractal analysis, real analysis, approximation theory, and functional analysis.