This book is the second volume of a series of three. The first one, entitled “Analysis in Banach spaces” [the authors, Analysis in Banach spaces. Volume I. Martingales and Littlewood-Paley theory. Cham: Springer (2016;
Zbl 1366.46001)], presented the essential techniques to handle Banach space-valued functions, namely, the theory of Bochner integration, vector-valued martingales and a study of certain properties such as the UMD property in order to deal with the Hilbert transform, the Littlewood-Paley theory or the Mihlin multiplier theorem in the Banach space-valued setting. In the present volume, the authors concentrate mainly on questions arising from evolution equations about the extension of classical \(L^2\)-estimates and techniques from Hilbert spaces to the setting of \(L^p\)-spaces or more general Banach spaces. This is done by the use of some fundamental randomisation techniques, in particular, by means of the concept of \(R\)-boundedness. The previous volume already provided a first glance at operator-valued versions of certain multiplier theorems where the uniform boundedness condition on certain families of operators appearing in the statement was replaced by the \(R\)-boundedness of the family for the analogues of classical theorems to hold in this setting. This method is now used again quite efficiently in many situations, allowing the authors to get a number of applications in harmonic analysis and stochastic analysis. A second tool to extend Hilbert space techniques to Banach spaces consists of using radonifying operators which, in particular, allow to generalise square function estimates to an abstract Banach space setting. Using the mentioned tools, the authors manage to get a far reaching extension of the theory of \(H^\infty\)-functional calculus to the \(L^p\)-setting, including several new characterizations and very useful applications in the theory of evolution equations in \(L^p\)-spaces.
The book will become an invaluable reference for graduate students and researchers. Those interested in functional analysis, harmonic analysis and stochastic analysis will appreciate the detailed treatment of the theory of \(R\)-boundedness, radonifying operators and holomorphic functional calculus developed over the last 20 years. Also, the preliminary chapters and the topics selected in the appendix contain some rather important notions for any analyst working in harmonic analysis and evolution equations on Banach space-valued functions.
The book under review is divided into five chapters, starting with Chapter 6, as a continuation of the previous chapters from Volume I. The first chapter (Chapter 6) deals mostly with general aspects of random sums that remain valid in arbitrary Banach spaces. The authors make a systematic investigation of two basic random sums: Rademacher and Gaussian sums. The three first sections are concerned with basic relations and estimates for finite sums, and results such as Kahane’s contraction principle, the covariance domination for Gaussian sums, the Kahane-Khintchine inequality, among others, are included. The last sections are devoted to two fundamental convergence results of random series: the Itô-Nisio theorem about the equivalence of modes of convergence and the Hoffmann-Jørgensen-Kwapień theorem concerning the equivalence between boundedness and convergence under the assumption that the space does not contain a copy of \(c_0\). The chapter finishes with a result by Pisier about the equivalence between Rademacher sums and trigonometric sums over Sidon sets.
The second chapter (Chapter 7) deals with basic properties in Banach space theory such as type, cotype or \(K\)-convexity. All of them are intimately connected with Rademacher and Gaussian sums. After introducing and discussing the notions of type and cotype, the authors undertake the study of \((p,q)\)-summing operators, mainly due to the connection with cotype, as is reflected in the fact that any operator from \(C(K)\) into a Banach space \(X\) of cotype \(q\) is \((q,1)\)-summing. In this chapter, several basic results are given: Pisier’s factorization theorem, which establishes that a bounded operator from \(C(K)\) into \(X\) is \((q,1)\)-summing if and only if it factors through a Lorentz space \(L^{q,1}(\mu)\), the equivalence between Rademacher and Gaussian cotype, Kwapień’s isomorphic characterization of Hilbert spaces in terms of type and cotype 2, and Maurey-Pisier’s characterization of non-trivial (co)type, among others. Also, the notion of \(K\)-convexity introduced as a boundedness assumption on Rademacher projections is studied in detail and several characterizations are provided. The chapter finishes with different contraction principles for double random sums, which is motivated by the fact that, in contrast to random sums, this principle fails for double sums.
The third chapter (Chapter 8) is concerned with the notion of \(R\)-boundedness and its relatives. This continues the investigation of random series with vector-valued coefficients, now studying the transformations of such series under the termwise action of families of bounded operators. Namely, inequalities of the following type \[\left\|\sum_{n=1}^N \xi_n T_n x_n\right\|_{L^2(\Omega, Y)}\leq \left\|\sum_{n=1}^N \xi_n x_n\right\|_{L^2(\Omega, X)},\] where \(T_n\) are drawn from a family \(\mathcal F\) of operators in \(\mathcal L(X;Y)\) for some Banach spaces \(X\) and \(Y\) and \(\xi_n\) are independent symmetric identically distributed random variables. When such an inequality holds for the case of Rademacher variables \((\varepsilon_n)\) or Gaussian variables \((\gamma_n)\), the family \(\mathcal F\) is called \(R\)-bounded or \(\gamma\)-bounded. When dealing with Banach lattices \(X\) and \(Y\), a related notion called \(\ell^2\)-boundedness is defined using the inequality \[\left\|\left(\sum_{n=1}^N |T_n x_n|^2\right)^{1/2}\right\|_Y\leq C \left\|\left(\sum_{n=1}^N | x_n|^2\right)^{1/2}\right\|_X.\] The chapter is divided into several sections, starting with the basic theory covering comparisons between related notions and examples, and developing this concept to cover several sources either in real analysis or in operator theory, for instance, its connection with pointwise domination by maximal operators or inequalities with Muckenhoupt weights. Special emphasis is put on how to use \(R\)-boundedness to provide results on Fourier multipliers such as Marcinkiewicz or Mihlin multiplier theorems. In the setting of operator theory, the relationship with duality, interpolation or unconditionality is addressed. Finally, a section on integral means and their effect on type and cotype is presented.
Chapter 9 deals with the second basic tool in the book. It is devoted to the notion of radonifying operator and its connection with square functions. Square functions provide a well-known representation of classical function space norms in terms of expressions involving a quadratic component, this is the case of the Littlewood-Paley inequality for functions in \(f\in L^p(\mathbb R)\) where the norm turns out to be equivalent to \(\|(\int_0^\infty |t \frac{\partial}{\partial t}u(t,\cdot)|^2\frac{}{})^{1/2}\|_{L^p}\), where \(u\) is the harmonic extension of \(f\) into the upper half plane, or the Burkholder-Davis-Gundy inequality for martingales \((f_n)\) in \(L^p(\Omega)\) for which the norm of \(f_N\) is equivalent to \(\|(\sum_{n=1}^N |df_n|^2)^{1/2}\|_{L^p(\Omega)}\), where \(df_n=f_n-f_{n-1}\) is the martingale difference. The main idea in this chapter is to get a formulation for the “square function” in an abstract Banach space \(X\) where “pointwise evaluation” does not make sense. This is done through the use of radonifying operators. For this purpose, the authors fix a Hilbert space \(H\) and define the \(\gamma\)-radonifying operators as the closure of the finite rank operators in \(\mathcal L(H, X)\) under the norm \(\|T\|_{\gamma_\infty}=\sup (\mathbb E \|\sum_{j=1}^k \gamma_j Th_j\|^2)^{1/2}\), where the supremum is taken over all finite orthonormal systems \(\{h_1,\dots, h_k\}\) in \(H\) and \(\gamma_j\) are independent Gaussian variables. Denoting by \(\gamma_\infty(H,X)\) the space of operators where \(\|T\|_{\gamma_\infty}<\infty\) and by \(\gamma(H,X)\) the space of \(\gamma\)-radonifying operators, the authors study in detail such spaces, analysing their connections with other classes of operators, the coincidence between both, and other aspects concerning, for instance, duality or interpolation. Once these families of operators are defined, the authors introduce the functions representing \(\gamma\)-radonifying operators and define the classes \(\gamma_\infty(S, X)\) or \(\gamma(S, X)\) consisting of those strongly \(\mu\)-measurable and weakly in \(L^2\) functions \(f:S \to X\) such that the Pettis integral operator \(g \mapsto \int g(s)f(s)d\mu(s)\) belongs to \(\gamma_\infty(L^2(S),X)\) or \(\gamma(L^2(S),X)\), respectively. In this setting, they manage to show versions of the dominated convergence theorem, Fatou’s lemma and Fubini’s theorem. The use of \(\gamma\)-radonifying operators and \(\gamma(S,X)\) allow to recover square functions in \(L^p\)-spaces, and even a formulation for general “square function” in Banach spaces can be provided. Also, multiplier theorems can be proved in this new context.
In the last chapter (Chapter 10), the authors investigate sectorial operators and their holomorphic functional calculus defined by the Dunford integral \(f(A)=\frac{1}{2\pi i}\int_{\partial \Sigma } f(z)(z-A)^{-1}dz\), where \(f\) is a bounded holomorphic function on a sector \(\Sigma\) about the positive real line containing the spectrum of \(A\), satisfying \(\|(z-A)^{-1}\|\leq C \frac{1}{|z|}\) on the complement of \(\overline{\Sigma}\). An operator \(A\) is said to have a bounded \(H^\infty\)-calculus on \(\Sigma\) whenever \(f(A)\) is bounded and satisfies \(\|f(A)\|\leq C \|f\|_\infty\) for all bounded holomorphic function on \(\Sigma\). With this model in mind, the authors make use of the probabilistic and operator-theoretic techniques from the previous chapters to develop the theory of \(H^\infty\)-calculus for general sectorial operators acting on Banach spaces, studying among other things their \(R\)-bounded properties and formulations in terms of the generalised square functions. In this chapter, it is shown that the Laplace operator on \(L^p(\mathbb R^n,X)\), \(1<p<\infty\), has a bounded \(H^\infty\)-calculus if and only if \(X\) is a UMD space, and the range of its \(H^\infty\)-calculus is \(\gamma\)-bounded if and only if \(X\) possesses Pisier’s contraction property. Special cases such as \(A\) being the generator of either a \(C_0\)-semigroup of contractions on a Hilbert space, or positive contractions on an \(L^p\)-space or a \(C_0\)-group on a UMD space, are shown to have a bounded \(H^\infty\)-calculus.
The volume ends with an interesting list of open problems and an appendix containing six sections including some basics on probability theory, Banach lattices, semigroups of linear operators, Hardy spaces of holomorphic functions, Akcoglu’s theory of positive contractions on \(L^p\) and Muckenhoupt weights.
