The book under review introduces and develops a complete functional analysis framework that allows one to use concepts and techniques of square function estimates in general Banach spaces. Square function estimates in \(L^p\) are inequalities of the form \[ \left\| \left(\sum_{j} |T_{j}f_{j}|^{2}\right)^{1/2} \right\|_{L^p} \leq C \left\| \left(\sum_{j} |f_{j}|^{2}\right)^{1/2} \right\|_{L^p}, \] involving functions \(f_{j}\) and operators \(T_j\). Such estimates are ubiquitous, and play a key role in a number of distinct areas: boundary value problems (for well posedness through the square function/maximal function equivalence), stochastic integration (to construct integrals using a variation of Itô isometry), and ergodic theory (as a step towards controling maximal functions and proving almost everywhere convergence), to name just a few.
The basic approach to freeing the concept of a square function estimate from the lattice structure of \(L^p\) spaces is to use randomisation. Using a sequence of i.i.d. \(\pm 1\) variables \(\varepsilon_{j}\), for instance, the estimate takes the form: \[ \mathbb{E} \left\| \sum_{j}\varepsilon_{j}T_{j}f_{j}\right\|_{X} \leq C \mathbb{E} \left\| \sum_{j} \varepsilon_{j} f_{j}\right\|_{X}, \] which makes sense in any Banach space \(X\), not just in \(X=L^p\). A family of operators \((T_{j})_{j \in \mathbb{N}}\) satisfying such estimates for any finite family \((f_{j})_{j \in \mathbb{N}}\) is called R-bounded. Over the past 20 years, a range of results have been extended from Hilbert spaces to Banach spaces using R-boundedness assumptions to replace uniform bounds in the Hilbertian case. A key starting point was Lutz Weis’ operator-valued Fourier multiplier theorem proven in [L. Weis, Math. Ann. 319, No. 4, 735–758 (2001; Zbl 0989.47025)].
For Banach spaces that are part of an interpolation scale containing a Hilbert space, such extensions are extrapolation results. It is thus a very interesting question to ask whether or not they relate to known extrapolation methods, such as weighted extrapolation (a square function estimate holds in \(L^p(\mathbb{R}^{d})\) for all \(1<p<\infty\) as soon as it holds in \(L^{2}(w)\) for all weights \(w \in A_{2}\)). The book under review gives a precise answer to this question, demonstrating that extending results from Hilbert to Banach spaces in a way that generalises weighted extrapolation is possible, not just in Banach spaces that are part of an interpolation scale, and not just through randomisation, but whenever appropriate Euclidean structures are available.
Euclidean structures are introduced here as families of norms \(\| \cdot \|_{\alpha}\) on powers of \(X\) such that \(\|(x_{1})\|_{\alpha} = \|x_{1}\|_{X}\) and \(\|A(x_{1},\dots,x_{n})\|_{\alpha} \leq \|A\| \|(x_{1},\dots,x_{n})\|_{\alpha}\) for all \(x_{1},\dots,x_{n} \in X\) and all matrices \(A\). By the end of Chapter 1, the authors prove the first major result about such structures: a family of operators \(\Gamma \subset \mathcal{L}(X)\) is \(\alpha\)-bounded (i.e., \(\|(T_{1}x_{1},\dots,T_{n}x_{n})\|_{\alpha} \leq C \|(x_{1},\dots,x_{n})\|_{\alpha}\) for all finite families \(T_{1},\dots,T_{n} \in \Gamma\)) if and only if it arises as the transfer of a family of operators in an operator algebra through a bounded algebra homomorphism. In simple words: operators are \(\alpha\)-bounded for some Euclidean structure \(\alpha\) if and only they come from operators on a Hilbert space! When \(X=L^p\) (or some more general Banach function space), then the Euclidean structure is unique and given by standard square functions, while the Hilbert space turns out to be a weighted \(L^2\) space. This means that the Euclidean structure theory developed here not only generalises weighted extrapolation, but also gives an explanation for why weighted extrapolation works.
This reflects the nature of the book under review, which develops a complete theory, and focuses on explaining why techniques work rather than merely on optimising their applicability. The content is described below via highlights of each chapter.
The technical heart of Chapter 1 is Lemma 1.2.5, a variation on the theme of factorisation through a Hilbert space à la Maurey-Kwapien, that explains how Euclidean structures underpin the proofs of factorisation results. The key theorem about transfer from an operator algebra is proven here in Theorem 1.4.6.
The link with weighted extrapolation is then made in Chapter 2. The insight gained through that link bears fruits in Theorem 2.4.4, which gives a strikingly simple approach to the boundedness of the Hardy-Littlewood maximal function in Banach function spaces.
Chapter 3 constructs Banach spaces \(\alpha(\mathbb{R};X)\) associated with Euclidean structures. For the Euclidean structure arising from randomisation with i.i.d Gaussians, this coincides with the space \(\gamma(L^{2}(\mathbb{R});X)\) of \(\gamma\) radonifying operators that play a key role in Banach space valued stochastic analysis. The authors also develop an interpolation method between two Banach spaces \(X, Y\) with an Euclidean structure \(\alpha\), using the spaces \(\alpha(\mathbb{R};X)\) and \(\alpha(\mathbb{R};Y)\) (Subsection 3.3). This means that, while an Euclidean structure may not originate from an interpolation scale, interpolation scales can be constructed for two Banach spaces with the same Euclidean structure.
Chapter 4 extends classical results about R-boundedness to general Euclidean structures, focusing on the foundational results from [N. J. Kalton and L. Weis, Math. Ann. 321, No. 2, 319–345 (2001; Zbl 0992.47005)]. Again, a key feature of the theory is that it does not only generalise R-boundedness results, but also explains why they work. Theorem 4.4.1, in particular, applies the Hilbertian representation theory of Chapter 3 to show that, given an operator \(A\) acting on a Banach space \(X\), and satisfying appropriate \(\alpha\) boundedness assumptions, one can construct an operator \(\tilde{A}\) acting on a Hilbert space \(H\) such that \(\|f(A)\|_{\mathcal{L}(X)} \lesssim \|f(\tilde{A})\|_{\mathcal{L}(H)}\). Many functional calculus results can then be reproved as corollaries of this theorem, rather than by designing a R-bounded version of the Hilbertian proof.
Chapter 5 further develops the idea of constructing Banach spaces from Euclidean structures, just like classical function spaces can be constructed from Littlewood-Paley square functions. This culminates in Theorem 5.4.3 which establishes \(H^{\infty}\) calculus on specific square function spaces without relying on an \(\alpha\) (almost) sectoriality assumption. Its proof showcases the entire theory: Theorem 1.4.6 connects \(\alpha\)-boundedness to operator algebras, this gives Theorem 4.4.1 that transfers functional calculus from Hilbert to Banach spaces, which then gives Theorem 4.5.6 that reduces boundedness of the \(H^{\infty}\) calculus to \(\beta\)-BIP for some \(\beta\). All the authors have left to do is to exhibit an appropriate (simple and natural, as it turns out) Euclidean structure \(\beta\).
Chapter 6 first revisits the famous counterexamples of sectorial operators that are not R-sectorial from [N. J. Kalton and G. Lancien, Math. Z. 235, No. 3, 559–568 (2000; Zbl 1010.47024)]. It is shown that these counterexamples can be tweaked to give sectorial operators that are not almost \(\alpha\)-sectorial, as well as almost \(\alpha\)-sectorial operators that are not \(\alpha\)-sectorial. This uses classical, but highly refined, techniques from the theory of Schauder bases, and gives counterexamples in many spaces, including all \(L^p\) spaces for \(p\neq 2\).
The memoir ends (Subsection 6.4) with a long awaited and stunning counterexample of an operator that has a bounded \(H^{\infty}\) functional calculus, but only for an angle larger than the angle of sectoriality. Such behaviour is highly pathological, as these angles usually coincide (see, e.g., [A. Carbonaro and O. Dragičević, Duke Math. J. 166, No. 5, 937–974 (2017; Zbl 1476.47015)]). The existence of counterexamples was known since 2003, but only on specifically constructed Banach spaces. Here, the authors use a difficult construction on subspaces of \(L^{p}\), based on counter-intuitive non-isomorphic properties of the square function spaces in the absence of almost \(\alpha\)-sectoriality.