Dunkl spectral multipliers with values in UMD lattices
The spectral multiplier problem is a basic harmonic analytic question: for which functions \(m \in L^{\infty}(\mathbb{R})\) and which \(p \in [1,\infty)\), does the operator \(m(\Delta)\) extends from \(L^{2}(\mathbb{R}^d)\cap L^{p}(\mathbb{R}^d)\) to a bounded operator on \(L^{p}(\mathbb{R}^d)\)? The Hörmander-Mihlin Fourier multiplier theorem provides satisfying sufficient conditions, and has been generalised in many directions. One can replace \(\mathbb{R}^{d}\) by another metric measure space, one can replace \(\Delta\) by another (usually differential) operator, and one can replace \(L^p\) by another function space. In any of these directions, one can find good motivation for generalisation, either coming from applications to PDE and/or geometry, or from harmonic analysis itself (such as the desire to obtain simpler and/or more flexible proofs). The paper under review deals with two levels of generalisation that have rarely been combined before: replacing \(\Delta\) by a Dunkl Laplacian (and weighting the Lebesgue measure accordingly), and replacing \(L^p\) by a Bochner space \(L^{p}(\mathbb{R}^d;X)\) where \(X\) is a Banach space. Dunkl Laplacians are differential-difference operators that arise from deformations of the directional derivatives through the action of a reflection group associated with a root system. They naturally occur in harmonic analysis on certain Lie groups. In the other direction of generalisation, the action of spectral multipliers such as \(m(\Delta \otimes I_{X})\) on Bochner spaces \(L^{p}(\mathbb{R}^d;X)\) has been extensively studied to better understand the interactions between harmonic analysis and probability. A typical example of such interactions is the fact that Hörmander-Mihlin's theorem extends to \(X\)-valued functions if and only if \(X\) has the Unconditional Martingale Differences property (UMD). See the books [\textit{T. Hytönen} et al., Analysis in Banach spaces. Volume I. Martingales and Littlewood-Paley theory. Cham: Springer (2016; Zbl 1366.46001); Analysis in Banach Spaces, Volume 2. Preprint available at \url{http://fa.its.tudelft.nl/~neerven/.}] for a modern and extensive reference in the field of harmonic analysis of Banach space valued functions, and its relationship with probability in Banach spaces. Developments in this field of UMD valued harmonic analysis have generated many new ideas, that, in turn, have helped advance the standard scalar valued theory. The paper under review can be seen as a successful attempt to follow the same path in Dunkl harmonic analysis. More precisely, the authors generalise the Dunkl version of the Hörmander-Mihlin multiplier theorem due to \textit{F. Dai} and \textit{H. Wang} [J. Funct. Anal. 258, No. 12, 4052--4074 (2010; Zbl 1246.42015)], to the context of UMD Banach lattice valued functions. To do so, they follow the strategy employed by Dai and Wang, and going back to \textit{A. Bonami} and \textit{J.-L. Clerc} [Trans. Am. Math. Soc. 183, 223--263 (1973; Zbl 0278.43015)], i.e. they first prove a multiplier result on \(L^p(S^{d-1})\) (endowed with an appropriate weighted surface measure), and then derive their main result by transference. Working in a Banach space valued setting, however, the authors have to face the difficulty that the scale of function spaces \(L^p(S^{d-1}, h_{\kappa}^2;X)\) does not include a Hilbert space (unless \(X\) itself is isomorphic to one). Therefore, they cannot rely on the spectral theorem in \(L^2\) as a starting point for their spectral multiplier theory. Instead, they use the fact that the generator of the relevant heat semigroup has a bounded holomorphic functional calculus. This follows from the general theory of diffusion semigroups, and gives square function estimates that act as a replacement for classical Littlewood-Paley inequalities (see e.g. [Hytönen et al., loc. cit.]). As explained in [\textit{F. Dai} and \textit{Y. Xu}, Analysis on \(h\)-harmonics and Dunkl transforms. Edited by Sergey Tikhonov. Basel: Birkhäuser/Springer (2015; Zbl 1323.33003)], one can then use these Littlewood-Paley estimates to prove more appropriate square function estimates involving Cesàro means, from which the multiplier theorem follows fairly easily. The catch here is that, to make this change of square function work in the Banach space valued setting, one first needs to prove R-boundedness of the Cesàro means (Lemma 3.5). The proof of this Lemma is particularly clever, even if it forces the authors to restrict their attention to UMD Banach lattices (as opposed to general UMD spaces). It is proven by dominating the Cesàro means by an ergodic maximal operator based on the underlying heat semigroup, and then using a Fefferman-Stein inequality for this maximal operator. The domination is proven in [loc. cit.] and the Fefferman-Stein inequality is a special case of a recent result of \textit{Q. Xu} [Int. Math. Res. Not. 2015, No. 14, 5715--5732 (2015; Zbl 1339.47055)].
scientific article; zbMATH DE number 6675463
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