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## Research School - Jean Morlet Chair - Frontiers in Interacting Particle Systems, Aggregation-Diffusion Equations & Collective Behavior / Ecole - Chaire Jean Morlet - Frontières dans les équations de systèmes de particules en interaction. Equations d'agrégation-diffusion et comportement collectif

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## Project cyan: $H^{\infty}$-calculus and square functions on Banach spaces

To solve the Kato conjecture in the lectures, we first reformulated the Kato property as a square function estimate. One of the main characters in this reformulation was McIntosh's theorem, which states that a sectorial operator $L$ on a Hilbert space $H$ has a bounded $H^{\infty}$-calculus if and only if for some (equivalently all) nonzero $f \in H_{0}^{\infty}\left(S_{\varphi}\right)$ the quadratic estimate$$\begin{equation²}\left(\int_{0}^{\infty}|f(t L) u|_{H}^{2} \frac{\mathrm{d} t}{t}\right)^{1 / 2} \approx|u|_{H}, \quad u \in H \tag{2.3}\end{equation²}$$holds. Since neither the definition of the $H^{\infty}$-calculus, nor the statement of McIntosh's theorem explicitly use the Hilbert space structure of $H$, one may wonder if this theorem is also true for Banach spaces. This would, for example, be a useful tool in the study of the Kato property in $L^{p}(\Omega)$ with $p \neq 2$.In [1], it was shown that for a sectorial operator $L$ on $L^{p}(\Omega)$ the quadratic estimates need to be adapted, taking the form$$\begin{equation²}\left|\left(\int_{0}^{\infty}|f(t L) u|^{2} \frac{\mathrm{d} t}{t}\right)^{1 / 2}\right|_{L^{p}(\Omega)} \approx|u|_{L^{p}(\Omega)}, \quad u \in L^{p}(\Omega) \tag{2.4}\end{equation²}$$Note that (2.3) and (2.4) coincide for $p=2$ by Fubini's theorem.The connection between $H^{\infty}$-calculus and quadratic estimates in [1] is not yet as clean as the statement we know in the Hilbert space setting. Only after introducing randomness, through a notion called $\mathscr{R}$-sectoriality, we arrive at a formulation in $L^{p}(\Omega)$ fully analogous to McIntosh's theorem [3]. In this project, we will explore the intricacies of McIntosh theorem in $L^{p}(\Omega)$. Moreover, we will discuss what happens in a general Banach space $X$ [2]. Note that (2.4) does not have an obvious interpretation in this case, as $|x|^{2}$ has no meaning for $x \in X$ !

### Données de citation

• DOI 10.24350/CIRM.V.20191103
• Citer cette vidéo Lorist, Emiel; Stojanow, Johannes; Sharma, Himani; Pritchard, Andrew (20/06/2024). Project cyan: $H^{\infty}$-calculus and square functions on Banach spaces. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.20191103
• URL https://dx.doi.org/10.24350/CIRM.V.20191103

### Bibliographie

• COWLING, Michael, DOUST, Ian, MICINTOSH, Alan, et al. Banach space operators with a bounded H∞ functional calculus. journal of the australian mathematical society, 1996, vol. 60, no 1, p. 51-89. - https://doi.org/10.1017/S1446788700037393
• KALTON, Nigel et WEIS, Lutz. The $H^{\infty}$-Functional Calculus and Square Function Estimates. arXiv preprint arXiv:1411.0472, 2014. - https://doi.org/10.48550/arXiv.1411.0472
• LE MERDY, Christian. On square functions associated to sectorial operators. Bulletin de la Société Mathématique de France, 2004, vol. 132, no 1, p. 137-156. - https://doi.org/10.24033/bsmf.2462

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