Auteurs : Lorist, Emiel (Coordinateur) ;
Stojanow, Johannes (Auteur de la Conférence) ;
Sharma, Himani (Auteur de la Conférence) ;
Pritchard, Andrew (Auteur de la Conférence)
CIRM (Editeur )
Résumé :
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$$\begin{equation*}\left(\int_{0}^{\infty}\|f(t L) u\|_{H}^{2} \frac{\mathrm{d} t}{t}\right)^{1 / 2} \approx\|u\|_{H}, \quad u \in H \tag{2.3}\end{equation*}$$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$$\begin{equation*}\left\|\left(\int_{0}^{\infty}|f(t L) u|^{2} \frac{\mathrm{d} t}{t}\right)^{1 / 2}\right\|_{L^{p}(\Omega)} \approx\|u\|_{L^{p}(\Omega)}, \quad u \in L^{p}(\Omega) \tag{2.4}\end{equation*}$$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 $x \in X$ !
Keywords : holomorphic functional calculus; square function; Banach space
Codes MSC :
42B25
- Maximal functions, Littlewood-Paley theory
47A60
- Functional calculus
47D06
- One-parameter semigroups and linear evolution equations
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Informations sur la Rencontre
Nom de la rencontre : Harmonic analysis techniques for elliptic operators / Techniques d'analyse harmonique pour des opérateurs elliptiques Organisateurs de la rencontre : Egert, Moritz ; Haller, Robert ; Monniaux, Sylvie ; Tolksdorf, Patrick Dates : 17/06/2024 - 21/06/2024
Année de la rencontre : 2024
URL Congrès : https://conferences.cirm-math.fr/2972.html
DOI : 10.24350/CIRM.V.20191103
Citer cette vidéo:
Lorist, Emiel ;Stojanow, Johannes ;Sharma, Himani ;Pritchard, Andrew (2024). Project cyan: $H^{\infty}$-calculus and square functions on Banach spaces. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.20191103
URI : http://dx.doi.org/10.24350/CIRM.V.20191103
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Voir aussi
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[Multi angle]
Project purple: $L^{p}$-extrapolation à la Blunck-Kunstmann
/ Coordinateur Vogt, Hendrik ; Auteur de la Conférence Heidrich, Erik ; Auteur de la Conférence Söder, Charlotte ; Auteur de la Conférence Qi, Siguang ; Auteur de la Conférence Lenz, Jonas.
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[Multi angle]
Project red: $\mathscr{R}$-sectorial Operators and Maximal Regularity
/ Coordinateur Klioba, Katharina ; Coordinateur Seifert, Christian ; Coordinateur Trostorff, Sascha ; Auteur de la Conférence Carvalho, Francisco ; Auteur de la Conférence Ruff, Maximilian.
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[Multi angle]
Project violet: T(1) and T(b) theorems and applications
/ Coordinateur Bortz, Simon ; Auteur de la Conférence Migliaccio, Alessandra ; Auteur de la Conférence Lauterbach, Sven ; Auteur de la Conférence van Dijk, Dann.
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[Multi angle]
Project orange: Parabolic maximal regularity and the Kato square root property
/ Coordinateur Arendt, Wolfgang ; Coordinateur Schlierf, Manuel ; Auteur de la Conférence Abahmami, Sofian ; Auteur de la Conférence Heister, Henning ; Auteur de la Conférence Jahandideh, Azam ; Auteur de la Conférence Leone, Vinzenzo.
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