q-variational Hörmander functional calculus and Schrödinger and wave maximal estimates - Archive ouverte HAL Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2024

q-Variationen des Hörmander-Funktionalkalküls und Schrödinger- und Wellen-Maximalabschätzungen

q-variational Hörmander functional calculus and Schrödinger and wave maximal estimates

q-variations du calcul fonctionnel de Hörmander et estimations maximales de Schrödinger et des ondes

Résumé

This article is the continuation of the work [DK] where we had proved maximal estimates \[ ||\sup_{t > 0} |m(tA)f| \: ||_{L^p(\Omega,Y)} \leq C ||f||_{L^p(\Omega,Y)} \] for sectorial operators $A$ acting on $L^p(\Omega,Y)$ ($Y$ being a UMD lattice) and admitting a H\"ormander functional calculus (a strengthening of the holomorphic $H^\infty$ calculus to symbols $m$ differentiable on $(0,\infty)$ in a quantified manner), and $m : (0, \infty) \to \C$ being a H\"ormander class symbol with certain decay at $\infty$. In the present article, we show that under the same conditions as above, the scalar function $t \mapsto m(tA)f(x,\omega)$ is of finite $q$-variation with $q > 2$, a.e. $(x,\omega)$. This extends recent works by [BMSW,HHL,HoMa1,HoMa,JSW,LMX] who have considered among others $m(tA) = e^{-tA}$ the semigroup generated by $-A$. As a consequence, we extend estimates for spherical means in euclidean space from [JSW] to the case of UMD lattice-valued spaces. A second main result yields a maximal estimate \[ ||\sup_{t > 0} |m(tA) f_t| \: ||_{L^p(\Omega,Y)} \leq C \norm||f_t||_{L^p(\Omega,Y(\Lambda^\beta))} \] for the same $A$ and similar conditions on $m$ as above but with $f_t$ depending itself on $t$ such that $t \mapsto f_t(x,\omega)$ belongs to a Sobolev space $\Lambda^\beta$ over $(\R_+, \frac{dt}{t})$. We apply this to show a maximal estimate of the Schr\"odinger (case $A = -\Delta$) or wave (case $A = \sqrt{-\Delta}$) solution propagator $t \mapsto \exp(itA)f$. Then we deduce from it variants of Carleson's problem of pointwise convergence [Car] \[ \exp(itA)f(x,\omega) \to f(x,\omega) \text{ a. e. }(x,\omega) \quad (t \to 0+)\] for $A$ a Fourier multiplier operator or a differential operator on an open domain $\Omega \subseteq \R^d$ with boundary conditions.
Fichier principal
Vignette du fichier
Maximal-Hoermander-q-Var-revised-v03.pdf (628.63 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-04526930 , version 1 (29-03-2024)

Identifiants

  • HAL Id : hal-04526930 , version 1

Citer

Luc Deleaval, Christoph Kriegler. q-variational Hörmander functional calculus and Schrödinger and wave maximal estimates. 2024. ⟨hal-04526930⟩
0 Consultations
3 Téléchargements

Partager

Gmail Facebook X LinkedIn More