$q$-variational H{ö}rmander functional calculus and Schr{ö}dinger and wave maximal estimates

Abstract

This article is the continuation of the work [DK] where we had proved maximal estimates $${\Vert \underset{t>0}{\sup }|m(tA)f|\Vert }_{L^p(\Omega ,Y)}\le C\Vert f{\Vert }_{L^p(\Omega ,Y)}$$ for sectorial operators $A$ acting on $L^p(\Omega ,Y)$ ($Y$ being a UMD lattice) and admitting a Hörmander functional calculus(a strengthening of the holomorphic $H^{\infty }$ calculus to symbols $m$ differentiable on $(0,\infty )$ in a quantified manner), and $m:(0,\infty )\to \mathbb{C}$ being a Hörmander 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 $${\Vert \underset{t>0}{\sup }|m(tA)f_t|\Vert }_{L^p(\Omega ,Y)}\le C\Vert f_t{\Vert }_{L^p(\Omega ,Y({\Lambda }^{\beta }))}$$ 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 $({\mathbb{R}}_{+},\frac{dt}{t})$.We apply this to show a maximal estimate of the Schrödinger (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]\[ \exp(itA)f(x,ω) \to f(x,ω) \text{ a. e. }(x,ω) \quad (t \to 0+)\]for $A$ a Fourier multiplier operator or a differential operator on an open domain $\Omega \subseteq {\mathbb{R}}^d$ with boundary conditions.

arXiv admin note: text overlap with arXiv:2203.03263