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

Abstract

This article is the continuation of the work [30] where we had proved maximal estimates

$$\begin{aligned} \left\| \sup _{t > 0} |m(tA)f| \, \right\| _{L^p(\Omega ,Y)} \leqslant C \left\| f\right\| _{L^p(\Omega ,Y)} \end{aligned}$$

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 $m : (0, \infty ) \rightarrow \mathbb {C}$ 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 [13, 44,45,46, 52, 61] 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 [52] to the case of UMD lattice-valued spaces. A second main result yields a maximal estimate

$$\begin{aligned} \left\| \sup _{t > 0} |m(tA) f_t| \, \right\| _{L^p(\Omega ,Y)} \leqslant C \left\| f_t\right\| _{L^p(\Omega ,Y(\Lambda ^\beta ))} \end{aligned}$$

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 solutions to variants of Carleson’s problem of pointwise convergence [18]

$$\begin{aligned} \exp (itA)f(x,\omega ) \rightarrow f(x,\omega ) \text { a. e. }(x,\omega ) \quad (t \rightarrow 0+) \end{aligned}$$

for A a Fourier multiplier operator or a differential operator on an open domain $\Omega \subseteq \mathbb {R}^d$ with boundary conditions.