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

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.