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, 44–46, 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.
