A maximal inequality for stochastic convolutions in 2-smooth Banach spaces

Let $(e^{tA})_{t\geq0}$ be a $C_0$-contraction semigroup on a $2$-smooth Banach space $E$, let $(W_t)_{t\geq0}$ be a cylindrical Brownian motion in a Hilbert space $H$, and let $(g_t)_{t\geq0}$ be a progressively measurable process with values in the space $\gamma(H,E)$ of all $\gamma$-radonifying operators from $H$ to $E$. We prove that for all $0<p<\infty$ there exists a constant $C$, depending only on $p$and $E$, such that for all $T\geq0$ we have $$E\sup_{0\leq t\leq T}\left\Vert\int_0^t\!e^{(t-s)A}\,g_sdW_s\right\Vert^p\leq CE\left(\int_0^T\!\left(\left\Vert g_t\right\Vert_{\gamma(H,E)}\right)^2\,dt\right)^{p/2}.$$ For $p\geq2$ the proof is based on the observation that $\psi(x)=\Vert x\Vert^p$ is Frechet differentiable and its derivative satisfies the Lipschitz estimate $\Vert \psi'(x)-\psi'(y)\Vert\leq C\left(\Vert x\Vert+\Vert y\Vert\right)^{p-2}\Vert x-y\Vert$; the extension to $0<p<2$ proceeds via Lenglart’s inequality.