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

Let (e^{tA})_{t \geq 0} be a C_0-contraction semigroup on a 2-smooth Banach space E, let (W_t)_{t \geq 0} be a cylindrical Brownian motion in a Hilbert space H, and let (g_t)_{t \geq 0} be a progressively measurable process with values in the space γ(H,E) of all γ-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 \geq 0 we have \E \sup_{0\le t\le T} || \int_0^t e^{(t-s)A} g_s dW_s \ ||^p \leq C \mathbb{E} (\int_0^T || g_t ||_{γ(H,E)}^2 dt)^\frac{p}{2}. For p \geq 2 the proof is based on the observation that ψ(x) = || x ||^p is Fréchet differentiable and its derivative satisfies the Lipschitz estimate || ψ'(x) - ψ'(y)|| \leq C(|| x || + || y ||)^{p-2} || x-y ||; the extension to 0<p<2 proceeds via Lenglart's inequality.