Maximal inequalities for stochastic convolutions driven by compensated Poisson random measures in Banach spaces

We consider a Banach space (E, vertical bar vertical bar center dot vertical bar vertical bar) such that, for some q = 2, the function x bar right arrow vertical bar vertical bar x vertical bar vertical bar(q) is of C-2 class and its kth, k = 1, 2, Frechet derivatives are bounded by some constant multiples of the (q -k) th power of the norm. We also consider a C-0-semigroup S of contraction type on (E, vertical bar vertical bar center dot vertical bar vertical bar). Finally we consider a compensated Poisson random measure (N) over tilde on a measurable space (Z, (Z) over bar). We study the following stochastic convolution process u(t) = integral(t)(0)integral(Z) S(t - s)xi(s, z) (N) over tilde (ds, dz), t >= 0, where xi : [0, infinity) x Omega x Z -> E is an F circle times (Z) over bar -predictable function. We prove that there exists a cadlag modification (u) over tilde of the process u which satisfies the following maximal type inequality E sup(0 <= s <= t) vertical bar vertical bar(u) over tilde (s)vertical bar vertical bar(q') <= CE (integral(t)(0)integral(Z) vertical bar vertical bar xi(s, z)vertical bar vertical bar(p) N(dz, dz))(q'/p), for all q' >= q and 1< p <= 2 with C = C(q, p).