Fractional smoothness for the generalized local time of the indefinite Skorohod integral ✩

Let Xt = t 0 us dWs be the indefinite Skorohod integral on Wiener space (Ω,H,P), and let Lt (x) be its the generalized local time introduced by Tudor in [C.A. Tudor, Martingale-type stochastic calculus for anticipating integral processes, Bernoulli 10 (2004) 313–325].We prove that the generalized local time, as a nonlinear functional of ω, is in the fractional Sobolev spaces Dα,p (α < 12 andp >2) under some conditions imposed on the anticipating integrand u via the technique of Malliavin calculus and the K-method in the real interpolation theory. The result is optimal for the fractional Brownian motion with the Hurst parameter h ∈ (0, 12 ). © 2006 Elsevier Inc. All rights reserved