The fractional nonlocal Ornstein--Uhlenbeck equation, Gaussian symmetrization and regularity

For 0 < s < 1, we consider the Dirichlet problem for the fractional nonlocal Ornstein– Uhlenbeck equation ((−∆ + x · ∇)^s u = f in Ω, u = 0 on ∂Ω, where Ω is a possibly unbounded open subset of Rn, n ≥ 2. The appropriate functional settings for this nonlocal equation and its corresponding extension problem are developed. We apply Gaussian symmetrization techniques to derive a concentration comparison estimate for solutions. As consequences, novel L^p and L^p(log L)^α regularity estimates in terms of the datum f are obtained by comparing u with half-space solutions.