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

• * Corresponding author: Bruno Volzone
• For $0<s<1$, we consider the Dirichlet problem for the fractional nonlocal Ornstein-Uhlenbeck equation

$\begin{cases} (-Δ+x·\nabla)^su = f,&\hbox{in}~Ω,\\ u = 0,&\hbox{on}~\partialΩ, \end{cases}$

where $Ω$ is a possibly unbounded open subset of $\mathbb{R}^n$, $n≥2$. 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.

Mathematics Subject Classification: Primary: 35R11, 35B65, 35A01; Secondary: 28C20, 35K08, 46E35, 60J35.

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