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We establish sharp interior and boundary regularity estimates for solutions to ∂ t u − L u = f ( t , x ) in I × Ω , with I ⊂ R and Ω ⊂ R n . The operators L we consider are infinitesimal generators of stable Lévy processes. These are linear nonlocal operators with kernels that may be very singular. On the one hand, we establish interior estimates, obtaining that u is C 2 s + α in x and C 1 + α 2 s in t , whenever f is C α in x and C α 2 s in t . In the case f ∈ L ∞ , we prove that u is C 2 s − ϵ in x and C 1 − ϵ in t , for any ϵ > 0 . On the other hand, we study the boundary regularity of solutions in C 1 , 1 domains. We prove that for solutions u to the Dirichlet problem the quotient u / d s is Hölder continuous in space and time up to the boundary ∂Ω, where d is the distance to ∂Ω. This is new even when L is the fractional Laplacian. [ABSTRACT FROM AUTHOR]

We study the regularity of minimizers of the functional E (u) : = [ u ] H s (Ω) 2 + ∫ Ω f u. This corresponds to understanding solutions for the regional fractional Laplacian in Ω ⊂ R N. More precisely, we are interested on the global (up to the boundary) regularity of solutions, both in the case of free minimizers in H s (Ω) (i.e., Neumann problem), or in the case of Dirichlet condition u ∈ H 0 s (Ω) when s > 1 2. Our main result establishes the existence of a constant α s ∈ (0 , 1 − s) satisfying 2 s + α s > 1 such that for all α ∈ (0 , α s) the solution u ∈ C 2 s + α (Ω ‾) in the Neumann case, and u / δ 2 s − 1 ∈ C 1 + α (Ω ‾) in the Dirichlet case. Here, δ is the distance to ∂Ω. We also show the optimality of our result: these estimates fail for α > α s , even when f and ∂Ω are C ∞. [ABSTRACT FROM AUTHOR]