Abstract. The fractional Laplacian can be obtained as a Dirichlet-to-Neumann map via an extension problem to the upper half space. In this paper we prove the same type of characterization for the fractional powers of second order partial differential operators in some class. We also get a Poisson formula and a system of Cauchy-Riemann equations for the extension. The method is applied to the fractional harmonic oscillator H σ = (−∆ + |x| 2 ) σ to deduce a Harnack's inequality. A pointwise formula for H σ f (x) and some maximum and comparison principles are derived.

Abstract. Let L = − divx(A(x)∇x) be a uniformly elliptic operator in divergence form in a bounded domain Ω. We consider the fractional nonlocal equationson ∂Ω, andHere L s , 0 < s < 1, is the fractional power of L and ∂ A u is the conormal derivative of u with respect to the coefficients A(x). We reproduce Caccioppoli type estimates that allow us to develop the regularity theory. Indeed, we prove interior and boundary Schauder regularity estimates depending on the smoothness of the coefficients A(x), the right hand side f and the boundary of the domain. Moreover, we establish estimates for fundamental solutions in the spirit of the classical result by Littman-Stampacchia-Weinberger and we obtain nonlocal integro-differential formulas for L s u(x).Essential tools in the analysis are the semigroup language approach and the extension problem.

Abstract. We develop the regularity theory for solutions to space-time nonlocal equations driven by fractional powers of the heat operatorThis nonlocal equation of order s in time and 2s in space arises in Nonlinear Elasticity, Semipermeable Membranes, Continuous Time Random Walks and Mathematical Biology. It plays for spacetime nonlocal equations like the generalized master equation the same role as the fractional Laplacian for nonlocal in space equations. We obtain a pointwise integro-differential formula for (∂t−∆) s u(t, x) and parabolic maximum principles. A novel extension problem to characterize this nonlocal equation with a local degenerate parabolic equation is proved. We show parabolic interior and boundary Harnack inequalities, and an Almgrem-type monotonicity formula. Hölder and Schauder estimates for the space-time Poisson problem are deduced using a new characterization of parabolic Hölder spaces. Our methods involve the parabolic language of semigroups and the Cauchy Integral Theorem, which are original to define the fractional powers of ∂t − ∆. Though we mainly focus in the equation (∂t − ∆) s u = f , applications of our ideas to variable coefficients, discrete Laplacians and Riemannian manifolds are stressed out.

The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size h > 0for u, f : Z h → R, 0 < s < 1, is performed. The pointwise nonlocal formula for (−∆ h ) s u and the nonlocal discrete mean value property for discrete s-harmonic functions are obtained. We observe that a characterization of (−∆ h ) s as the Dirichlet-to-Neumann operator for a semidiscrete degenerate elliptic local extension problem is valid. Regularity properties and Schauder estimates in discrete Hölder spaces as well as existence and uniqueness of solutions to the nonlocal Dirichlet problem are shown. For the latter, the fractional discrete Sobolev embedding and the fractional discrete Poincaré inequality are proved, which are of independent interest. We introduce the negative power (fundamental solution)which can be seen as the Neumann-to-Dirichlet map for the semidiscrete extension problem. We then prove the discrete Hardy-Littlewood-Sobolev inequality for (−∆ h ) −s .As applications, the convergence of our fractional discrete Laplacian to the (continuous) fractional Laplacian as h → 0 in Hölder spaces is analyzed. Indeed, uniform estimates for the error of the approximation in terms of h under minimal regularity assumptions are obtained. We finally prove that solutions to the Poisson problem for the fractional Laplacian (−∆) s U = F, in R, can be approximated by solutions to the Dirichlet problem for our fractional discrete Laplacian, with explicit uniform error estimates in terms of h.2010 Mathematics Subject Classification. Primary: 35R11, 49M25. Secondary: 35K05, 65N15. Key words and phrases. Nonlocal discrete diffusion equations, fractional discrete Laplacian, regularity and extension problem, Sobolev and Poincaré inequalities, error of approximation, semidiscrete heat equation.

Abstract. We extend results of Caffarelli-Silvestre and Stinga-Torrea regarding a characterization of fractional powers of differential operators via an extension problem. Our results apply to generators of integrated families of operators, in particular to infinitesimal generators of bounded C 0 semigroups and operators with purely imaginary symbol. We give integral representations to the extension problem in terms of solutions to the heat equation and the wave equation.