The Harnack inequality for a class of nonlocal parabolic equations

Abstract

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators [Formula: see text]. Our focus is on non-Euclidean situations.

Regularity of solutions to the fractional Cheeger-Laplacian on domains in metric spaces of bounded geometry

“…To achieve the goals described above, we use the line of investigation implemented by Caffarelli and Silvestre in [9]. Additional tools and structures from [2,3,19,20,27] are also key components in our proofs. The following are the main results of this paper.…”

“…They show that in this case, the solution on 0 := R n × (0, ∞) has an extension to all of × R that is (weighted) harmonic on × R. Then the knowledge that the harmonic functions are locally Hölder continuous and satisfy a Harnack inequality can be used to verify the corresponding property for the Besov energy minimizer on . This approach was extended in [15] to Carnot groups and in [2] to the parabolic setting. For a related non-local problem in the manifold setting, see [10].…”

“…The work [2] studied scale-invariant Harnack inequalities for fractional powers of smooth parabolic and elliptic operators, extending the result of [8] to this generality. Indeed, the smoothness assumption seems to be cosmetic there, and it is not difficult to see that the work of [2] extends also to the setting of sub-Riemannian manifolds. It was pointed out in [2] that their methods extend to a general class of Dirichlet forms and associated infinitesimal generator as the elliptic operator.…”

“…Indeed, the smoothness assumption seems to be cosmetic there, and it is not difficult to see that the work of [2] extends also to the setting of sub-Riemannian manifolds. It was pointed out in [2] that their methods extend to a general class of Dirichlet forms and associated infinitesimal generator as the elliptic operator.…”

“…During the preparation of this manuscript, we became aware of the concurrent work by Baudoin, Lang, and Sire [3], which established the Harnack principle of solutions to fractional Laplace problems in the context of strongly local Dirichlet forms that satisfy a 2-Poincaré inequality, and further studied an analog of the boundary Harnack principle for the case that is an inner uniform domain in X. Their approach, as well as that of [2], based on spectral theory, gave us valuable tools to use in the study undertaken here. In our setting, we consider a specific Dirichlet form given by the measurable inner product structure on a choice of Cheeger differential structure available on the metric space.…”