Robust Hölder Estimates for Parabolic Nonlocal Operators

Abstract

In this work we study parabolic equations determined by nonlocal operators in a general framework of bounded and measurable coefficients. Our emphasis is on the weak Harnack inequality and Hölder regularity estimates for solutions of such equations. We allow the underlying jump measures to be singular with a singularity that depends on the coordinate direction. This approach also allows to study several classes of non-singular jump measures that have not been dealt with so far. The main estimates are robust in the sense that the constants can be chosen independently of the order of differentiability of the operators.

Regularity estimates for fractional orthotropic $p$-Laplacians of mixed order

“…The proof is analog to the proof of the Poincaré inequality for the case p = 2, see [15,Theorem 4.2].…”

“…This subsection is devoted to the proofs of a Sobolev and a Poincaré-type inequality. We start our analysis by first proving a technical lemma, see also [15,Lemma 4…”

“…This assumption allows us for instance to study operators of the form (1.2) for µ axes in the general framework of bounded and measurable coefficients. We emphasize that further examples of families of measures satisfying (1.5) can be constructed similarly to the case p = 2 (see [15,Section 9]). In this paper, we study nonlocal operators of the form (1.2) for families of measures that satisfy the previously given assumptions.…”

“…Our approach follows mainly the ideas of [25,14] and [15]. In [25], the authors develop a local regularity theory for a class of linear nonlocal operators which covers the case s = s 1 = • • • = s n ∈ (0, 1) and p = 2.…”

“…, s n to be different. In [15] parabolic equations in the case p = 2 and possible different orders of differentiability are studied. That paper provides robust regularity estimates, which means the constants in the weak Harnack inequality and Hölder regularity estimate do not depend on the orders of differentiability but on their lower only.…”