Local Behavior of Fractional Equations in Grushin-type Spaces

TL;DR

This work extends De Giorgi-Nash-Moser regularity theory to nonlinear, nonlocal equations driven by integro-differential operators on Grushin-type spaces, using the fractional $p$-Laplacian as a prototypical model. It adopts a variational approach, showing that $p$-minimizers of a nonlocal energy are equivalent to weak solutions of the Euler-Lagrange problem, and proves core regularity results: local boundedness and Hölder continuity of solutions. Central technical contributions include nonlocal Caccioppoli-type inequalities with tail, a Logarithmic Lemma, and a precise oscillation-decay argument adapted to the sub-Riemannian, center-dependent volume growth of Grushin-type spaces, yielding Hölder exponents $\alpha<\frac{sp}{p-1}$. Together, these results extend the DG-M theory to a degenerate, nonlocal setting and clarify how tail terms and Grushin geometry influence regularity in this context.

Abstract

In this paper, we establish the De Giorgi-Nash-Moser theory for a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, with the fractional $p$-Laplacian operator in Grushin-type spaces $\mathbb{G}^n$ serving as a prototypical example. Among other results, we prove that the weak solutions to this class of problems are both bounded and Hölder continuous, while also establishing general estimates, such as fractional Caccioppoli-type estimates with tail terms and logarithmic-type bounds.

Theorems & Definitions (19)

• Theorem 1.1: Local boundedness
• Theorem 1.2: Hölder continuity
• Theorem 1.3: Caccioppoli estimates with tail
• Theorem 1.4: Logarithmic Lemma
• Remark 1.5
• Remark 1.6
• Lemma 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.4