Local Behavior of Fractional Equations in Grushin-type Spaces

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 H\"{o}lder continuous, while also establishing general estimates, such as fractional Caccioppoli-type estimates with tail terms and logarithmic-type bounds.