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 Hölder continuous, while also establishing general estimates, such as fractional Caccioppoli-type estimates with tail terms and logarithmic-type bounds.