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We extend the celebrate De Giorgi-Nash-Moser theory to a wide class of nonlinear equations driven by nonlocal, possibly degenerate, integro-differential operators, whose model is the fractional p-Laplacian operator on the Heisenberg-Weyl group H n . Among other results, we prove that the weak solutions to such a class of problems are bounded and Hölder continuous, by also establishing general estimates as fractional Caccioppoli-type estimates with tail and logarithmic-type estimates. [ABSTRACT FROM AUTHOR]

We deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group H n , whose prototype is the Dirichlet problem for the p-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is p = 2 , we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent s goes to 1. [ABSTRACT FROM AUTHOR]