On H-type sub-Riemannian manifolds we establish sub-Hessian and sub-Laplacian comparison theorems which are uniform for a family of approximating Riemannian metrics converging to the sub-Riemannian one. We also prove a sharp sub-Riemannian Bonnet-Myers theorem that extends to this general setting results previously proved on contact and quaternionic contact manifolds.The above formulas show that the Hladky connection defined relative to g ε in (2.1) will coincide for all choices of ε > 0.Let (M, g) be a Riemannian manifold, equipped with an orthogonal splitting T M = H ⊕ V. If V is integrable and metric, then (M, H, g) is a Riemannian foliation with bundlelike metric and totally geodesic leaves, tangent to V. We simply refer to these structures as totally geodesic foliations. In this foliation context, Hladky connection is referred to as the Bott connection (see [17]). The H-type and the J 2 conditionsWe introduce a condition that will play a prominent role in the following. For any Z ∈ T M, let J Z : T M → T M be defined asWe remark that J is defined for any Riemannian manifold (M, g) equipped with an orthogonal splitting T M = H ⊕ V.Remark 2.3. If (M, g) is a totally geodesic foliation, it holdsDefinition 2.4. We say that the H-type condition is satisfied ifDefinition 2.5 ([26, 22]). We say that J 2 condition holds if for all Z, Z ′ ∈ Γ(V), X ∈ Γ(H) with Z, Z ′ = 0 there exists Z ′′ ∈ Γ(V) such that

In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators [Formula: see text]. Our focus is on non-Euclidean situations.

Introduction 1 2. Preliminaries 3 2.1. Horizontal Laplaceans 5 2.2. Intrinsic distance and gauge pseudo-distance 6 2.3. Homogeneous polynomials 7 2.4. The Folland-Stein Hölder classes 8 2.5. The characteristic set 9 3. Some basic regularity estimates 9 4. Proof of Theorem 1.1 14 References 25

In this paper, we obtain gradient continuity estimates for viscosity solutions of ∆ N p u = f in terms of the scaling critical L(n, 1) norm of f , where ∆ N p is the normalized p−Laplacian operator defined in (1.2) below. Our main result, Theorem 2.2, corresponds to the borderline gradient continuity estimate in terms of the modified Riesz potentialĨ f q . Moreover, for f ∈ L m with m > n, we also obtain C 1,α estimates, see Theorem 2.3 below. This improves one of the regularity results in [3], where a C 1,α estimate was established depending on the L m norm of f under the additional restriction that p > 2 and m > max(2, n, p 2 ) (see Theorem 1.2 in [3]). We also mention that differently from the approach in [3], which uses methods from divergence form theory and nonlinear potential theory in the proof of Theorem 1.2, our method is more non-variational in nature, and it is based on separation of phases inspired by the ideas in [36]. Moreover, for f continuous, our approach also gives a somewhat different proof of the C 1,α regularity result, Theorem 1.1, in [3].

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