Abstract. We show that the conformally invariant fractional powers of the sub-Laplacian on the Heisenberg group are given in terms of the scattering operator for an extension problem to the Siegel upper halfspace. Remarkably, this extension problem is different from the one studied, among others, by Caffarelli and Silvestre.

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Abstract. In this paper we introduce the notion of Einstein-type structure on a Riemannian manifold (M, g), unifying various particular cases recently studied in the literature, such as gradient Ricci solitons, Yamabe solitons and quasi-Einstein manifolds. We show that these general structures can be locally classified when the Bach tensor is null. In particular, we extend a recent result of Cao and Chen [8].

MSC: 35J10 35B30 35D05 35P05 46E35 Keywords: Degenerate elliptic operators Weak solutions Maximum principles Principal eigenvalueFor second order linear equations and inequalities which are degenerate elliptic but which possess a uniformly elliptic direction, we formulate and prove weak maximum principles which are compatible with a solvability theory in suitably weighted versions of L 2 -based Sobolev spaces. The operators are not necessarily in divergence form, have terms of lower order, and have low regularity assumptions on the coefficients. The needed weighted Sobolev spaces are, in general, anisotropic spaces defined by a non-negative continuous matrix weight. As preparation, we prove a Poincaré inequality with respect to such matrix weights and analyze the elementary properties of the weighted spaces. Comparisons to known results and examples of operators which are elliptic away from a hyperplane of arbitrary codimension are given. Finally, in the important special case of operators whose principal part is of Grushin type, we apply these results to obtain some spectral theory results such as the existence of a principal eigenvalue.

Communicated by Enzo Mitidieri MSC: 35J70 35J60 35B65 Keywords: Quasilinear equations Degenerate elliptic partial differential equations Degenerate quadratic forms Weak solutions Regularity Harnack's inequality Hölder continuity Moser method a b s t r a c tWe continue to study regularity results for weak solutions of the large class of second order degenerate quasilinear equations of the form div (A(x, u, ∇u)) = B(x, u, ∇u) for x ∈ Ω as considered in our paper Monticelli et al. (2012). There we proved only local boundedness of weak solutions. Here we derive a version of Harnack's inequality as well as local Hölder continuity for weak solutions. The possible degeneracy of an equation in the class is expressed in terms of a nonnegative definite quadratic form associated with its principal part. No smoothness is required of either the quadratic form or the coefficients of the equation. Our results extend ones obtained by J. Serrin (1964) andN. Trudinger (1967) for quasilinear equations, as well as ones for subelliptic linear equations obtained in Wheeden (2006, 2010).