We establish sharp regularity and Fredholm theorems for the ∂̄b-Neumann problem on domains satisfying some non-generic geometric conditions. We use these domains to construct explicit examples of bad behaviour of the Kohn Laplacian: it is not always hypoelliptic up to the boundary, its partial inverse is not compact and it is not globally subelliptic.

We prove a Liouville-type theorem for bounded stable solutions v ∈ C(R) of elliptic equations of the type (−∆)v = f(v) in R, where s ∈ (0, 1) and f is any nonnegative function. The operator (−∆) stands for the fractional Laplacian, a pseudo-differential operator of symbol |ξ|.

On a manifold with a projective connection we canonically assign a second order differential operator acting on the algebra of all densities to any tensor density S of fixed weight λ. In particular, this implies that on any projectively connected manifold, a ‘bracket’ (symmetric biderivation) on the algebra of functions extends canonically to the algebra of densities.

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. Here {λk} ∞ k=1 are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ := max(0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as λ...

We review some results about nonlocal advection-diffusion equations based on lower bounds for the fractional Laplacian. To Haim, with respect and admiration.

We prove that CR lines in an exponentially degenerate boundary are propagators of holomorphic extension. This explains, in the context of the CR geometry, why in this situation the induced Kohn-Laplacian b is not hypoelliptic (Christ [3]). MSC: 32F10, 32F20, 32N15, 32T25

We address the problem of setting the kernel bandwidth used by Manifold Learning algorithms to construct the graph Laplacian. Exploiting the connection between manifold geometry, represented by the Riemannian metric, and the Laplace-Beltrami operator, we set by optimizing the Laplacian’s ability to preserve the geometry of the data. Experiments show that this principled approach is effective an...

A bounded domain in C with connected Lipschitz boundary is pseudoconvex if the bottom of the essential spectrum of the Kohn Laplacian on the space of (0, q)-forms, 1 ≤ q ≤ n− 1, with L-coefficients is positive.

We prove Lp-bounds for the Riesz transforms d/ √ −∆, δ/ √ −∆ associated with the Hodge-Laplacian ∆ = −δd − dδ equipped with absolute and relative boundary conditions in a Lipschitz subdomain Ω of a (smooth) Riemannian manifoldM, for p in a certain interval depending on the Lipschitz character of the domain.

In this paper we prove that a function u ∈ C(Ω) is the continuous value of the Tug-of-War game described in [19] if and only if it is the unique viscosity solution to the infinity laplacian with mixed boundary conditions