Andrew Raich

Regularity Results for on CR-Manifolds of Hypersurface Type

We introduce a class of CR-manifolds of hypersurface type called weak Y(q)-manifolds that includes Y(q) manifolds and q-pseudoconvex manifolds. We develop the L 2-regularity theory of the complex Green operator on weak Y(q) manifolds and show that and the Kohn Laplacian have closed range at all Sobolev levels, the space of harmonic forms is finite dimensional, the Szegö kernel is continuous and can be solved in C ∞ on the appropriate forms levels. Our argument involves building a weighted norm from a microlocal decomposition.

One-parameter families of operators in ℂ

AbstractWe introduce classes of one-parameter families (OPF) of operators on Cct8(ℂ) which characterize the behavior of operators associated to the

DEFINING FUNCTIONS FOR UNBOUNDED C m DOMAINS

\bar \partial - problem

Lp estimates for the

problem in the weighted space L2 (ℂ, e−2p) where p is a subharmonic, nonharmonic polynomial. We prove that an order 0 OPF operator extends to a bounded operator from Lq (ℂ) to itself, 1 < q < ∞, with a bound that depends on q and the degree of p but not on the parameter τ or the coefficients of p. Last, we show that there is a one-to-one correspondence given by the partial Fourier transform in τ between OPF operators of order m ≤ 2 and nonisotropic smoothing (NIS) operators of order m ≤ 2 on polynomial models in ℂ2.

\bar\partial

Let L = −1/4 The purpose of this note is to present a simplified calculation of the Fourier transform of fundmental solution of theb-heat equation on the Heisenberg group. The Fourier transform of the fundamental solution has been computed by a number of authors (Gav77, Hul76, CT00, Tie06). We use the approach of (CT00, Tie06) and compute the heat kernel using Hermite functions but differ from the earlier approaches by working on a different, though biholomorphically equivalent, version of the Heisenberg group. The simplification in the computation occurs because the differential operators on this equivalent Heisenberg group take on a simpler form. Moreover, in the proof of Theorem 1.2, we reduce the n-dimensional heat equation to a 1-dimensional heat equation, and this technique would also be useful when analyzing the heat equation on the nonisotropic Heisenberg group (e.g., see (CT00)). We actually use the same version of the Heisenberg group as Hulanicki (Hul76), but he computes the fundamental solution of the heat equation associated to the sub-Laplacian and not the Kohn Laplacian acting on (0,q)-forms.

-equation on a class of infinite type domains

For a domain

A remark on boundary estimates on unbounded Z(q) domains in

\Omega\subset\mathbb R^n

Schrödinger Operators with \(A_{\infty }\) Potentials

, we introduce the concept of a uniformly