Date of Graduation

5-2014

Document Type

Dissertation

Degree Name

Doctor of Philosophy in Mathematics (PhD)

Degree Level

Graduate

Department

Mathematical Sciences

Advisor

Andrew S. Raich

Committee Member

John R. Akeroyd

Second Committee Member

Marco M. Peloso

Third Committee Member

Philip S. Harrington

Keywords

Pure sciences; Fourier analysis; Schrodinger operators; Several complex variables; Tangential cauchy-riemann complex

Abstract

We present two different results on operator kernels, each in the context of its relationship to a class of CR manifolds M={z,w1,...wn) element of Cn⁺¹ : Im wifi(Re z)} where n d 2 and (phi)i( x) is subharmonic for i = 1,...,n. Such models have proven useful for studying canonical operators such as the Szegö projection on weakly pseudoconvex domains of finite type in C², and may play a similar role in work on higher codimension CR manifolds in C³. Our study in Part II concerns the Szegö kernel on M for which the (empty set)i are subharmonic nonharmonic polynomials. We wish to develop, for n = 2, an approach based on [36] Nagel's estimation of the Szegö kernel through an explicit integral formula when n = 1. After a careful review of his methods and the related control geometry, we write out the analogous integral formula in codimension two. For the "degenerate'' case of M subset C³ with phi1(x) = a(phi)1(x) for a element of R , we prove a simple relationship between the Szegö kernel on M and the kernel on the codimension one CR manifold defined by phi1(x). Part III of the dissertation considers only n = 1. Identifying M with CxR with coordinates (x, y, t) and taking a partial Fourier transform in the y and t directions, (delta)b on L²(M) is transformed to a two parameter family of differential operators Dbar(etatau) = (delta)x - eta +phi '1tau on L² (R). For tau > 0 we study Dbar(etatau)D(etatau) and D(etatau) Dbar(etatau) as real Schrödinger operators on L²(R) . Using Auscher and Ben Ali's work [4] on reverse Hölder potentials, we obtain new upper bounds on the heat kernels associated to these operators for a large class of phi1(x). In fact, our estimates apply to the heat kernel of any Schrödinger operator on L²(Rn) whose potential satisfies a reverse Holder inequality. For Schrödinger operators with potentials in the supremum reverse Holder class, we also prove heat kernel lower bounds derived from van den Berg's estimates on the Dirichlet Laplacian.

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