| 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 ∈Cn+1 :Im wifiRe z where n ≤ 2 and &phis;i( x) is subharmonic for i = 1,...,n. Such models have proven useful for studying canonical operators such as the Szegő 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 Szegő kernel on M for which the &phis;i are subharmonic nonharmonic polynomials. We wish to develop, for n = 2, an approach based on [36] Nagel's estimation of the Szegő 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 ⊂ C3 with &phis;2(x) = a&phis; 1(x) for a ∈ R , we prove a simple relationship between the Szegő kernel on M and the kernel on the codimension one CR manifold defined by &phis; 1(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, 6b&d1; on L2(M) is transformed to a two parameter family of differential operators D¯ etatau = ∂x - eta + &phis; '1tau on L2 R . For tau > 0 we study D¯ etatauDetatau and D etatauD¯etatau as real Schrodinger operators on L2 R . Using Auscher and Ben Ali's work on reverse Holder potentials, we obtain new upper bounds on the heat kernels associated to these operators for a large class of &phis;1(x). In fact, our estimates apply to the heat kernel of any Schrodinger operator on L2 Rn whose potential satisfies a reverse Holder inequality. For Schrodinger 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. |
