On maximal Sobolev and Hoelder estimates for the tangential Cauchy-Riemann operator and boundary Laplacian

Let M be the boundary of a (smoothly bounded) pseudoconvex domain in Cn (n ≥ 3), or more generally any compact, pseudoconvex CR-manifold of dimension 2 n − 1 for which the range of $6b$ is closed in L2. In this dissertation, we study the Lp-Sobolev and Hölder regularity properties of $6b$ and $\square b$ near a point of finite type under a comparable eigenvalues condition on the Levi form.; We show that if all possible sums of q₀ eigenvalues of the Levi matrix are comparable to its trace near a point of finite commutator type (“Condition D(q₀)”), then the inverse K_q of $\square b$ on (0, q)-forms for q₀ ≤ q ≤ n−1−q ₀ satisfies sharp kernel estimates in terms of the quasi-distance associated to the Hörmander sum of squares operator. In particular, we obtain the “maximal Lp estimates” for $\square b$ which were conjectured in the 1980s. We also prove sharp estimates for certain parts of the kernels of K_q0−1 and K_n−q0 and give some applications concerning domains with at most one degenerate eigenvalue.; In the second part of this work, we establish the composition and mapping properties of a class of singular integral (nonisotropic smoothing) operators that arises naturally in complex analysis. These results yield optimal regularity of K_q (and related operators) in the nonisotropic Sobolev and Lipschitz norms.