On maximal Sobolev and Holder estimates for the tangential Cauchy-Riemann operator and boundary laplacian

Let M be the boundary of a (smoothly bounded) pseudoconvex domain in C-n (n greater than or equal to 3), or more generally any compact pseudoconvex CR-manifold of dimension 2n - 1 for which the range of partial derivative(b) is closed in L-2. In this article. we study the L-P-Sobolev and Holder regularity. properties of ab 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 q(0) eigenvalues of the Levi matrix are comparable to its trace near a point of finite commutator type ("Condition D(q(0))"), then the inverse K-q Of square(b) on (0, q)-forms for q(0) less than or equal to q less than or equal to n - 1 - q(0) satisfies sharp kernel estimates in terms of the quasi-distance associated to the Hormander sum of squares operator. In particular, we obtain the "maximal L-P estimates" for square(b) which were conjectured in the 1980s. we also prove sharp estimates for certain parts of the kernels of Kq(0)-1 and K-n-q(0). and give some applications concerning domains with at most one degenerate eigenvalue. Finally, 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 Sobolev and Lipschitz norms, both isotropic and nonisotropic.