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 2n - 1 for which the range of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is closed in L2. In this article, we study the Lp-Sobolev and Hölder regularity properties of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] and □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 Kq of □b on (0, q)-forms for q0qn - 1 - q0 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 □b which were conjectured in the 1980s. We also prove sharp estimates for certain parts of the kernels of Kq0-1 and Kn-q0 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 Kq (and related operators) in the Sobolev and Lipschitz norms, both isotropic and nonisotropic.


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pp. 129-197
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