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

Abstract

Let M be the boundary of a (smoothly bounded) pseudoconvex domain in C n ( n ≥ 3), or more generally any compact pseudoconvex CR-manifold of dimension 2 n - 1 for which the range of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] is closed in L 2 . In this article, we study the L p -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 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 □ b on (0, q )-forms for q 0 ≤ 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 L p estimates" for □ b which were conjectured in the 1980s. We also prove sharp estimates for certain parts of the kernels of K q 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.