TAILIEUCHUNG - Đề tài " The ¯ ∂b-complex on decoupled boundaries in Cn "

Introduction The purpose of this paper is to prove optimal estimates for solutions of the Kohn-Laplacian for certain classes of model domains in several complex variables. This will be achieved by applying a type of singular integral operator whose novel features (related to product theory and flag kernels) differ essentially from the | Annals of Mathematics The db-complex on decoupled boundaries in Cn By Alexander Nagel and Elias M. Stein Annals of Mathematics 164 2006 649 713 The Ỡ5-complex on decoupled boundaries in Cn By Alexander Nagel and Elias M. Stein Contents 1. Introduction 2. Definitions and statement of results 3. Geometry and analysis on Mj and on M1 X X Mn 4. Relative fundamental solutions for on M1 X X Mn 5. Transference from M1 X X Mn to M and Lp regularity of K 6. Pseudo-metrics on M 7. Differential inequalities for the relative fundamental solution K 8. Holder regularity for K 9. Examples References 1. Introduction The purpose of this paper is to prove optimal estimates for solutions of the Kohn-Laplacian for certain classes of model domains in several complex variables. This will be achieved by applying a type of singular integral operator whose novel features related to product theory and flag kernels differ essentially from the more standard Calderon-Zygmund operators that have been used in these problems hitherto. . Background. We consider the Kohn-Laplacian on Q-forms b2 dbdb dfrdb defined on the boundary M ỠQ of a smooth bounded pseudo-convex domain Q c Cn. Our objective is the study of the relative inverse operator K and the corresponding Szego projection S when it exists which satisfy è K K è I S. By definition S is the orthogonal projection on the L2 null-space of . Research supported in part by grants from the National Science Foundation. 650 ALEXANDER NAGEL AND ELIAS M. STEIN In formulating the questions of regularity pertaining to the above it is useful to recall Fefferman s hierarchy Fef95 of the levels of understanding of the problem which we rephrase as follows 1 Proof of C x regularity. 2 Derivation of optimal Lp Holder and Sobolev-space estimates of solutions. 3 Analysis of singularities of the distribution kernels of the operators K and S and derivation of the estimates in 2 from a corresponding theory of singular integrals. Now as far as the C x regularity .

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