Optimal 𝐿^{𝑝} and Hölder estimates for the Kohn solution of the \overline∂-equation on strongly pseudoconvex domains

Abstract

Let Ω \Omega be an open, relatively compact subset in C n + 1 {{\mathbf {C}}^{n + 1}} , and assume the boundary of Ω \Omega , ∂ Ω \partial \Omega , is smooth and strongly pseudoconvex. Let Op ⁡ ( K ) \operatorname {Op}(K) be an integral operator with mixed type homogeneities defined on Ω ¯ \overline \Omega : i.e., K K has the form as follows: \[ ∑ k , l ≥ 0 E k H l , \sum \limits _{k,l \geq 0} {{E_k}{H_l},} \] where E k {E_k} is a homogeneous kernel of degree − k - k in the Euclidean sense and H l {H_l} is homogeneous of degree − l - l in the Heisenberg sense. In this paper, we study the optimal L p {L^p} and Hölder estimates for the kernel K K . We also use Lieb-Range’s method to construct the integral kernel for the Kohn solution ∂ ∗ ¯ N \overline {{\partial ^\ast }} {\mathbf {N}} of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to ∂ ∗ ¯ N \overline {{\partial ^\ast }} {\mathbf {N}} . On the other hand, we prove Lieb-Range’s kernel gains 1 1 in "good" directions (hence gains 1 / 2 1/2 in all directions) via Phong-Stein’s theory. We also discuss the transferred kernel from the Siegel upper-half space to Ω \Omega .