TY - JOUR

T1 - Optimal lpand holder estimates for the kohn solution of the ∂-equation on strongly pseudoconvex domains

AU - Ruan, Zhong Jin

N1 - Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 1989/9

Y1 - 1989/9

N2 - Let Ω be an open, relatively compact subset in Cn+1, and assume the boundary of Ω, ∂Ω, is smooth and strongly pseudoconvex. Let Op(K) be an integral operator with mixed type homogeneities defined on Ω: i.e., K has the form as follows: where Ekis a homogeneous kernel of degree -K: in the Euclidean sense and H1is homogeneous of degree -1 in the Heisenberg sense. In this paper, we study the optimal U and Holder estimates for the kernel K. We also use Lieb- Range's method to construct the integral kernel for the Kohn solution ∂N of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to ∂N. On the other hand, we prove Lieb-Range's kernel gains 1 in "good" directions (hence gains 1/2 in all directions) via Phong-Stein's theory. We also discuss the transferred kernel from the Siegel upper-half space to fi.

AB - Let Ω be an open, relatively compact subset in Cn+1, and assume the boundary of Ω, ∂Ω, is smooth and strongly pseudoconvex. Let Op(K) be an integral operator with mixed type homogeneities defined on Ω: i.e., K has the form as follows: where Ekis a homogeneous kernel of degree -K: in the Euclidean sense and H1is homogeneous of degree -1 in the Heisenberg sense. In this paper, we study the optimal U and Holder estimates for the kernel K. We also use Lieb- Range's method to construct the integral kernel for the Kohn solution ∂N of the Cauchy-Riemann equation on the Siegel upper-half space and then apply our results to ∂N. On the other hand, we prove Lieb-Range's kernel gains 1 in "good" directions (hence gains 1/2 in all directions) via Phong-Stein's theory. We also discuss the transferred kernel from the Siegel upper-half space to fi.

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U2 - 10.1090/S0002-9947-1989-0937241-0

DO - 10.1090/S0002-9947-1989-0937241-0

M3 - Article

AN - SCOPUS:77951028779

VL - 315

SP - 273

EP - 304

JO - Transactions of the American Mathematical Society

JF - Transactions of the American Mathematical Society

SN - 0002-9947

IS - 1

ER -