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

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Abstract

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.

Original languageEnglish (US)
Pages (from-to)273-304
Number of pages32
JournalTransactions of the American Mathematical Society
Volume315
Issue number1
DOIs
StatePublished - Sep 1989
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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