On the Romanov kernel and Kuranishi’s L²-estimate for ∂_b over a ball in the strongly pseudo convex boundary

Takao Akahori and Harunori Ameku

Vol. 149 (1991), No. 1, 1–12

DOI: 10.2140/pjm.1991.149.1

Abstract

As is proved by Kerzman-Stein, over a compact strongly pseudo convex boundary in Cⁿ, Szegö projection S is the operator defined by Henkin-Ramirez modulo compact operators. While, over a special ball, U_𝜀, in the strongly pseudo convex boundary, in order to obtain a local embedding theorem of CR-structures, Kuranishi constructed the Neumann type operator N_b for ∂_b and so we have a local Szegö operator by

where ∂_b^∗ means the adjoint operator of ∂_b. There might be a relation between S_U𝜀 and the Romanov kernel like the case of the Szegö operator and the Henkin-Ramirez kernel. We study this problem and show some estimates for the Romanov kernel.

Mathematical Subject Classification

Primary: 32F20

Milestones

Received: 26 October 1989

Revised: 2 January 1990

Published: 1 May 1991