ABSTRACT

A domain Ω ⊂ Cⁿ⁺¹ and its boundary δΩ  are set to be decoupled of finite type if there exists sub-harmonic, non-harmonic polynomials {P_j}_j=1,...,n with P_j(0)=0 such that 

Ω={(z₁,...,z_n,z_n+1):Im(z_n+1)>∑ⁿ_j=1p_j(z_j)}

We call the integer m_j=2+degree(ΔP_j) the degree of P_j. The "type" of Ω is m = max{m₁,...,m_n}.

In the first part of this talk, we briefly recalled the method developed by Greiner-Stein and Chang-Nagel-Stein to construct the "fundamental solution" N for the δ-Neumann problem: Given a (0,1) -form f=∑ⁿ_j=1 f_j ω_{j ,} find a (0,1)-form μ=N(f) such that 

(δδ*+δ*δ) μ = f        in    Ω

μ _{n+1                      =}0        in    δΩ  

 Z

for  j=1 ,...,n. Here                      is the  complex normal with t=Re(z_n+1) and 

ρ=Im(z_n+1) - ∑ⁿ_j=1P_j(z_j)

is the "height function" defined on Ω. New classes of singular integral operators arrive naturally. Then we will discuss possible sharp estimates for the operator N in the second part of my talk.