Singular integrals along Lipschitz curves with holomorphic kernels

Abstract

If γ(x)=x+iA(x),tan ⁻¹‖A′‖_∞<ω<π/2,S ⁰_ω ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S ⁰_ω and$\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}$, denotingT f (z)= ∫ϕ(z-ζ)f(ζ)dζ, ∀f∈C ₀(γ), ∀z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

• 10 T can be extended to be a bounded operator on L²(γ);

• 20 there exists a function ϕ₁ ∈H ^∞(S ⁰_ω ) such that ϕ′₁(z)=ϕ(z)+ϕ(-z), ∀z∈S ⁰_ω ∀z∈S ⁰_w .