Singular integrals along Lipschitz curves with holomorphic kernels |
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Authors: | A McIntosh T Qian |
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Institution: | 1. Macquarie University, Australia 2. Academia Sinica, China
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Abstract: | If γ(x)=x+iA(x),tan ?1‖A′‖∞<ω<π/2,S ω 0 ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S ω 0 and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C 0(γ), ?z∈suppf, where Cc(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent: - T can be extended to be a bounded operator on L2(γ);
- there exists a function ?1 ∈H ∞(S ω 0 ) such that ?′1(z)=?(z)+?(-z), ?z∈S ω 0 ?z∈S w 0 .
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