L2-boundedness of the cauchy transform on smooth non-Lipschitz curves

Z,-boundedness of the Cauchy Transform on Smooth Non-lipschitz Curves

If A is a Lipschitz function, i.e., || A |L < °°, then %A makes a very significant example of non-convolution type singular integral operators. The problem of L -boundedness of the Cauchy transform was raised and solved when || A |L is small by A. P. Calderόn in relation to the Dirichlet problem on Lipschitz domains [Call, Cal2]. Since then, it has been a central problem in the theory of singul...

Two Elementary Proofs of the L 2 Boundedness of Cauchy Integrals on Lipschitz Curves

Of the L 2 Boundedness of Cauchy Integrals on Lipschitz Curves

On the Boundedness of the Bilinear Hilbert Transform along “non-flat” Smooth Curves

We are proving L(R) × L(R) → L(R) bounds for the bilinear Hilbert transform HΓ along curves Γ = (t,−γ(t)) with γ being a smooth “non-flat” curve near zero and infinity.

L2-boundedness of the Cauchy Integral Operator for Continuous Measures

1. Introduction. Let µ be a continuous (i.e., without atoms) positive Radon measure on the complex plane. The truncated Cauchy integral of a compactly supported function f in L p (µ), 1 ≤ p ≤ +∞, is defined by Ꮿ ε f (z) = |ξ −z|>ε f (ξ) ξ − z dµ(ξ), z ∈ C, ε > 0. In this paper, we consider the problem of describing in geometric terms those measures µ for which |Ꮿ ε f | 2 dµ ≤ C |f | 2 dµ, (1) f...