Cauchy integral on Lipschitz curves

Remarks

The operator $C_\varepsilon f$ is the truncated Cauchy integral $\frac{1}{\pi i}\int \frac{F(\zeta)}{\zeta - z} \, d\zeta$ over $\Gamma$, written in the parameter $x$, with $F(y + iA(y)) = f(y)$ and $z = x + iA(x)$. A Lipschitz function is differentiable almost everywhere, with $A'$ bounded, so the formula makes sense. Let $Hf(x) = \frac{1}{\pi} \lim_{\varepsilon \to 0} \int_{\abs{x - y} > \varepsilon} \frac{f(y)}{x - y} \, dy$ be the Hilbert transform. For $A = 0$ the curve is the real line and the operator is $iH$. So the theorem generalizes the $L^2$ bound of the Hilbert transform.

The case of small $\norm{A'}_{L^\infty}$ is due to Calderón. Expanding the kernel in powers of $A(y) - A(x)$ writes the operator as a series of Calderón commutators, and the series converges when $\norm{A'}_{L^\infty}$ is small. The general case, for every Lipschitz function $A$, was proved by Coifman, McIntosh and Meyer. Since the kernel is a Calderón–Zygmund kernel, the $L^2$ bound also gives boundedness on $L^p(\R)$ for $1 < p < \infty$.

Going deeper

For a student

The theorem is a central example of a singular integral operator whose kernel is not of convolution type. It is the starting point for the method of layer potentials on Lipschitz domains, which solves the Dirichlet problem for the Laplacian on domains with rough boundary.