A general question about Cauchy operators

Question

I want to familiarize myself more with the Cauchy operators. As soon as I say "operator" I have to specify on which space, Okay, that should be my first question:

On which spaces the Cauchy operators are usually considered ? $L^{1}([a,b]), L^{1}(\mathbb{R})$, $L^{2}([a,b]), L^{2}(\mathbb{R})$, $\ldots$ , $L^{\infty}([a,b])$, $L^{\infty}(\mathbb{R})$ ?

As a simple check, I wanted to check $L^{\infty}([a,b])$, let $f \in L^{\infty}([a,b])$ ,$f \neq 0$, and consider its Cauchy transform $$C[f](x)=\frac{1}{2 \pi i } \int^{b}_{a} \frac{f(y)}{y-x}dy$$ $$\left| \ C[f](x) \ \right| = \frac{1}{2 \pi } \left| \int^{b}_{a} \frac{f(y)}{y-x}dy \right| \leq \frac{||f||_{L^{\infty}}}{2 \pi } \left| \int^{b}_{a} \frac{1}{y-x}dy \right|= \frac{||f||_{L^{\infty}}}{2 \pi } \left|\ln|\frac{b-x}{a-x}| \right| $$ but the right hand side is not bounded.

Does it mean that $C$ is not an operator on $L^{\infty}([a,b])$ ? How about other spaces ?

Answer 1

As mentioned in the comments, the operator you are looking at is usually called the Hilbert transform (a generalization of the Cauchy integral, and a first example of a singular integral operator).

I am not sure I have a full answer for you, but I would suggest to take a look at Chapter 3 of "Fourier Analysis" by Duoandikoetxea. As also mentioned in the comments, you need to define your operator using principal values. I won't go too much into details, since you can look them up in the book I have mentioned.

The Hilbert transform $H$ of a Schwartz function $f$ is $$ H(f)= \frac{1}{\pi} \operatorname{p.v.}\frac{1}{x} *f, $$ where $\operatorname{p.v.}\frac{1}{x}$ is a tempered distribution defined by $$ \operatorname{p.v.}\frac{1}{x}(\phi)=\lim_{\varepsilon \to 0} \int_{|x|>\varepsilon}\frac{\phi(x)}{x}\, dx, $$ where $\phi$ is a Schwartz function. Then the above definition for $H$ works for Schwartz functions $\mathcal{S}(\mathbb{R})$ and can be easily extended to $L^2(\mathbb{R})$. Then we have Riesz-Kolmogorov theorem:

For $f \in \mathcal{S}(\mathbb{R})$, the following assertions are true:

1. $H$ is weak $(1,1)$: $$ \left|\{x \in \mathbb{R} \mid |Hf(x)|>\lambda\}\right|\leq \frac{C}{\lambda} \|f\|_1 $$

2. $H$ is strong $(p,p)$, for $p \in (1,\infty)$: $$ \|H(f)\|_p \leq C \|f\|_p. $$

This allows us to extend $H$ to $L^p(\mathbb{R})$, for $p\in [1,\infty)$. For a definition of weak $L^p$, I suggest to look at Folland's book "Real Analysis", Chapter 6.