This is an exercise in Lax.
The Hilbert-Hankel operator is defined to be $f\mapsto g(r)=\int^\infty_0 \frac{f(t)}{t+r}dt$. The question is to show the operator is a bounded map of $L^p(\mathbb{R^+})\rightarrow L^p(\mathbb{R^+})$.
Here is what I have so far:
I need to show that $\frac{||g||_p}{||f||_p}\le C$ for all $f\in L^p(\mathbb{R^+})$. Assume $f\ge 0$, by Minkowski's inequality for integrals, $||g||_p\le c\int^\infty_0 f(t)t^{1/p-1}dt=cI$.
Now I have no idea to show the bound of $I/||f||_p$, or equivalently the bound of $I^p/||f||_p^p$. Or do it need to divide the integral for different treatments?