Hilbert transform and maximal function

Question

What is the relation between the Hilbert transform (in $\mathbb{R}$) and maximal functions?

Answer 1

For example, the real Hardy space $H^1(\mathbb R)$ can be described in two equivalent ways:

• the space of functions $f\in L^1(\mathbb R)$ such that the Hilbert transform of $f$ also belongs to $L^1$
• the space of functions $f\in L^1(\mathbb R)$ such that the maximal function $\sup_t (f*\Phi_t)$ also belongs to $L^1(\mathbb R)$. Here $\Phi_t=t^{-n}\Phi(x/t)$ and $\Phi$ can be any Schwarz function.

The Hardy-Littlewood maximal function does not have such a direct relation, since it looks only at the amplitude $|f|$, ignoring the sign.