## A little-known definition of Hardy spaces

In the paper by Sedleckiĭ, A. M.
An equivalent definition of the $H\sp{p}$ spaces in the half-plane, and some applications. (Russian)
Mat. Sb. (N.S.) 96(138) (1975), 75–82, 167, the following is proved: given an analytic function $f$ on the upper-half plane ${\mathbb H}=\{z: {\mathcal Im} z>0\}$ consider the following two quantities for $0:

$\displaystyle \|f\|_{H_p}^p=\sup_{y>0}\int_{-\infty}^{+\infty}|f(x+iy)|^pdx$

and

$\displaystyle \|f\|_{H_p^\star}^p=\sup_{0<\theta<\pi}\int_0^\infty |f(re^{i\theta})|^pdr.$

Then

$\displaystyle A_p\|f\|_{H_p}\leq \|f\|_{H_p^\star}\leq B_p\|f\|_{H_p}$

for some constants $0.