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<p<\infty:

\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<A_p<B_p<\infty.

Advertisements
This entry was posted in Complex Variables. Bookmark the permalink.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s