# Questions tagged [hardy-spaces]

For questions about Hardy spaces. Use the other related tag like (tag: complex-analysis) or (operator-theory).

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### Hardy Spaces - outer function is holomorphic

I am studying Banach Spaces of Analytic Functions by Hoffman. In Chapter 5 Page 61, the textbook claims that If $u \in L^1 (\mathbb T)$ then the function $F: \mathbb D \to \mathbb C$ \begin{align*} F(...
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1 vote
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### What's the BMO norm of the sine function?

In other words, what is the value of the maximum mean oscillation of the sine function over an interval, \begin{align} || \sin ||_{\mathrm{BMO}} &= \sup_{a,b\in\mathbb{R}} \frac{1}{|b-a|}\int_a^b ...
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### Harmonic majorants for non-negative sub-harmonic functions with a certain norm condition

This is a detailed proof of my latest post. Is there a mistake anywhere in the proof? Let $s = s(t,x)$ be a non-negative sub-harmonic function on $\mathbb{R}_+^{n+1} = (0,\infty) \times \mathbb{R}^n$ ...
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1 vote
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### The Mellin transform as a mapping from a Hardy space to a weighted space

As far as I know, the following fact must be published somewhere, and I would like to find a reference. The Mellin transform is defined by $f\mapsto\frac1{\sqrt{2\pi}}\int_0^\infty f(x)x^{s-1}\,dx$. ...
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### M. Riesz's Theorem for $L^2(T)$

Suppose that $f \in L^2(T)$ and define \begin{equation} F(z)=\frac{1}{2\pi}\int_T \frac{e^{it}+z}{e^{it}-z} f(e^{it}) dt \end{equation} for $z \in U$. M. Riesz's Theorem states that there exists a ...
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### Why $\|f\|_p=u_f(0)^\frac{1}{p}$?

Suppose that $f \in H^p$, where $0<p<\infty$. Let $u_f$ be the least harmonic majorant of $|f|^p$. Why is it true that \begin{equation} \|f\|_p=u_f(0)^\frac{1}{p}? \end{equation} I was hinted to ...
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