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

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a mulitplication theorem in Hardy space

$\newcommand{\H}{\mathcal{H}}$ Suppose $f\in \H$ (Hardy space), $g\in C_0^\infty$, can we say that $fg\in \H$? If not, then what the best expected space of $g$?
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$L^1(\mathbb{R}^n)$ functions not in $\mathcal{H}^1(\mathbb{R}^n)$

I am wondering how to imagine the Hardy space on $\mathcal{H}^1(\mathbb{R}^n)$ and in particular what sort of functions are in $L^1(\mathbb{R}^n)\backslash\mathcal{H}^1(\mathbb{R}^n)$. Furthermore, is ...
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Why p>1, $L^p$ and $H^p$ are essentially the same?

the conclusion comes from http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183538894 (the first page) $L^{p}$ is Lebesgue integral function space, ...
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Showing atomic $H^{1,p}$ is a Banach Space

Consider $1<p\leq\infty$ and let $H^{1,p}=H^{1,p}(\mathbb{R}^n)$ be an atomic Hardy space, that is $H^{1,p}\subset L^1(\mathbb{R}^n)$, and $f\in H^{1,p}$ if there exists a sequence of $p$-atoms ...
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Composition of $\mathrm H^p$ function with Möbius transform

Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$ Consider a Möbius ...
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Prove that $S$is a closed subspace of $H^2$ invariant under multiplication by $z$. Find the inner function $F$ such that $S=FH^2$

Let ${\alpha_n}$ be a sequence of points in the open unit disc such that $\sum(1-|\alpha_n|)<\infty$. Let $S$ be the set of all functions $f$ in $H^2$ spaces such that $f(\alpha_n)=f'(\alpha_n)=0$ ...
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How to show that $ν(z)$ is Carleson measure

If $ν(z)=|1+z|^ β dμ(z)$, $β\in \!\ R^-$. How to show that $ν(z)$ is Carleson? I know that $ν$ is Carleson measure if $ν(Q_I)\leqq \!\ c.I$ But how to apply this?
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Hardy space question

Let $T$ be the unit circle. Let $\phi\in C(T)$ and let $\psi$ be a function in $L^2(T)$ such that $\phi+i\psi\in H^2$. Assume both $\psi$ and $\phi$ are real-valued. Show $e^{\phi+i\psi}\in ...
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invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
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Removable sets for harmonic functions and Hardy spaces of general domains

Let $\Omega$ be a domain of the complex plane. The Hardy space $H^p(\Omega)$ is defined, for $1 \leq p<\infty$, as the class of functions $f$ that are holomorphic on $\Omega$ such that $|f|^p$ has ...