Questions tagged [hardy-spaces]
For questions about Hardy spaces. Use the other related tag like (tag: complex-analysis) or (operator-theory).
178
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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(...
- 3,605
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11
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Question on Hardy Spaces Definition
I am self studying The Theory of $H^p$ spaces by Duren. The author writes the following in page 2:
A function $f(z)$ analytic in the unit disk $\lvert z \rvert < 1$ is said to be class $H^p \; (0 &...
- 3,605
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Hardy Spaces - Proving that the norm is well defined
I am self studying Banach Spaces of Analytic Function by Hoffman. In the Chapter 3 titled "Analytic and Harmonic Functions in the Unit Disc", the author defines the class $H^p$ for $1\le p \...
- 3,605
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Hardy class of bounded analytic functions is Banach space
I need help with the following:
Let $H^{\infty}(\mathbb{D})$ denote Hardy class of bounded analytic functions on unit disc $\mathbb{D} = \{z \in \mathbb{C}: |z|<1\}$. Prove that $$||f|| = \...
- 161
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Prove that, $H^\infty+C(\Bbb{T})$ is not a $C^*$-algebra
Here $H^\infty:=\left\{f\in L^\infty(\Bbb{T}):\ \frac{1}{2\pi}\int\limits_0^{2\pi} f\chi_{-n}\ dt=0\ \forall n<0\right\}$ i.e. all those $L^\infty$ functions whose negative fourier coefficients are ...
- 2,813
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Show $\lim_{t \to 0} \frac{\sigma\left( \left\{ e^{is} : s \in (\theta - t, \theta + t \right\} \right)}{2t}=0$ for singular measure $\sigma$ of $T$
I am self studying "Representation Theorems in Hardy Spaces" by Javad Mashreghi. Here's a claim [in Page 72 of the textbook] that I am unable to prove:
If $\sigma$ is a singular measure of $\...
- 3,605
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Inverting series with logs and W
You've all heard it: what does a drowning analytic number theorist say? Log log log log....
I very frequently deal with the sorts of functions that one comes across and want to invert them. Generally ...
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For some Banach space $B$, find a $u \in h^p(\mathbb{D};B)$ but $u \notin \tilde{h}^p(\mathbb{D};B)$
Let $B$ is a Banach space and $1\le p < \infty$, $\tilde{h}^p(\mathbb{D};B) = \left\{P_z * f : f \in L_p(\mathbb{T},m;B) \right\}$, which is a subspace of $h^p(\mathbb{D};B) = \left\{ u : \mathbb{D}...
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If sequence of finite Blaschke Products $(B_n)_{n\in N}$ converges uniformly to $B$ then $\frac{B_n(z)}{z^m}$ converges uniformly to $\frac{B}{z^m}$
I am having problem with showing that if sequence of finite Blaschke Products
$$B_n=z^m\prod_{k=1}^{n}\frac{|z_k|}{z_k}\frac{z_k - z}{1 - \overline{z}_k z}$$
where $(z_n)_{n\in \mathbb{N}} \subset \...
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Approximations of BMO functions
I couldn't think of a good title. There was a small detail when I was trying to prove something about the norm of BMO functions. This small detail is not directly related to the problem itself, but ...
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Let $f\in L^2(\Bbb{T}),\ f\ne0$ and $M=\overline{f\wp_+}$. Prove that, $M$ is simply invariant for $\chi_1$ iff $f\notin \chi_N M$ for some $N>0$
Here we denote $\chi_n$ to be the functions on $\Bbb{T}$ defined as $\chi_n(z)=z^n\ \forall z\in\Bbb{T}$ for $n\in\Bbb{Z}$. We define $\wp_+=\{\sum\limits_{n=0}^N a_n\chi_n|\ a_n\in\Bbb{C},N\ge0\}$ be ...
- 2,813
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Weighted Hardy spaces and $L^p$ isomorphism
I'm working on weighted Hardy spaces, which are related to $L^p$ spaces. In a paper, I read that there is an isomorphism between weighted Hardy spaces $H^p$ and $L^p$, says:
Let $1<p<\infty$ and ...
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Analytic extension in the Smirnov Hardy class of holomorphic functions
I have the following question, which I need help addressing.
If $U\subset \mathbb{C}$ and $D\subset\mathbb{C}$, with $\partial U\cap\partial D=\Gamma\neq\emptyset$, are open connected domains and $G:U\...
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$\int_{-\pi}^{\pi}{\frac{d\theta}{[(1-r)^2 + 2\pi^{-2}\theta^2]^p}} < \frac{1}{(1-r)^{2p-1}}\int_{-\infty}^{\infty}{\frac{dt}{[1+2\pi^{-2}t^2]^p}}$
I found this inequality from Duren's book Theory of Hp-spaces(2000) page 66. According to the book inequality should be true for $\frac{1}{2}\leq r < 1$ and $p>\frac{1}{2}$. I have tried to ...
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What is the domain of functions in $\mathcal H_2$?
In a section of a book that I'm reading about Hardy spaces, the author writes
$\mathcal H_2$ is a (closed) subspace of $\mathcal L_2(j \mathbb R)$ with matrix functions $F(s)$ analytic in $\text{Re}(...
- 988
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$H^p$ is not a Banach space for $p<1$?
As in E. Steins's "Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals" consider the space $H^p$ (with $p < 1$), which contains all tempered distributions $f ...
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Are Hardy spaces Banach algebras?
Let $\mathbb D= \{z\in\mathbb{C}: |z|<1\}$. Let $Hol(\mathbb {D})$ denote the space of holomorphic functions on $\mathbb D$. The Hardy spaces on $\mathbb D$ are defined as follows.
$$H^p=\{f\in ...
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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 ...
- 41
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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 $\mathbb{R}_+^{n+1} = (0,\infty) \times \mathbb{R}^n$ ...
- 40
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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 $f\mapsto\frac1{\sqrt{2\pi}}\int_0^\infty f(x)x^{s-1}\,dx$. ...
- 11
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Absolute value of functions in $H^1(\mathbb R^n)$
Let $f$ be a function in the atomic Hardy space $H^1_{at}(\mathbb R^n)$. That is, there exists a sequence of atoms $a_j$ satisfying
supp $a_j \subset B_j$ for some ball $B_j$,
$\int a_j dx = 0$,
$\...
- 345
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$H^2$ does not contain any rational functions with poles on the unit circle
The Hardy-hilbert space, $H^2$, consists of all analytic functions having power series representations with square-summable complex coefficients. That is,
$$H^2=\{f : f(z)=\sum_{n=0}^{\infty} a_n z^ns....
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Why does the Laplace transform of a function in L2 belong in the Hardy space?
Does anyone have a proof / mathematical explanation to why the Laplace transform of a function of $\mathbb{L}^2(0,\infty )$ belongs to the Hardy space?
Any guidance would be fantastic! I cant find ...
- 11
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a generalized Jensen's equality for Nevanlinna class
Define the Nevanlinna class of holomorphic functions on unit disk by $\displaystyle N(\mathbb{D})=\left\{f\in\text{Hol}(\mathbb{D}):\sup_{0<r<1}\int_\mathbb{T}\log^+|f_r|dm<\infty\right\}$ ...
- 1,894
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Showing $\arcsin(z)-i\operatorname{Log}(-iz)-i\log 2 \in H^2(\mathbb{H})$
In order to prove a result about Hilbert transforms, I need to show the complex function $F(z) = \arcsin(z)-i\operatorname{Log}(-iz)-i\log 2$ lies in $H^2(\mathbb{H})$, the Hardy Space for the upper ...
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containment of $H^1$ spaces into $L^1$ spaces
Let $\mathbb{D}$ denote the unit disk and $\mathbb{T}$ the unit circle
, if $f\in H^1(\mathbb{D})$ is outer , then necessarily $\log|f|\in L^1(\mathbb{T})$ ?
We know from theory of Hardy spaces that ...
- 1,894
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Introductory reference for Hardy spaces on the disk
I'm looking for a good introductory reference for $H^p$ spaces on the unit disk. I already have a good background in real/complex analysis and functional analysis. I currently have the text Funtion ...
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atomic decomposition of Hardy spaces: support question
Consider the Hardy space $H^1$. We can define this space by means of an atomic decomposition: A function $a$ is called an $L^2$-atom for $H^1$ if it is supported in a ball $B$, one has $\|a\|_2 \leq |...
- 1,959
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shift invariant subspaces and absolute continuity
Let $\mu$ be a finite complex Borel measure on unit circle $\mathbb{T}$ ,
and let $E$ be a shift invariant subspace of $L^2(\mu)$ . Show that
$zE\subseteq E\subseteq L^2(\mu)$ implies $zE=E$ iff the ...
- 1,894
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Orthogonal basis for $L^2$-martingales (martingale-Hardy space)
Let $(X,\mathcal{B}(X),(\mathcal{A}_n)_{n=1}^N,\mu)$ be a filtered probability space with $\mathcal{A}_N=\mathcal{B}(X)$ and let $H$ be the space of $\mathcal{A}_{\cdot}$-adapted martingales $m_{\cdot}...
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Decay at 0 and $\infty$ of functions in Hardy space $H^1(\Sigma_\varphi)$ on a sector
Let $\Sigma_\phi = \{z \in \mathbb C : |\arg(z)|<\phi\}, 0<\phi<\pi$ be a sector and let $f \colon \Sigma_\phi \to \mathbb C$ be holomorphic. If
$$
\|f\|_{H^1(\Sigma_\phi)} = \sup_{|\omega|&...
- 280
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51
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Hardy space of Banach space valued analytic functions
For $p \in [1, \infty)$, let $H^{p}(\mathbb{D},X)$ be the space of analytic functions from $\mathbb{D}$ into a complex Banach space $X$ such that
\begin{equation} \label{him-p7-e-1.11}
||f||_{H^{p}(\...
- 11
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Convergence in Hardy spaces
I consider for $p \in [1, \infty)$ the Hardy Space
$$ H^p(\mathbb{D}) = \\ \left\lbrace f:\mathbb{D} \rightarrow \mathbb{C} \, : f \, \text{is holomorphic and} \, \sup_{0\leq r < 1} \left(\frac{...
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$L^p$ norm of $\frac{1}{1-\overline{\alpha }e^{i\theta}}.$
I am trying to calculate the $L^p$ norm of the function $f_\alpha(e^{i\theta}) = \frac{1}{1-\overline{\alpha }e^{i\theta}},$ where $\alpha$ is a complex number with $|\alpha| < 1.$ Id est, for ...
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Why to prove $f\in H^1\mathbb{(D)}$, it is enough to prove that $f\in L^1(|z|=1)$?
Prove that $$\sqrt{\frac{1+z}{1-z}}$$ is in the Hardy space $H^1(\mathbb{D})$, where we take the principal branch of the square root.
Answer- let $f(z) = \frac{(1+z)^{1/2}}{(1-z)^{1/2}}$ analytic on $|...
user922763
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If$\int_0^{2\pi} |f^*(e^{i\theta})|^{1/3}d\theta <\infty$ then $f\in H^{1/3}(D) $ [closed]
Question If$\int_0^{2\pi} |f^*(e^{i\theta})|^{1/3}d\theta <\infty$ then $f\in H^{1/3}(D) $ where $f$ is analytic and $f^*$ is non tangential limit of $f$
$H^{1/3}(D)$ is Hardy Space.
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Scalar product on Hardy spaces on the strip $\mathcal{S} = \mathbb{R} + i(-1,1)$
I want to learn more about the Hardy space $H^2(\mathcal{S})$ on the complex strip $\mathcal{S} = \{ x+iy \in \mathbb{C} ~|~ x,y \in \mathbb{R},~ |y|<1 \}$. In particular, I am interested in it as ...
- 382
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Prove $|| f||_{H^1}=||f^*||_{L^1}$
The Hardy space $H^1(\mathbf{D})$ is the vector space of holomorphic functions $f$ on the open unit disk that satisfy:
$$
\sup_{0\leq r<1}\left(\frac{1}{2\pi} \int_0^{2\pi}\left|f \left (re^{i\...
user865029
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Hardy's inequality dual with power weights
We have that hardy inequality dual with power weights is
$$\hspace{20mm}\int_{0}^{\infty} \left( \frac{1}{x}\int_{x}^{\infty} f(t) dt \right)^{p} x^{\alpha} dx \leq \left( \frac{p}{\alpha+1-p} \right)^...
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Boundedness of differential operator on the space of analytic functions on a disk.
I need to show that the operator of taking the derivative is continuous, but at a certain point, I got stuck. This must be a simple problem but I'm a layman in complex analysis.
Denote $\mathbb{D}=\{z\...
- 55.4k
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1
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217
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Proof that polynomials are dense in $H^p$ spaces
In some papers and books I read they used that the polynomials are dense in $H^p$ for $1\le p<\infty$, but I could not find a proof for this stament. I tried to proof it myself but failed. My idea ...
- 15
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Hardy space and boundary regularity
Let $H^\infty(D)$ be the Hardy-space of bounded, holomorphic functions $\Phi:D\rightarrow \mathbb{C}$, defined on the unit disk $D\subset \mathbb{C}$. It is a standard result that $\Phi\in H^\infty(D)$...
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Is $\sqrt{i \cdot \frac{1+z}{1-z}}$ in the Hardy space $H^1(\mathbb{D})$
I am trying to prove that
$$\sqrt{i \cdot \frac{1+z}{1-z}}$$ is in the Hardy space $H^1(\mathbb{D})$, where we take the principal branch of the square root. I think that this is true but I am stuck at ...
- 55
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Why do atoms belong to the real Hardy space $\mathcal{H}^{1}(\mathbb{R}^n).$
Define atom $a\in L^1(\mathbb{R}^n)$ associated to the ball $B_r(x_0)$ such that it satisfies
$\text{supp}(a)\subset B_r(x_0).$
$|a|\leq 1/|B_r(x_0)|$ and so $||a||_{L^1}\leq 1.$
$\int_{B_r(x_0)} a ...
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1
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55
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Analytic function on $\mathbb{D}$ belongs to $H^2$ iff $\int_{\mathbb{D}}|f'(z)|^2(1-|z|^2)dA(z)<\infty$
Analytic function on $\mathbb{D}$ belongs to Hardy space $H^2$ iff $\int_{\mathbb{D}}|f'(z)|^2(1-|z|^2)dA(z)<\infty$ where $dA$ is the area measure.
By definition, $f\in H^2$ if $$\|f\|^2_{H^2}=:\...
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$\hat{g}(n)=\hat{f}(n)$ if $n \ge 0$ and $\hat{g}(n)=0$ otherwise. Prove there is a $M_p$ depending only on $p$ such that $\|g\|_p \le M_p\|f\|_p$
Given $1<p<\infty$, $f \in L^p(T)$ and $T$ is the unit circle. Let $\hat{f}(n)$ be the Fourier coefficients of $f$. If there exists an $g \in L^p(T)$ such that $\hat{g}(n)=\hat{f}(n)$ for $n \ge ...
- 317
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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 ...
- 317
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0
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42
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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 ...
- 317
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168
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Confusing step in the proof of Beurling projection theorem
I have been reading Harmonic Measure by John Garnett and Donald Marshall. Their formulation of Beurling theorem (page 105) is:
If $E\subset\overline{\mathbb{D}}\setminus\{0\}$, and $E^*=\{|z|: z\in ...
- 3
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53
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Prove a norm equality (Composition operator on Hardy space)
For $\lambda\in \mathbb{D}$ fixed, let $C_{\phi}$ denote the composition operator on Hardy space $H^2(\mathbb{D})$ (I.e. $C_{\phi}f:=f(\phi)$) with symbol $\phi=\frac{\lambda-z}{1-\bar{\lambda}z}$ (a ...
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