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

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Moments of positive Carleson measure

A result of Widom from the 60's shows that a measure $\mu$ on the unit disc $\mathbb D$ concentrated on $(-1,1)$ is a Carleson measure if and only if $$ \int_{(-1,1)} t^k\,d\mu(t) = O(1/k)\quad\text{ ...
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1answer
73 views

Does this function belong to $H^1(\mathbb D)$?

$\mathbb D$ is the unitary disk centered at $0$. Does the following function belong to $H^1(\mathbb D)$? .$$f_\epsilon(z) = \frac{1}{(1-z)\left(\frac{1}{z}\log\frac{1}{1-z}\right)^{1+\epsilon}}, z\in\...
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21 views

Interchanging Limit and Integral sign

I'm reading a book on composition operators, and the author makes the following claim: Given a self-map of the unit disc, and a $H^2$ function $f$, where $H^2$ is the Hardy space, if we fix a radius $...
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27 views

A $H^p$ function

Set $\mathbb U=\{x+iy|\;y>0\}$. A function $f:\mathbb U\to\mathbb C$ is called a $H^p$ function if $f(z)$ is holomorphic and $\|f\|_{H^p}:=\sup_{y>0} \left(\int_{-\infty}^{\infty} |f(x+iy)|^p dx\...
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Invariant subspace shift operator $\psi H^2$

$H^2$ - Hardy space analytics function on unit disk; $M_z:H^2 \to H^2$, $M_z f = zf(z)$. We know Beurling theorem: Every invariant subspace shift operator other than $\{0\}$ has the form $\phi H^2$, ...
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Hardy space on unit disk $\mathbb{D}$ and Hardy space on the upper half plane $\mathbb{H}$

I am studying Hardy space on the upper half plane $\mathbb{H}$ recently. Wiki says that there is a isometic isomorphism between Hardy space on the upper half plane and Hardy space on the unit disk ...
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1answer
42 views

Compostion on $H^2(U)$

Below is a question that I'm attempting to do but so far have made no progress. Any suggestions would be helpful. Show that whenever $0 < \alpha < \frac{1}{2}$, then $\left( \frac{1+z}{1-z} \...
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1answer
60 views

Is Riemann–Lebesgue lemma valuble in $L2(\mathbb{R})$

If $f\in L_1$ on $\mathbb{R}$, that is to say, if the Lebesgue integral of $|f|$ is finite, then the Fourier transform of $f$ satisfies $$\hat{f}(z):= \int_{\mathbb{R}} f(x)e^{-izx} dx \rightarrow 0, \...
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1answer
42 views

Hardy space on the upper plane

Recently,I need to study something about Hardy space. However, many books only contain Hardy space on the unit disk. Is there any book having detailed description about Hardy space on the upper plane ...
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1answer
35 views

How to prove Mellin transform on $L^2[0,1]$ is unitary?

Let $\{Im (s)\lt 0\}=\{s\in \mathbb{C}\mid Im(s)\lt 0\}$, and $H^2(\{Im (s)\lt0\})$ is the Hardy space on $\{Im (s)\lt 0\}$. I know a classical theorem of Paley and Wiener Fourier transform $\...
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351 views

Cyclic vector in $\ell^2(\mathbb{Z})$ space

Suppose we look at $\ell^2(\mathbb{Z})$, which contains vectors $c=(\dots,c_{-2},c_{-1},c_0,c_1,\dots)$ with $\sum|c(n)|^2<\infty$. Define the right-translation operator by $$R:\{c(n)\}\mapsto \{c(...
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1answer
45 views

Inverse error function, its analytic continuation and Hardy space

Let $\mathrm{erf}(x) := \frac{2}{\sqrt{\pi}} \int_{-\infty}^x \exp(-t^2) \, dt$ be the error function $\mathrm{erf}: \mathbb{R} \to (-1,1)$. It is monotonously increasing and therefore has an inverse $...
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44 views

belong to $H^1$

‎‎‎ I can't prove the function $‎\dfrac{1}{1-z}‎ ‎\left( ‎\dfrac{1}{z}‎ log \dfrac{1}{1-z} ‎\right)‎^{-c}‎$ belong to $H^1$ (Hardy space) if $c>1$ . for this I write $z=x+iy$ but the problem ...
2
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1answer
42 views

Hardy-Littlewood theorem about the Poisson integral for $p=1$

(Hardy-Littlewood Theorem) : Let ‎$ u(r,‎\theta)‎$‎ be the Poisson integral of ‎$ ‎\varphi ‎\in L‎^{p}‎‎$‎, ‎$ 1<P ‎\leqslant‎ ‎\infty‎$‎ , and let $ U(‎\theta)=\sup‎_{r<1}‎|u(r,‎\theta)|‎ $‎. ...
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Invariant Subspace Vs Reducing Subspace in some special Hilbert space

Recently I was reading a book "Operator Function and system" written by Nikolski. There I found this statement. Let $\mu$ be a finite positive measure on the unit circle in complex plane which is ...
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2answers
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$L^{1}$ Boundedness of Hilbert Transform on $\left\{f\in L^{1}(\mathbb{R}) : \int_{\mathbb{R}}f=0\right\}$

It is well-known that the Hilbert transform $H(f)$ of a bounded, compactly supported function $f:\mathbb{R}\rightarrow\mathbb{C}$ belongs to $L^{1}(\mathbb{R})$ precisely when $\int f=0$. One can ...
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Criterion for Membership in Hardy Space $H^{1}(\mathbb{R}^{n})$

Let $H^{1}(\mathbb{R}^{n})$ denote the real Hardy space (I am agnostic about the choice of characterization). It is known that if $f:\mathbb{R}^{n}\rightarrow\mathbb{C}$ is a compactly supported ...
8
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1answer
208 views

Error in Stein Shakarchi Exercise on $H^{1}(\mathbb{R})$ and $L\log L$

In Stein and Shakarchi's Functional Analysis (Princeton Lectures in Analysis Vol. 4), the authors claim in Section 2 Exercise 17 that the function $$f(x):=\dfrac{\chi_{|x|\leq 1/2}}{x(\log|x|)^{2}}...
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1answer
38 views

Boundary function of product of $H^\infty$ and $H^2$ function

If $f\in H^2$ and $\phi \in H^\infty$ is it true that $\widetilde{\phi f} = \tilde{\phi}\tilde{f}$? It is is easy to see that $ \widetilde{z^nf} = \tilde{z^n}\tilde{f} $ and so for all polynomials $p$...
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1answer
53 views

Hardy Hilbert space: Multiplication by $H^\infty$ function [duplicate]

If $H^2(\mathbb{D})$ is space of all analytic functions whose series of taylor coefficient is absolute square summable and $\phi$ is analytic in $\mathbb{D}$,then $\phi H^2 \subset H^2$ if and only ...
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Inner Functions in Annuli: Not Likely!

The other day someone reminded me of something I'd thought about some years ago. As back then it took me a little while to see why there was any problem; this time I got much farther on a solution ...
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2answers
70 views

Integrability of Maximal Convolution Operator

Let $f\in L^{\infty}(\mathbb{R}^{n})$ be supported in the unit ball and have mean zero. Let $\phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n})$ be a Holder continuous function with exponent ...
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2answers
28 views

Radial Limits of Singular Inner Functions

Given a positive singular measure $\mu$ on $[-\pi,\pi]$, we define a singular inner function by $$S(z)=\exp\left(-\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(\theta)\right).$$ It is ...
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31 views

Show that the ideal generated by an inner function is closed.

Suppose $H^\infty=\{f\in H(U):\exists M<\infty,\,\forall z\in U,\,|f(z)|\le M\}$ is the set of bounded holomorphic functions in the unit disk. It is a Banach algebra with pointwise multiplication. ...
0
votes
1answer
47 views

Show that the inclusion relation of Hardy spaces is proper

The definition of Hardy spaces for the unit disk is here. It is clear that for $0<p<q\le\infty$, $H^q\subseteq H^p$, by Hölder's inequality. I'm asked to show that the inclusion relation is ...
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3answers
103 views

Alternative definition of Hardy spaces

Classically, Hardy spaces $H^p$on the disk are introduced as set of functions analytic on $\mathbb{D} = \{z \in \mathbb{C}: |z|<1\}$, which has bounded $H^p$ norm: $$ \|f\|_{H^p} = \sup_{0\leq r &...
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Tensor product of bounded analytic functions

Let $H^\infty(\mathbb{D})$ denote the set of functions holomorphic and bounded on $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Conseqently, $H^\infty(\mathbb{D}^n)$ denotes the set of bounded ...
4
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1answer
291 views

Are bounded analytic functions on the unit disk continuous on the unit circle?

Let $f(z)$ be holomorphic on the open disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$. Moreover, let $f$ be bounded on the boundary of $\mathbb{D}$, i.e. $$ \sup_{\varphi \in [0,2\pi]} |f(e^{i\...
2
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1answer
27 views

$MN-\langle M,N \rangle$ is uniformly integrable when M, N are $H^2$?

This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too
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1answer
211 views

On the Hilbert Transform of a Bounded Function

Let $f: \mathbb R \rightarrow \mathbb R$ be a bounded function that is smooth and also in $L^1(\mathbb R) \cap L^2(\mathbb R)$. I want to prove that the Cauchy transform of this function $Kf$ is in $H^...
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0answers
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Holomorphic function in unit disc?

I am stuck on this exercise from Stein and Shakarchi's Real Analysis: suppose $F$ is holomorphic in the unit disc, and $$\sup_{0\leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi \log^+ |F(re^{i\theta})|\,...
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How to prove $f$ is outer, when $Re f$ >0?

This is an exercise problem, which can be found in most of the book which deals with Hardy Space. Any kind of suggestion or hint is also welcome. The question is following: Let $f$ be a function in ...
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1answer
39 views

When the singular inner part disappear in inner outer factorization?

I saw this remark in Hoffman's book - "Banach space of analytic function". If $f$ is analytic in a neighborhood of $\bar{\mathbb{D}}$, the closure of $\mathbb{D}$; then in the inner-outer ...
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0answers
33 views

How to check if a matrix transfer function is in Hardy-infinity space?

Just like the question says. For instance if I have a matrix transfer function $$\mathbf{G}(s) = \begin{bmatrix}s & -s \\ T & s \\ \end{bmatrix}$$ where $T$ is a positive constant, how can I ...
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1answer
36 views

Find a sequence of complex polynomials with certain properties. (Hardy spaces over unit circle)

Let $\lambda\in \Bbb S^1$. Find a sequence of complex polynomials $p_n(z)$ such that for any $c>0$ the following inequality does not hold: $$|p_n(\lambda)|\le c\cdot \|p_n\|$$ where $$\|p_n\|=\...
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1answer
23 views

Show that for every real-valued $L^2$ function $u$ on $S^1$ there is a real-valued $v$ in the same space such that $u + iv\in \widetilde{\mathbf H}^2$

For a homework exercise ($1.8$ in the book An Introduction to Operators on the Hardy-Hilbert Space) I am asked to show Let $u$ be a real-valued function in $L^2(S^1)$. Show that there exists a ...
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1answer
33 views

How to decide whether a function is in $H^p_+$.

$H^p_+$ is the Hardy space on the upper half-plane of $\mathbb C$ consisting of functions $f$ satisfying $$ \sup_{y_0>0}\left(\int_{-\infty}^{\infty} |f(x+iy_0)|^p dx\right)^{\frac 1p}<\infty $$ ...
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1answer
86 views

Interpolation sequences and open mapping theorem

I'm using Garnett's "Bounded Analytic Functions" as a course text and looking at interpolation sequences. $z_n$ is a sequence of interpolation if for each sequence $a_n \in l^{\infty}$ there exists $f ...
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1answer
14 views

A function in $H^\infty(\Delta^2)$

Can you give an example of a function $f\in H^\infty(\Delta^2)$ with $f^{-1}\in L^\infty(T^2)$ but not inner? Here $H^\infty(\Delta^2)$ is the space of all bounded analytic functions defined on bi-...
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0answers
56 views

Multiplication operator is bijective?

Let $H^2(\Delta^2)$ denote the Hardy space on the polydics and $M_f:H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be a multiplication operator by $f\in H^\infty(\Delta^2)$. Is the operator is bijective? ...
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1answer
86 views

A simple function with no tangential limits but with non-tangential limits

I am going to be teaching a course about the Hardy space and I would like to show the students that the non-tangential limit is a necessary concept BEFORE telling them about Blaschke products. Is ...
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1answer
77 views

Literature on interpolation in Hardy spaces

I'm an undergraduate mathematics student and I'm searching for notes and books on Hardy spaces $H^p$, in particular interpolation theory including topics like Carleson measures, Carleson's $H^\infty$ ...
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1answer
184 views

Showing a certain sequence is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$

The problem is to show $$\left\{\frac{1}{\pi^{1/2}(i+z)}\left(\frac{i-z}{i+z}\right)^n\right\}_{n=1}^{\infty}$$ is an orthonormal basis of $H^2(\mathbb{R}_{+}^{2}).$ In another exercise, I have ...
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The definition of Hardy space

The definition of the Hardy space consists functions $u(z)$ that are analytic a. outside the closed unit disc or b. inside the open unit disc What is the difference between the two definition ...
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Real Hardy space question

Please see this related question for the definition of the grand maximal function and the class of normalised test functions $\mathcal{T}$. I will refer to them in this question. Let $f\in L^{\infty}(...
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1answer
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Constructing a specific normalised test function.

Given the following class of normalised test functions: \begin{equation} \mathcal{T}=\{\phi\in C_c^{\infty}(\mathbb{R}^n)\ |\ \text{supp }\phi \subseteq B(0, 1),\ \|D\phi\|_{\infty}\leq 1\} \end{...
3
votes
1answer
124 views

Hardy Space Cancellation Condition

I have been reading the chapter on Hardy spaces in Stein's Harmonic Analysis book, and I am having a lot of trouble figuring something out. The setting here is $\mathbb{R}^n.$ Let $f \in L^q$ be ...
5
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1answer
72 views

$L^1(\mathbb{R}^n)$ 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 ...
1
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1answer
70 views

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, $H^...
2
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1answer
264 views

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 $\{...