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

learn more… | top users | synonyms

1
vote
2answers
60 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 ...
1
vote
2answers
19 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 ...
1
vote
0answers
25 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
23 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 ...
0
votes
3answers
59 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 ...
2
votes
0answers
62 views

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 ...
0
votes
0answers
25 views

Set of Functions that is dense in space of bounded analytic functions

Let $H^\infty$ denote the set of all functions holomorphic and bounded in the open unit disk $\mathbb{D} = \{z \in \mathbb{C}: |z| < 1\}$, i.e. $\|f\|_{H^\infty} = \sup_{0<r<1} \sup_{\varphi ...
4
votes
1answer
54 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]} ...
2
votes
1answer
19 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
1
vote
1answer
82 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 ...
0
votes
0answers
16 views

Step of proof Hardy's Inequality [duplicate]

I trying to prove the Hardy's Inequality, by Evans book. I need of a little help in the step: If $u\in L^1(B(0,r))$ satisfying $$(2-n)\int_{B(0,r)}\frac{u^2}{|x|^2}dx=2\int_{B(0,r)}u ...
0
votes
0answers
39 views

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^+ ...
1
vote
0answers
29 views

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 ...
2
votes
1answer
21 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 ...
0
votes
0answers
24 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 ...
0
votes
1answer
29 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 ...
1
vote
1answer
15 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 ...
1
vote
1answer
30 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 $$ \sup_{y_0>0}\left(\int_{-\infty}^{\infty} |f(x+iy_0)|^p dx\right)^{\frac 1p}<\infty $$ ...
1
vote
1answer
60 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 ...
0
votes
1answer
11 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 ...
0
votes
0answers
32 views

Bounded-below multiplication operator on Hardy space

Let $H^2(\Delta^2)$ denotes the Hardy space on the bi-disc $\Delta^2$ and $M_f :H^2(\Delta^2)\rightarrow H^2(\Delta^2)$ be multiplication operator by $f\in H^\infty(\Delta^2)$ defined by ...
1
vote
0answers
35 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? ...
0
votes
1answer
40 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 ...
0
votes
1answer
45 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 $H^\infty$ ...
3
votes
1answer
157 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 ...
1
vote
0answers
44 views

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 ...
1
vote
0answers
62 views

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 ...
1
vote
1answer
47 views

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\} ...
3
votes
1answer
93 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
votes
1answer
59 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
vote
1answer
64 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, ...
2
votes
1answer
131 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 ...
1
vote
1answer
85 views

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 ...
1
vote
1answer
54 views

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 $f(\alpha_n)=f'(\alpha_n)=0$ ...
1
vote
0answers
47 views

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?
4
votes
1answer
117 views

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 ...
6
votes
1answer
150 views

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 ...
10
votes
1answer
236 views

Suppose $\phi$ is a weak solution of $\Delta \phi = f \in \mathcal{H}^1$. Then $\phi\in W^{2,1}$

I'm trying to prove the statement in the title in as simple a way as possible. It is Theorem 3.2.9 in Helein's book "Harmonic maps, conservation laws, and moving frames", although it is not proved ...
6
votes
1answer
255 views

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 ...