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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2
votes
0answers
41 views

$L^p$ 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 ...
2
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0answers
61 views

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 $|...
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0answers
69 views

If, $0<r<1$ then prove that $|(1-r^2)^3F(r)|$ is bounded. [closed]

Let, $$f(z)=(s-1)\zeta(s) $$ where $s=\frac{1}{1-z}$ and $\zeta(s)$ denotes the Riemann zeta function. Let, $H^{p}(D)$ , $p>0$ denote the Hardy space. Then, $f\in H^{1/3}(D).$ So there exists a ...
0
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1answer
63 views

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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28 views

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 ...
1
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1answer
66 views

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\...
1
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1answer
35 views

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)^...
0
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1answer
20 views

Boundedness of differential operator on the space analytic funcitons 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\...
1
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1answer
67 views

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 ...
0
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0answers
32 views

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 $\Phi\in H^\infty(D)$...
2
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1answer
42 views

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 ...
1
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1answer
64 views

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 ...
0
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0answers
43 views

Equivalent statement of M. Riesz's Therorem

Let $1<p<\infty$ and $f \in L^p(T)$, where $T$ is the unit circle. Let $\hat{f}(n)$ be the Fourier coefficients. Prove that \begin{equation*} \sum_{n=0}^\infty \hat{f}(n)e^{int} \end{equation*} ...
0
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0answers
24 views

The Cauchy Transform and convergence of functions on the disk compactly vs. in Hardy space norm

Let $\left\{ f_{m}\right\} _{m\geq1}$ be a sequence of holomorphic functions on the open unit disk $\mathbb{D}$ that converge compactly (i.e., uniformly on every compact subset of $\mathbb{D}$) to a ...
0
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1answer
41 views

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}=:\...
2
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1answer
79 views

$\hat{g}(n)=\hat{f}(n)$ 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 ...
1
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0answers
54 views

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 ...
0
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0answers
38 views

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 ...
0
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0answers
21 views

Dense subset of $1-\text{Bernstein}$ space

My question rises from a problem encountered while studying Paley-Wiener spaces. We define the Bernstein spaces $$ B^1_\sigma=\left\lbrace f \text{ entire such that support(} \hat f)\subset[-\sigma,\...
0
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1answer
124 views

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 ...
0
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1answer
31 views

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 ...
2
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0answers
79 views

Toeplitz Kernel Question

$$\newcommand{\kern}{\operatorname{Ker}}$$When looking at a Toeplitz Operator (on $\mathbb{H}^2(\mathbb{C^+}))$ with unimodular symbol, $\phi=e^{i \psi}$, what can we say about $\kern T_{\phi}$ when $$...
0
votes
1answer
87 views

$\text{Log}(1-z) \in H^p(\mathbb{D})$ for all $0<p<\infty$

Apparently, it is well known that $\text{Log}(1-z) \in H^p(\mathbb{D})$ for all $0<p<\infty,$ where $\text{Log}$ denotes the principal complex logarithm. How can this be proven?
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0answers
35 views

Questions while reading “$H^p$ spaces of several variables(Fefferman and Stein)”.

I have two questions while reading this paper: Hp spaces of several variables." Acta math 129 (1972): 167-193. Question 1. Line $7$ on page $147$ to line $13$ on page $148$ is a proof of "Theorem 3$\...
0
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1answer
51 views

Criteria for function to be of Hardy class $H^p$

I was reading some papers, and I don't know how to prove following statement: Let $f$ be conformal map of $\mathbb{D}$ onto an unbounded domain,such that, $f(0)=0$ and $dA$ denote Lebesgue measure on ...
1
vote
1answer
30 views

Let $f\in L^{\infty}$, its that true $Pf\in H^{\infty}$?

My question is: Let $f\in L^{\infty}[S^1]\subseteq L^2[S^1]$, is that always true for $Pf\in H^{\infty}[S^1]\subseteq H^2[S^1]$? Here, $S^1$ is the unit circle in complex plane, and $H^{\infty}$ ...
2
votes
1answer
75 views

Functions on the unit circle having finitely many zeroes

I actually consider functions on the strip $S(0,\pi) = \{ z | 0 < Im(z)<\pi \}$ in the complex plane, but by a biholomorphism, I can map to the unit disc. In particular, the functions I consider ...
1
vote
2answers
51 views

Weighted Hardy space composition

Let $H^2(\omega)$ be the closure of the set of analytic on the unit circle $T$ polynomials, with respect to the scalar product $$ \langle f, g\rangle_\omega = \oint_T f \bar{g} \omega \,\mathrm{d}z, $$...
-1
votes
1answer
34 views

Is $H^\infty(\mathbb{C_+}) \subset H^2(\mathbb{C_+})$?

I know that $H^\infty(\mathcal{D}(0,1)) \subset H^2(\mathcal{D}(0,1))$, and that there is a mapping $$f: \mathcal{D}(0,1)\to \mathbb{C}_+ : z\mapsto s=\frac{1-z}{1+z}.$$ However in $\mathbb{C_+}$ I ...
0
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0answers
84 views

Hardy inner product for the right half plane

The Hardy space $H^\infty(\mathbb{C_+})$ is defined to be the space of all functions $f$ >analytic and bounded on the complex right half plane $\mathbb{C_+}$ with the norm $\lVert \cdot \rVert_H$ ...
1
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1answer
81 views

Question on Douglas, Banach Algebra, Hardy spaces

In the text by Douglas, Banach Algebra techniques in Operator Theory, pg. 25, 1.47 Hardy spaces was introduced. If $\mathbb {T}$ denotes the unit circle in the complex plane, let $\chi_n $ denote ...
4
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1answer
87 views

Wiener's tauberian theorem for Hardy space

For $a>0$ let us define $$H^2(-a,a)=\{f \ \mbox{is analytic in the strip $|\Im(z)|<a$}: \sup_{y\in [-a,a]}\int_{\mathbb{R}}|f(x+iy)|^2\,dx<\infty\}.$$ For $f\in H^2(-a,a)$, define $\|f\|=\...
1
vote
1answer
28 views

How much can the hypotheses of the Schwarz integral formula be weakened?

According to https://en.wikipedia.org/wiki/Schwarz_integral_formula, which I will paraphrase here, if $f$ is a holomorphic on the closed unit disk $\{ \lvert z \rvert \leq 1 \}$, then $f(z) = \frac{1}...
0
votes
1answer
96 views

Is a de-zeroed Hardy function still Hardy?

Suppose a Hardy function $f(z)$ on the upper half complex plane or $f\in H^{2+}$ (Chapter II, p.45 of Fulvio Ricci, Hardy Spaces in One Complex Variable) has a zero of order $m$ at $\omega$ with $\...
5
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0answers
66 views

What is the closure of $h^2(U)$ in the topology of the uniform convergence on upper half planes?

Let $U:=\{z\in\mathbb{C}\ |\ \operatorname{Im}(z)>0\}$. Let $h(U,\mathbb{R})$ be the set of real harmonic functions defined on $U$. Define $$h_0(U,\mathbb{R}):=\left\{u\in h(U,\mathbb{R})\ |\ \left(...
1
vote
1answer
41 views

If $f$ is in a weighted Bergman space for the upper half plane, then $\forall\varepsilon>0, z\mapsto f(z+i\varepsilon)$ is in the Hardy space.

Let $U:=\{z\in\mathbb{C}\ |\ \operatorname{Im}(z)>0\}$ and denote with $H(U)$ the set of holomorphic functions on $U$. Define: $$H^2(U):=\{f\in H(U) \ |\ \sup_{y>0}\int_\mathbb{R} |f(x+iy)|^2\...
2
votes
0answers
70 views

Looking for an explicit $f\in C(\mathbb{T})$ such that $t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{int}\notin L^\infty(\mathbb{T})$.

Denote the 1-torus with $\mathbb{T}$ and if $f\in L^1(\mathbb{T})$ denote the Fourier transform of $f$ by $\hat{f}$. I know that $$\exists f\in C(\mathbb{T}), t\mapsto\sum_{n=0}^{\infty}\hat{f}(n)e^{...
3
votes
0answers
133 views

Convergence of Fourier series in Bergman norm

Let $D$ be the unit disk, let $\nu>-1$, let $\varphi_\nu(w):=(1-|w|^2)^\nu$, let $\operatorname{d}\mu_\nu:=\varphi_\nu\operatorname{d}\mu$ where $\mu$ is the Lebesgue measure on the unit disk. If $...
1
vote
0answers
64 views

Range of the Bergman projection.

If $D$ is the unit disk in the complex plane, $\mu$ is the Lebesgue measure on $D$ and $H(D)$ is the space of holomorphic functions on $D$, define the Bergman projection by: $$B:L^1(\mu)\rightarrow H(...
1
vote
2answers
203 views

Regarding Poisson kernel being in $h^p(\mathbb{D})$

Let $\mathbb{D}$ denote the open unit disc in the complex plane. Let $h^p(\mathbb{D})$ be the space of all harmonic functions $f$ on $\mathbb{D}$ such that $$\sup_{0\leq r<1}\left(\frac{1}{2\pi}\...
2
votes
1answer
61 views

The spectrum of the Hardy Banach algebra $(H^1(\mathbb{T}),+,*,\|\|_1)$.

Let $\mathbb{T}$ be the $1$-torus and define: $$H^1(\mathbb{T}):=\{f\in L^1(\mathbb{T})\ | \ \forall n<0, \hat{f}(n)=0\},$$ where if $f\in L^1(\mathbb{T})$ we have denoted by $\hat{f}$ the Fourier ...
3
votes
1answer
314 views

Does a holomorphic function on the unit disk with continuous radial limits have a continuous extension to the closed disk?

Question: does there exist a holomorphic function $f$ defined on the unit disk $D$ such that $\forall t \in \mathbb{R}, \exists \lim _{r\rightarrow 1^-} f(re^{it})\in\mathbb{C}$; the periodic ...
1
vote
1answer
121 views

Cauchy Integral theorem and Hardy Spaces

I am working through Theory of $H^p$ Spaces by Peter L. Duren, and I am having some difficulty with proof of the following theorem: Theorem. Every function $f \in H^1$ can be expressed as the ...
3
votes
2answers
324 views

Is harmonic Hardy space $h^p (D)$ complete for $0<p<1$?

Let $D$ be the unit disk and $0<p<1$. Let $h^p(D)$ be the harmonic Hardy space on the unit disk (i.e. $u\in h^p(D)$ if and only if $u$ is harmonic on the disk and $\sup_{0<r<1}\int_{-\pi}^\...
2
votes
1answer
134 views

A (counter)example in harmonic Hardy spaces.

I'm looking for an example of an harmonic function in the unit disk $D$, say $u$, such that the family $$(t\mapsto u(re^{it}))_{r\in[0,1)}$$ is uniformly integrable on the torus $\mathbb{T}$ and such ...
9
votes
1answer
248 views

Convergence of Taylor series in Hardy norm

If $0< p<\infty$, let $H^p(D)$ denote the Hardy space on the unit disk. We know from Hardy spaces theory that, for every $f\in H^p(D)$, the family of functions $$f_r :D\rightarrow \mathbb{C}, z\...
2
votes
1answer
153 views

Norm convergence for Fourier series in Hardy spaces with $0<p<1$

Fix $0<p<1$. Let $D$ be the complex unit disk and let $\mathbb{T}$ be the torus. Let ${\frak{H}}^p(\mathbb{T})$ denotes the $L^p(\mathbb{T})$ closure of the holomorphic trigonometric polynomials,...
2
votes
1answer
418 views

Analytic function on the right half-plane, continuous and bounded on the closed right-half plane

Let $\mathbb{C}_+$ be the right half-plane defined by \begin{align} \mathbb{C}_+ \stackrel{\text{def}}{=} \{z \in \mathbb{C} \ \colon \Re(z) > 0\}. \end{align} I am trying to understand why for ...
0
votes
1answer
65 views

Regarding invertible functions being outer

I am reading these notes titled Hardy spaces, inner and outer functions, shift-invariant subspaces, Toeplitz and Hankel operators In the second last paragraph, it says that any function that is ...
3
votes
1answer
276 views

Are polynomials dense in $H^\infty$

Let $H^\infty(D)=\{f:D\longrightarrow \mathbb{C}: f \;\text{is bounded and analytic on}\; {D}\}$ where $D=\{z\in\mathbb{C}: |z|<1\}$. Are the polynomials dense in $H^\infty(D)$? I know that they ...