Questions tagged [hardy-spaces]
For questions about Hardy spaces. Use the other related tag like (tag: complex-analysis) or (operator-theory).
203
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An estimate in Hardy space.
Consider function $u=(u_1,u_2):\mathbb{R}^2\to \mathbb{R}^2$, and define $\nabla^\perp=(\partial_2,-\partial_1)$, I attempt to show that
$$\lVert \nabla u_1\cdot \nabla^\perp u_2 \rVert_{\mathcal{H}^1}...
- 11
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Is a Toeplitz operator just a multiplication operator with a bounded function on the unit disk?
I have been recently studying Hardy spaces on the unit disk (most specifically, $H^2$) and I have come across the so-called Toeplitz operators.
In every book I have found about the topic they define a ...
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Equivalence of Fourier Transform on $\ell_2(\mathbb{Z}_+)$ and $L_2(\mathbb(R)_+)$ via equivalence of $H_p( \mathbb{D})$ and $H_p(\mathbb{C}_+)$?
Throughout I'll use the fact that the Hardy space $H_2$ is the set of $L_2$ functions on the boundary with vanishing Fourier coefficients.
We know that the Fourier Transform is an isometric ...
- 286
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Does $f'$ have an infinite unrestricted limit at 1?
Let $f$ be an analytic function defined in the unit disc $\mathbb{D}=\{z:|z|<1\}$. If the unrestricted limit of $f$ at the boundary point $1$ equals to $\infty$, i.e.
$$
\lim_{\substack{z \...
- 95
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Bounding $f'(z)$ with $O(\log(\frac{1}{1-r}))$ for an Analytic Series
I am working with an analytic function defined within the unit disk $|z| < 1$ as follows:
$$
f(z) = \sum_{n=1}^{\infty} a_{n} z^{n},
$$
where I have the condition that $\sum_{n=1}^{N} n|a_{n}| = O(\...
- 81
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Boundedness of derivative related to hardy spaces
Assume $f:[0,1]\to [0,1]$ is an orientation-preserving homeomorphism, and $f$ is absolutely continuous on $[0,1]$, $\log f'\in H^{1/2}([0,1])$, that is to say,
$$\int_{0}^{1}\int_{0}^{1}\frac{|\log f'(...
- 1,834
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26
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About the algebra $H^\infty + C(\mathbb{T})$
We know that $H^\infty + C(\mathbb{T})$ is the closed subalgebra of $L^\infty(\mathbb{T})$ containing $H^\infty$.
How to show that
$H^\infty + C(\mathbb{T})$ = clos$[\cup_{n\geq 0} \chi_{-n} H^\infty$]...
- 23
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Fejer-Riesz Theorem for polynomials
Let $\mathbb{D}=\{z: |z| <1\}$ and $\mathbb{T}=\{z: |z|=1\}$. Suppose $D$ and $E$ are polynomials of degree atmost $n$ with complex coefficients such that
$|E(z)| \leq |D(z)|$ for all $z \in \...
- 973
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Why is this assumption necessary on the proof of duality of $H^1$ and BMO? (Stein)
I was going through the proof on Stein's Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals of the fact that BMO is the dual of $H^1$ (this is on Chapter IV, section 1.2....
- 283
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Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?
The whole theorem goes as follows:
Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying:
$$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...
- 41
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Closure of $H^{\infty}$ with BMO-norm
Let $H^{\infty}$ denote the space of bounded analytic functions on the unit disk $\mathbb{D}$. Let $BMOA$ be the subspace of Hardy Space $H^2$, such that
$$\|f\|_{BMOA}=|f(0)|+\sup_{a\in\mathbb{D}}\|f\...
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2
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Is there an "easy" proof of the existence of radial limits of functions in the hardy spaces $H^p$?
Let $HOL(\mathbb{D})$ be the analytic functions defined on the unit disk and for $1\leq p \leq \infty$,
$$H^p = \{f\in HOL(\mathbb{D}) : \lim_{r\nearrow 1} \int_{rC}|f(z)|^p \frac{dz}{2\pi i} < \...
- 286
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Boundedness of functions in the Hardy space $H^2(\mathbb H)$
Let $F: \mathbb H \to \mathbb C$ be a function in the Hardy space $H^2(\mathbb H)$. In other words, we have
$$\sup_{y>0} \int_{-\infty}^\infty |F(x+iy)|^2 \, dx < \infty.$$
Let $f(x) = \lim_{y \...
- 2,108
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29
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Does these conditions ensure Hardy space membership?
Let D be a disk of center $c$ and radius $R>0$ located in the complex plane, $\partial D$ be the boundary of the disk, $H(D)$ denote the collection of analytic functions defined on $D$, $$H^p(D)=\...
- 659
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Example of a function which is in Hardy space $H^1(\mathbb{D})$ but not in $H^2(\mathbb{D})$
We know that the Hardy spaces on unit disc $\mathbb{D}$, $H^2(\mathbb{D})\subset H^1(\mathbb{D})$. I need to find an example to show that the containment is proper. I was trying to use the fact that ...
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