Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [hardy-spaces]

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

Hardy spaces are classes of holomorphic functions on the unit disc which satisfy some integrability conditions. Namely, if $p>0$, then $H^p$ is the space of holomorphic functions such that $$\sup_{r\in (0,1)}\int_0^{2\pi}|f(re^{it})|dt<\infty.$$

211 questions
35
votes
1 answer

Major error in classic "Banach Spaces of Analytic Functions"

"Banach Spaces of Analytic Functions" by Kenneth Hoffman is an excellent introduction to $H^{p}$ spaces and is considered a classic mathematical analysis textbook. I was therefore surprised to find major mistakes in the proof of a central theorem…
SihOASHoihd
  • 1,536
12
votes
1 answer

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 there. The statement is as follows. Suppose…
Glen Wheeler
  • 2,975
11
votes
1 answer

Showing atomic $H^{1,p}$ is a Banach Space

Consider $1
Moss
  • 2,030
9
votes
1 answer

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 $$f_r :D\rightarrow \mathbb{C}, z\mapsto f(rz)$$ converges in $H^p(D)$ to $f$ for…
Bob
  • 5,995
9
votes
1 answer

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 $$f(x):=\dfrac{\chi_{|x|\leq 1/2}}{x(\log|x|)^{2}}$$ does not belong to the real Hardy space…
Matt Rosenzweig
  • 4,311
9
votes
1 answer

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. $$ \sup_{\varphi \in [0,2\pi]} |f(e^{i\varphi})| < \infty $$ (This class of functions is…
User133713
  • 1,000
7
votes
1 answer

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 $\chi_1 M\subset M$ if and only if $M=\phi H^1$ for some…
john
  • 1,191
7
votes
2 answers

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 function (say in a ball $B$) such that $\int f=0$…
Matt Rosenzweig
  • 4,311
7
votes
1 answer

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 than I did back then. This is supposed to be a…
David C. Ullrich
  • 92,839
6
votes
1 answer

Riesz representation theorem for functionals acting on Hölder $C^\alpha$ functions

Assume you have a linear functional $F:C^\alpha(\mathbb R^n) \mapsto\mathbb R$ such that $$ |F(f)| \leq \vert f \vert_{C^\alpha(\mathbb R^n)} $$ but only depending on the Holder seminorm, that is, $$ |f|_{C^\alpha} = \sup_{x,y}|x-y|^{-\alpha}…
HHN
  • 367
6
votes
0 answers

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 $$R:\{c(n)\}\mapsto \{c(n+1)\}$$ and the left translation operator is…
Erick Jerald
  • 61
6
votes
1 answer

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 a harmonic majorant on $\Omega$, i.e. there is a…
Malik Younsi
  • 3,968
5
votes
1 answer

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 compactly supported, and let $\phi$ be a Schwartz…
Rob F
  • 83
5
votes
1 answer

$L^1(\mathbb{R}^n)$ 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 it possible to find explicit examples of…
David
  • 537
5
votes
0 answers

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 half plane. In other words, I need to show…
chessman
  • 51
1
2 3
…
13 14