# Tagged Questions

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

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### Intuition regarding $\lim \lVert u_r - u \rVert_{p}=0$

I have some trouble intrepreting the following statment If $u$ is harmonic in $D$ and has bouned means for order p on circles of radius $< 1$ then $\lVert u \rVert_{p}=\lVert u \rVert_{L^{p}}$ ...
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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 $... 0answers 44 views ### belong to$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$but the problem ... 1answer 43 views ### Hardy-Littlewood theorem about the Poisson integral for$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)|‎ $‎. ... 0answers 24 views ### Invariant Subspace Vs Reducing Subspace in some special Hilbert space Recently I was reading a book "Operator Function and system" written by Nikolski. There I found this statement. Let$\mu$be a finite positive measure on the unit circle in complex plane which is ... 2answers 82 views ###$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$. One can ... 0answers 98 views ### Criterion for Membership in Hardy Space$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}$is a compactly supported ... 1answer 210 views ### Error in Stein Shakarchi Exercise on$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 $$f(x):=\dfrac{\chi_{|x|\leq 1/2}}{x(\log|x|)^{2}}... 1answer 39 views ### Boundary function of product of H^\infty and H^2 function If f\in H^2 and \phi \in H^\infty is it true that \widetilde{\phi f} = \tilde{\phi}\tilde{f}? It is is easy to see that \widetilde{z^nf} = \tilde{z^n}\tilde{f} and so for all polynomials p... 1answer 55 views ### Hardy Hilbert space: Multiplication by H^\infty function [duplicate] If H^2(\mathbb{D}) is space of all analytic functions whose series of taylor coefficient is absolute square summable and \phi is analytic in \mathbb{D},then \phi H^2 \subset H^2 if and only ... 0answers 164 views ### Inner Functions in Annuli: Not Likely! The other day someone reminded me of something I'd thought about some years ago. As back then it took me a little while to see why there was any problem; this time I got much farther on a solution ... 2answers 70 views ### Integrability of Maximal Convolution Operator Let f\in L^{\infty}(\mathbb{R}^{n}) be supported in the unit ball and have mean zero. Let \phi\in L^{1}(\mathbb{R}^{n})\cap C^{\alpha}(\mathbb{R}^{n}) be a Holder continuous function with exponent ... 2answers 29 views ### Radial Limits of Singular Inner Functions Given a positive singular measure \mu on [-\pi,\pi], we define a singular inner function by$$S(z)=\exp\left(-\int_{-\pi}^{\pi}\frac{e^{i\theta}+z}{e^{i\theta}-z}\,d\mu(\theta)\right).$$It is ... 0answers 31 views ### Show that the ideal generated by an inner function is closed. Suppose H^\infty=\{f\in H(U):\exists M<\infty,\,\forall z\in U,\,|f(z)|\le M\} is the set of bounded holomorphic functions in the unit disk. It is a Banach algebra with pointwise multiplication. ... 1answer 47 views ### Show that the inclusion relation of Hardy spaces is proper The definition of Hardy spaces for the unit disk is here. It is clear that for 0<p<q\le\infty, H^q\subseteq H^p, by Hölder's inequality. I'm asked to show that the inclusion relation is ... 3answers 108 views ### Alternative definition of Hardy spaces Classically, Hardy spaces H^pon the disk are introduced as set of functions analytic on \mathbb{D} = \{z \in \mathbb{C}: |z|<1\}, which has bounded H^p norm:$$ \|f\|_{H^p} = \sup_{0\leq r &... 0answers 103 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 ... 1answer 310 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. $$\sup_{\varphi \in [0,2\pi]} |f(e^{i\... 1answer 28 views ### MN-\langle M,N \rangle is uniformly integrable when M, N are H^2? This proof below really seems to have little to do with uniform integrability, what does the DCT application give exactly? Any alternative proofs would be good too 1answer 214 views ### On the Hilbert Transform of a Bounded Function Let f: \mathbb R \rightarrow \mathbb R be a bounded function that is smooth and also in L^1(\mathbb R) \cap L^2(\mathbb R). I want to prove that the Cauchy transform of this function Kf is in H^... 0answers 55 views ### Holomorphic function in unit disc? I am stuck on this exercise from Stein and Shakarchi's Real Analysis: suppose F is holomorphic in the unit disc, and$$\sup_{0\leq r < 1} \frac{1}{2\pi} \int_{-\pi}^\pi \log^+ |F(re^{i\theta})|\,... 0answers 42 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 ... 1answer 40 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 ... 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 ... 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 $$\|p_n\|=\... 1answer 24 views ### Show that for every real-valued L^2 function u on S^1 there is a real-valued v in the same space such that u + iv\in \widetilde{\mathbf H}^2 For a homework exercise (1.8 in the book An Introduction to Operators on the Hardy-Hilbert Space) I am asked to show Let u be a real-valued function in L^2(S^1). Show that there exists a ... 1answer 33 views ### How to decide whether a function is in H^p_+. H^p_+ is the Hardy space on the upper half-plane of \mathbb C consisting of functions f satisfying$$ \sup_{y_0>0}\left(\int_{-\infty}^{\infty} |f(x+iy_0)|^p dx\right)^{\frac 1p}<\infty $$... 1answer 87 views ### Interpolation sequences and open mapping theorem I'm using Garnett's "Bounded Analytic Functions" as a course text and looking at interpolation sequences. z_n is a sequence of interpolation if for each sequence a_n \in l^{\infty} there exists f ... 1answer 14 views ### A function in H^\infty(\Delta^2) Can you give an example of a function f\in H^\infty(\Delta^2) with f^{-1}\in L^\infty(T^2) but not inner? Here H^\infty(\Delta^2) is the space of all bounded analytic functions defined on bi-... 0answers 57 views ### Multiplication operator is bijective? Let H^2(\Delta^2) denote the Hardy space on the polydics and M_f:H^2(\Delta^2)\rightarrow H^2(\Delta^2) be a multiplication operator by f\in H^\infty(\Delta^2). Is the operator is bijective? ... 1answer 89 views ### A simple function with no tangential limits but with non-tangential limits I am going to be teaching a course about the Hardy space and I would like to show the students that the non-tangential limit is a necessary concept BEFORE telling them about Blaschke products. Is ... 1answer 78 views ### Literature on interpolation in Hardy spaces I'm an undergraduate mathematics student and I'm searching for notes and books on Hardy spaces H^p, in particular interpolation theory including topics like Carleson measures, Carleson's H^\infty ... 1answer 184 views ### Showing a certain sequence is an orthonormal basis of H^2(\mathbb{R}_{+}^{2}). The problem is to show$$\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 ... 0answers 75 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 ... 0answers 85 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 L^{\infty}(...
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 ...