# Talk:Hardy space

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## "Big" vs. "Little" Hardy spaces.

On the unit circle we have a bit of a problem. Some of the theorems we quote are about the "little" Hardy spaces (of harmonic extensions), while some of the theorems we quote are about the "big" hardy space (of holomorphic extensions.) And maybe we should find a way to warn the reader of the difference between the two. Thenub314 (talk) 06:40, 5 November 2008 (UTC)

I have seen that some work was already done to address the problem, but I have tried to clarify it even more. It would be good that a "native English speaker" looks and corrects my words. However, some repetitions and incoherences remain. Bdmy (talk) 14:38, 10 November 2008 (UTC)

## Real Hardy spaces

What should one understand under the (too) vague sentence "Hp is essentially the same as Lp"? I understand: same vector space with an equivalent norm. I am correct? And if so, why not say it? Bdmy (talk) 11:24, 10 November 2008 (UTC)

Yes. your correct, we should just say it. Thenub314 (talk) 02:25, 11 November 2008 (UTC)

## Factorization

There is something wrong in the sentence "for some real value φ and some real-valued function g(z) that is integrable on the unit circle. In particular, G∈ H1". According to what follows (and seems OK to me) the function G is integrable iff exp(g) is integrable, which is not the same. Bdmy (talk) 20:08, 10 November 2008 (UTC)

I made some changes, let me know what you think. Thenub314 (talk) 12:43, 11 November 2008 (UTC)

At the first glance I agree with all the changes you have made, although I may (a little) regret my sentence (in any equivalent "true English" form) "After this correspondence is established, one can write Hp to denote either Hp(T) or the space Hp from the first section." Bdmy (talk) 14:14, 11 November 2008 (UTC)
The description of the exterior function is OK, but now it is (log φ) that is integrable (classical, OK), and |G| on the boundary is now φ, so you are in H1 under the additional assumption that φ is in L1. After this small point is fixed, it will be a better point of view to have |G| = φ on the boundary, I believe. Bdmy (talk) 14:34, 11 November 2008 (UTC) Bdmy (talk) 14:45, 11 November 2008 (UTC)
Good catch, I made a small change to reflect this. Thenub314 (talk) 15:18, 11 November 2008 (UTC)

## Introduction

The introduction says "For 1 ≤ p ≤ ∞ these real Hardy spaces Hp are certain closed subsets of Lp": I don't think that this is true when p = 1. The unit ball of real-H1 is perhaps closed in L1, but not the whole space I think. Bdmy (talk) 21:07, 10 November 2008 (UTC)

I believe this statement is correct. I think I can prove it, but I will look around for a reference. Thenub314 (talk) 13:52, 11 November 2008 (UTC)

I still believe it is wrong. On the circle, real-H1 is the space of functions such that f and its conjugate tilde-f are integrable, with the norm the sum of the two L1-norms; this norm is not equivalent (as you know from the Hilbert transform properties), and the space can't be closed, because of the general Banach theorems for example. Bdmy (talk) 14:18, 11 November 2008 (UTC)
You may be correct. I looking back at my "proof," I certainly see I did not get enough sleep last night. I say we remove for the moment. If a reference turns up with a proof all the better, but it sounds to me like your thinking is in the right direction. Thenub314 (talk) 15:10, 11 November 2008 (UTC)

I have two difficulties with the sentence:

"In real analysis Hardy spaces are certain spaces of distributions on the real line, which are (more or less) boundary values of the holomorphic functions of the complex Hardy spaces, and are related to the Lp spaces of functional analysis."

The first is holomorphic. The objects that are usually studied in real analysis (I think) are what this article calls the real Hardy spaces which morally boundary values of harmonic functions, and can be written as the sum of a function in complex Hardy space and the conjugate of another function in complex Hardy space (from Stein's Harmonic analysis).

The second is "functional analysis" seems a bit limiting of a role to put Lp spaces in. Are there any objections to:

"In real analysis Hardy spaces are certain spaces of distributions on Euclidean space, which are (more or less) boundary values of the harmonic functions, these spaces are closely to the complex Hardy spaces, and are closely related to the Lp spaces."

Though by saying they are morally boundary values I feel like I am cheating a bit, the uninformed reader my not recognize that I am thinking of the upper half plane, and that article is rather specific to one dimension. Which maybe I should be as well. Thenub314 (talk) 12:05, 11 November 2008 (UTC)

## Real Hardy spaces on the circle.

The article states that

The space real-Hp consists of the real parts of the functions in Hp.

which does not agree with what I would think of as the definition. Also, we compare these to the real Hardy spaces below, but there is no restriction there of the functions in the real Hardy Spaces on Rn being real valued. Am I being dense again? Thenub314 (talk) 15:43, 11 November 2008 (UTC)

This is a simple way to say things, and that is related to the matters of this section. After this, you may if you like consider complex functions like h = f + ig where f and g are both in this real H1 (take p = 1, say). If you have observed in the meantime that f and g being the real part of H1 holomorphic functions yields (actually, is equivalent to the fact) that f, tilde-f, g and tilde-g are in L1 (here, tilde-f is the conjugate function of f), then you may define the "complex real space" by saying that
$\|h\|_{H^1} = \|h\|_1 + \|\tilde h\|_1$
which is equivalent to the definition by convolution given later, but I see no easy way to relate this norm of h to an holomorphic function in the disk (disc?).
By the way I think you are wrong about real-Hp for p < 1, I'll try to explain later (in the article). Bdmy (talk) 19:54, 11 November 2008 (UTC)
My point about real-H^p was not that I didn't understand how to get at the complex functions from this definition. It is simply that we should give a definition that corresponds to what people usually consider on the circle. It is nice to keep the connection to real parts of complex-H^p functions. I simply wanted to give a more complete definition. I was a bit careless simply saying p could go down to 0. Seeing we have a separate section now, I don't really see any changes necessary though. I did the title though "First approach" let me now what you think.
I don't mind about your successive versions for the title of the subsection; the last (for now) with "connection.." seems OK to me. Thanks for the English improvements!
I moved the section "Real Hardy spaces on the circle" to be at the end of the discussion of "all about the disk-circle". Better?
My problem with having complex functions in the "Real Hardy space" on the circle is that it may get very confusing for the reader, and hard to get notation for! Bdmy (talk) 09:26, 13 November 2008 (UTC)
Fair enough. But, in some sense, it is a confusing point of the subject. I should first point out that the only book I know of to use the phrase "real Hardy spaces" is Stein's "Harmonic Analysis". Here the spaces certainly do include complex functions. Hence saying that "real Hardy space consists of real parts of functions in Hp" disagrees with the definitions of how this phrase is usually used. While I don't want to confuse the readers, I don't want to give an impression that real Hardy spaces consist of real valued functions.
The ordering of the section seems fine to me. Thenub314 (talk) 13:13, 13 November 2008 (UTC)
I guess Stein is not talking about the circle, but about R^n; there, I have no problem with having complex functions, as there is no other candidate in that case. But on the circle the complex functions in the Hardy spaces refer I believe, for most people, to the analytic ones. Maybe changing "Real Hardy spaces on the circle" to "Real parts of H^p functions on the circle"? (but just to please you; actually, I think that people working in the classical setting of the circle would understand "Real Hardy space" my way). Bdmy (talk) 14:08, 13 November 2008 (UTC)
I am a bit confused by the comment about no other candidate, you could certainly restrict the definition to real valued functions, but maybe you meant something different. But, I disagree with your your comment about complex functions in Hardy spaces on the circle. In the GTM text "Harmonic function theory" by Axler, Bourdon, and Ramey they define real Hardy spaces (harmonic Hardy spaces by their terminology) as complex valued functions, as does Schlag in his notes here [1] (which he calls the "little" Hardy spaces). Stein was not (in the particular section I have in mind) talking about Rn, but rather R1 so he could explain the connection between real Hardy spaces and the Hardy spaces in complex analysis. Thenub314 (talk) 15:26, 13 November 2008 (UTC)
I am affraid that you are right about the most common use. I'll try to do something about it, but you'll probably need to come after and clean.. Bdmy (talk) 20:09, 13 November 2008 (UTC)

## Wow!

I'm gone for a few days, and the article has magically doubled in size! Bravo! Good job! linas (talk) 03:00, 17 November 2008 (UTC)

Thanks! But I also introduced wrong statements, see below (and see my last edit of the article). Bdmy (talk) 08:56, 17 November 2008 (UTC)

## H∞

We have to clarify what is done in the references about real Hardy spaces when p = ∞; Clearly the maximal function stays in L for every function in L, but I don't believe that "people" say that H = L. Bdmy (talk) 08:56, 17 November 2008 (UTC)

Reference Folland (Encyclopaedia of M.) excludes p = ∞. Bdmy (talk) 09:01, 17 November 2008 (UTC)

Interesting question. I just checked Stein, which explicitly includes the statement that for 1<p≤∞ that Hp = Lp, does folland define H? Thenub314 (talk) 15:58, 17 November 2008 (UTC)
In his Encyclopaedia article, Folland starts with 1 < p < ∞ and does not talk about H. I checked the basic article on the question, Fefferman-Stein (1972); they have p < ∞ all the way. My feeling is that you may state that real-H = L, but (due to his trivial definition) it has to be useless and we better skip it. Bdmy (talk) 18:18, 17 November 2008 (UTC)

## H^1 and L^1 have inequivalent norms

$f_k(x) = \mathbf{1}_{[0, 1]}(x - k) - \mathbf{1}_{[0, 1]}(x - k).$

According to the article (in the Real Hardy Spaces for R^n section), the sequence of functions defined above is bounded in L^1 but not H^1. This sequence is just the sequence of zero functions on the line, making it trivially bounded in both L^1 and H^1. —Preceding unsigned comment added by 165.112.77.70 (talk) 20:49, 20 January 2011 (UTC)