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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.
513:
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
846:
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
995:
This section is confusing because it's misplaced. The part about the connection (the comparison) should really be later, after the definition of real Hardy spaces, but the important example (z+1)/(z-1) could also come earlier in the purely complex context. (It could then be referenced in the "real"
894:
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
162:
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
929:
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
803:
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.
478:
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.
1328:
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.
178:
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.
847:
of real parts of functions in H" 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.
895:
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).
1371:
I don't think p can be zero since Hardy spaces are not defined for p=0. And yes, radial limits exist for p= infinity as well, see Rudins Real and complex analysis(Fatous theorem states that radial limits exist for all
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505:"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
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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
140:
1352:
930:
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
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460:-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.
814:
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!
820:
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!
529:, 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
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283:
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) "
540:
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
1010:
And point out "There are two versions: the target space can be either R or in C." Which one is more common? I only know the one with real values.
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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..
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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
817:
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?
1017:, where the two characterizations are compared, is moved after this section, then the comparisons will make a lot more sense.
33:
933:(which he calls the "little" Hardy spaces). Stein was not (in the particular section I have in mind) talking about
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for some real value Ο and some real-valued function g(z) that is integrable on the unit circle. In particular, Gβ H
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which is equivalent to the definition by convolution given later, but I see no easy way to relate this norm of
1404:). Btw I don't think H^p(T)is defined for p<1 since Fourier coefficients might not make sense for p<1.
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so he could explain the connection between real Hardy spaces and the Hardy spaces in complex analysis.
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I believe this statement is correct. I think I can prove it, but I will look around for a reference.
203:"? I understand: same vector space with an equivalent norm. I am correct? And if so, why not say it?
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It's pretty wild to talk about "real" Hardy spaces that might consist of complex-valued functions!
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Interesting question. I just checked Stein, which explicitly includes the statement that for 1<
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Thanks! But I also introduced wrong statements, see below (and see my last edit of the article).
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I'm gone for a few days, and the article has magically doubled in size! Bravo! Good job!
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We have to clarify what is done in the references about real Hardy spaces when
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After this correspondence is established, one can write H to denote either H(
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The second is "functional analysis" seems a bit limiting of a role to put
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holomorphic functions yields (actually, is equivalent to the fact) that
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For 1 β€ p β€ β these real Hardy spaces H are certain closed subsets of L
525:"In real analysis Hardy spaces are certain spaces of distributions on
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If these issues are spelled out more fully, and the section called
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The description of the exterior function is OK, but now it is (log
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Maybe call them "Hardy spaces defined via maximal functions"?
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757:{\displaystyle \|h\|_{H^{1}}=\|h\|_{1}+\|{\tilde {h}}\|_{1}}
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What should one understand under the (too) vague sentence "
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complex Hardy space (from Stein's
Harmonic analysis).
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635:= 1, say). If you have observed in the meantime that
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Connection to real Hardy spaces on the circle, redux
784:< 1, I'll try to explain later (in the article).
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375:Good catch, I made a small change to reflect this.
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1107:Reference Folland (Encyclopaedia of M.) excludes
456:are integrable, with the norm the sum of the two
444:I still believe it is wrong. On the circle, real-
268:I made some changes, let me know what you think.
571:consists of the real parts of the functions in
864:The ordering of the section seems fine to me.
772:to an holomorphic function in the disk (disc?).
1085:, but I don't believe that "people" say that
1015:Connection to real Hardy spaces on the circle
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776:By the way I think you are wrong about real-
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501:I have two difficulties with the sentence:
1077:= β; Clearly the maximal function stays in
313:) that is integrable (classical, OK), and |
242:There is something wrong in the sentence "
220:Yes. your correct, we should just say it.
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1198:H^1 and L^1 have inequivalent norms
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38:It is of interest to the following
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1426:Mid-priority mathematics articles
115:Knowledge:WikiProject Mathematics
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353:) 14:34, 11 November 2008 (UTC)
158:"Big" vs. "Little" Hardy spaces.
118:Template:WikiProject Mathematics
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1367:Hardy Spaces on the unit circle
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559:Real Hardy spaces on the circle
509:spaces of functional analysis."
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1351:It's obviously corrected now.
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1103:08:56, 17 November 2008 (UTC)
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1000:Real Hardy spaces terminology
981:20:09, 13 November 2008 (UTC)
951:15:26, 13 November 2008 (UTC)
905:14:08, 13 November 2008 (UTC)
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830:09:26, 13 November 2008 (UTC)
794:19:54, 11 November 2008 (UTC)
671:is the conjugate function of
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425:21:07, 10 November 2008 (UTC)
385:15:18, 11 November 2008 (UTC)
363:14:45, 11 November 2008 (UTC)
301:14:14, 11 November 2008 (UTC)
278:12:43, 11 November 2008 (UTC)
264:20:08, 10 November 2008 (UTC)
230:02:25, 11 November 2008 (UTC)
213:11:24, 10 November 2008 (UTC)
189:14:38, 10 November 2008 (UTC)
163:difference between the two.
109:and see a list of open tasks.
1421:B-Class mathematics articles
1361:19:47, 17 January 2020 (UTC)
1345:20:49, 20 January 2011 (UTC)
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345:on the boundary, I believe.
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407:= 1. The unit ball of real-
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