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At WillieWong's request, I'm cleaning up (i.e. deleting) the comments on http://mathoverflow.net/questions/69589/carlesons-theorem-on-the-adeles-and-other-exotic-groups, as they all refer to a previous incarnation of the question which has been significantly rewritten. I'm dumping the deleted comments here for reference.
Please see mathoverflow.net/howtoask – David Roberts Jul 6 at 4:35
David: Is this question inappropriate? Is asking for references more answerable? – Taylor Dupuy Jul 6 at 6:29
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No, I meant phrasing the question(s) with a little more padding/explanation/motivation. The content of the question is fine, and indeed I think it might receive more attention if not written just as a block of text. – David Roberts Jul 6 at 7:27
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Taylor, when you say "where can I find a proof for nonabelian compact groups", does that mean you know the result is true for nonabelian compact groups? or is that part of your question? – Yemon Choi Jul 6 at 16:16
Yemon: The validity is part of my question: Is it true that for every f in Lp(G) that fˆ in Lp(Gˆ). I can't find a reference but that doesn't necessarily mean it isn't true. – Taylor Dupuy Jul 7 at 9:35
I removed this part from the post: In interested in the following questions: - Does there exists some type of Calderon-Zymund decomposition for f in L^1(G) when G is a locally compact topological group? - What are some references for Calderon-Zygmund theory over locally compact topological groups? – Taylor Dupuy Jul 7 at 9:47 1
@Dupuy: Could you please clarify what you mean by Fourier transform of a function on a non-abelian group (for example, the symmetric group Sn)? Thanks! – SGP Jul 7 at 11:52
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Taylor, you should seriously edit your question : Fourier transform is not continuous from Lp(Rn) to itself if p≠2. Instead, it sends Lp to Lq if p∈[1,2] (as usual 1/p+1/q=1) by Hausdorff-Young inequality, and Lp outside L1loc if p>2. Also, you should precise what you mean by Calderon-Zygmund decomposition. – BS Jul 7 at 12:13
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Taylor, do you mean that the partial sum operator, or some associated maximal operator, is bounded from Lp to itself? If so, that is not, repeat NOT, what you wrote... – Yemon Choi Jul 7 at 16:11
Willie: Uh, thanks for the insightful comment, derp. SGP: en.wikipedia.org/wiki/… BS: en.wikipedia.org/wiki/… you would need a generalization of Cubes in the general setting. Boundedness of the Fourier Transform comes from boundedness of the Hilbert Transform + properties of translating the fourier transform, and the proof I know uses a Calderon-Zygmund decomposition. Yemon: I like your bold. Lp Boundedness of the operator Cf(x)=supR>0|∫R−Re−2πixξf(x)dx| implies bddness of Fouri – Taylor Dupuy Jul 8 at 6:42
I vote against closing. We should let the OP an opportunity to edit and clarify his question. – Gil Kalai Jul 8 at 8:55
Yemon and BS: That was a really silly and embarrasing mistake of me and I made an edit once I made the mistake that was still wrong. Forgive me for not seeing this right away. – Taylor Dupuy Jul 8 at 13:59
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Gil and Other: The question should have been about a Carleson operator (which I messed up the statement of) and about whether it held for other groups (I'm in the process of learning harmonic analysis and wanted to know the statement in general). Is the "Carleson Operator" Strong (p,p) for locally compact abelian groups and compact nonabelian groups. I know the answer to my question is no now because of what Fefferman proved jstor.org/pss/1970864. I also know now that a statement of what a "Carleson operator" is for general groups isn't easy to straight forward. – Taylor Dupuy Jul 8 at 14:04
What should I do with this one? Should I rephrase the question in a new post? I'm not exactly sure what your guys policy is since I botched this up pretty badly. – Taylor Dupuy Jul 8 at 14:06 1
I think underneath there is a good question in there. I used the answer page instead of the comment spot because I don't have enough space in the comments to clearly write down the argument. What you should do now, is, I think, edit your question to the correct form (the one about Carleson operators), because when you started asking the question you didn't know the result. Then you can post an answer yourself stating that you've actually found out about Fefferman's paper on ball multiplier, after you've formulated the question. – Willie Wong Jul 8 at 14:53
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