From: David Ullrich
Subject: Re: Lipschitz and AC references and other questions.
Date: Fri, 18 Dec 1998 13:19:30 -0600
Newsgroups: [missing]
To: Dave Rusin
Keywords: What's new in Singular Integrals [42-XX]
Dave Rusin wrote:
>
> >I know the singular integral stuff there is a little dated
>
> Would you care to comment about how the field has changed, or to suggest
> a place I might look to learn this?
> dave
Well golly, this is stuff I'm supposed to know but I haven't
been paying attention to the last few years (sigh, it happens). None
of the theorems in Stein have become false, there's just a lot known
that wasn't known then. (I guess I'm wondering whether I'm misunderstanding
your question, because I don't see how it can be less than obvious that
a lot of progress has been made since Stein.)
Maybe "singular integral stuff" was too specific; I was
really talking about "harmonic analysis" in general. Except that
"harmonic analysis" is much too broad... You could try Torchinsky
"Real-Variable Methods in Harmonic Analysis" for an idea of what
the story was about ten years ago; it's a very nice book with
lots of good stuff inside. A large aspect of _how_ things
have changed would definitely be a larger reliance on real-variable
methods: You can learn a lot about the circle in Zugmund and
Garnett, turns out that proofs using real methods are easier to
extend to R^n than proofs using Blaschke products.
(Hmm, come to think of it I suppose "these days we use
real-variable methods when possible" isn't really an indication
of how things have changed since Stein "Singular Integrals",
that's a lot of real-variable stuff already.)
Whatever field it is that I'm having a hard time naming
precisely, it certainly includes Littlewood-Paley theory and
Hardy spaces (atomic decompositions, etc). Which then gets us
into wavelets, which I really don't know much about: I'm certain
that a lot of the groundbreaking research in wavelets is just
rephrasings of things that Calderon knew 40 years ago, but I
suspect there may be some interesting new stuff in there as
well.
There's no BMO = (H^1)*in Stein; no atomic decompositions
(at least not explicitly). Leads to another "hmm": if you want
a paper that was really a _big_ influence on a lot of the things
I'm babbling about it would probably be Fefferman-Stein
"H^p Spaces of Several Variables", Acta Math 1972.
Etc etc. If your "Would you care to comment about how
the field has changed" was a polite way of expressing skepticism
the answer is of _course_ the field's changed tremendously in the
last 28 years.
--
David Ullrich
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