2,966 research outputs found
Lectures on Nehari's Theorem on the Polydisk
We give a leisurely proof of a result of Ferguson--Lacey (math.CA/0104144)
and Lacey--Terwelleger (math.CA/0601192) on a Nehari theorem for "little"
Hankel operators on a polydisk. If H_b is a little Hankel operator with symbol
b on product Hardy space we have || H_b || \simeq || b ||_{BMO} where BMO is
the product BMO space identified by Chang and Fefferman. This article begins
with the classical Nehari theorem, and presents the necessary background for
the proof of the extension above. The proof of the extension is an induction on
parameters, with a bootstrapping argument. Some of the more technical details
of the earlier proofs are now seen as consequences of a paraproduct theory.Comment: 35 pages. 65 Reference
Maximal smoothness of the anti-analytic part of a trigonometric null series
We proved recently math.CA/0510403 that the anti-analytic part of a
trigonometric series, converging to zero almost everywhere, may be square
integrable on the circle. Here we prove that it can even be infinitely
differentiable, and we characterize precisely the possible degree of smoothness
in terms of the rate of decrease of the Fourier coefficients. This sharp
condition might be viewed as a "new quasi-analyticity".Comment: 6 pages. Announcement of math.CA/0406261 and sketch of the proof of
one half of it. Should be identical to journal version except the French
abstrac
An "Analytic" Version of Menshov's Representation Theorem
Every measurable function f on the circle can be represented as a sum of
harmonics with positive spectrum, converging in measure. For convergence almost
everywhere this is not true. We discuss several other subsets of Z for which
one might get a Menshov type representation converging almost everywhere or in
measure.Comment: 4 pages. Announcement of math.CA/0510616. Should be identical to
journal version except for the French abstrac
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