Two Remarks on Marcinkiewicz Decompositions by Holomorphic Martingales

DOI10.1112/BLMS/28.2.149zbMATH Open0863.30042arXivmath/9210209OpenAlexW2094041185MaRDI QIDQ4893633

Publication date: 18 September 1996

Published in: Bulletin of the London Mathematical Society (Search for Journal in Brave)

Abstract: The real part of

is not dense in

. The John-Nirenberg theorem in combination with the Helson-Szeg"o theorem and the Hunt Muckenhaupt Wheeden theorem has been used to determine whether

can be approximated by

or not:

d i s t (f, R e H^{i} n f t y) = 0

if and only if for every

e > 0

there exists

l_{0} > 0

so that for

l > l_{0}

and any interval

.

|{xin I:| ilde f-( ilde f)_I|>l}|le |I|e^{-l/ e},

where

i l d e f

denotes the Hilbert transform of

f

. See [G] p. 259. This result is contrasted by the following �egin{theor} Let

f i n L^{i} n f t y_{R}

and

e > 0

. Then there is a function

and a set

so that

and

f=Re gquadmbox{ on } E.

end{theor} This theorem is best regarded as a corollary to Men'shov's correction theorem. For the classical proof of Men'shov's theorem see [Ba, Ch VI S 1-S4]. Simple proofs of Men'shov's theorem -- together with significant extensions -- have been obtained by S.V. Khruschev in [Kh] and S.V. Kislyakov in [K1], [K2] and [K3]. In [S] C. Sundberg used

-techniques (in particular [G, Theorem VIII.1. gave a proof of Theorem 1 that does not mention Men'shov's theorem. The purpose of this paper is to use a Marcinkiewicz decomposition on Holomorphic Martingales to give another proof of Theorem 1. In this way we avoid uniformly convergent Fourier series as well as

-techniques.

Full work available at URL: https://arxiv.org/abs/math/9210209

zbMATH Keywords

Brownian motion holomorphic martingales Marcinkiewicz decomposition

Mathematics Subject Classification ID

Martingales and classical analysis (60G46)