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Two Remarks on Marcinkiewicz Decompositions by Holomorphic Martingales

Two Remarks on Marcinkiewicz Decompositions by Holomorphic Martingales TWO REMARKS ON MARCINKIEWICZ DECOMPOSITIONS BY HOLOMORPHIC MARTINGALES PAUL F. X. MULLER 1. Introduction The real part of H™(T) is not dense in L$(T). The John-Nirenberg theorem, in combination with the Helson-Szego theorem and the Hunt-Muckenhaupt-Wheeden theorem, has been used to determine whether or not a given feL%(T) can be approximated by Re//°°(T): dist(/, Rei/ ) = 0 if and only if for every e > 0 there exists A > 0 so that for X > X and any interval / £ T, o o *dt xsl: >x where /denote s the Hilbert transform of/; see [6, p. 259]. This result can be compared to the following. THEOREM 1. LetfeLg and e > 0. Then there is a function ge//°°(T) and a set Eczl so that \1\E\ < e and f= Reg on E. Moreover, g satisfies WgW^ = <9(log(l/e)). This theorem is best regarded as a corollary to Men'shov's correction theorem. For the classical proof of Men'shov's theorem, see [1, Chapter VI §1—§4]. Simple proofs of Men'shov's theorem—together with significant extensions—have been obtained by S. V. Khrushchev in [8] and S. V. Kislyakov in [9], [10] and [11]. S. V. Kislyakov's work on correction theorems has led http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

Two Remarks on Marcinkiewicz Decompositions by Holomorphic Martingales

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References (2)

Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/28.2.149
Publisher site
See Article on Publisher Site

Abstract

TWO REMARKS ON MARCINKIEWICZ DECOMPOSITIONS BY HOLOMORPHIC MARTINGALES PAUL F. X. MULLER 1. Introduction The real part of H™(T) is not dense in L$(T). The John-Nirenberg theorem, in combination with the Helson-Szego theorem and the Hunt-Muckenhaupt-Wheeden theorem, has been used to determine whether or not a given feL%(T) can be approximated by Re//°°(T): dist(/, Rei/ ) = 0 if and only if for every e > 0 there exists A > 0 so that for X > X and any interval / £ T, o o *dt xsl: >x where /denote s the Hilbert transform of/; see [6, p. 259]. This result can be compared to the following. THEOREM 1. LetfeLg and e > 0. Then there is a function ge//°°(T) and a set Eczl so that \1\E\ < e and f= Reg on E. Moreover, g satisfies WgW^ = <9(log(l/e)). This theorem is best regarded as a corollary to Men'shov's correction theorem. For the classical proof of Men'shov's theorem, see [1, Chapter VI §1—§4]. Simple proofs of Men'shov's theorem—together with significant extensions—have been obtained by S. V. Khrushchev in [8] and S. V. Kislyakov in [9], [10] and [11]. S. V. Kislyakov's work on correction theorems has led

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Mar 1, 1996

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