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B. Maurey (1980)
Isomorphismes entre espaces H1Acta Mathematica, 145
M. Ramanujan, N. Bary, Margaret Mullins (1966)
Treatise of Trigonometric Series
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
Bulletin of the London Mathematical Society – Wiley
Published: Mar 1, 1996
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