On a weak type (1,1) inequality for a maximal conjugate function

Nakhlé Asmar; Stephen Montgomery-Smith

Studia Mathematica (1997)

  • Volume: 125, Issue: 1, page 13-21
  • ISSN: 0039-3223

Abstract

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In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of H p spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.

How to cite

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Asmar, Nakhlé, and Montgomery-Smith, Stephen. "On a weak type (1,1) inequality for a maximal conjugate function." Studia Mathematica 125.1 (1997): 13-21. <http://eudml.org/doc/216417>.

@article{Asmar1997,
abstract = {In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of $H^p$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.},
author = {Asmar, Nakhlé, Montgomery-Smith, Stephen},
journal = {Studia Mathematica},
keywords = {conjugate function; maximal function; weak type (1,1); Brownian motion; harmonic conjugation; martingale difference},
language = {eng},
number = {1},
pages = {13-21},
title = {On a weak type (1,1) inequality for a maximal conjugate function},
url = {http://eudml.org/doc/216417},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Asmar, Nakhlé
AU - Montgomery-Smith, Stephen
TI - On a weak type (1,1) inequality for a maximal conjugate function
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 1
SP - 13
EP - 21
AB - In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of $H^p$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.
LA - eng
KW - conjugate function; maximal function; weak type (1,1); Brownian motion; harmonic conjugation; martingale difference
UR - http://eudml.org/doc/216417
ER -

References

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  1. [1] N. Asmar and S. J. Montgomery-Smith, Hahn's Embedding Theorem for orders and analysis on groups with ordered dual groups, Colloq. Math. 70 (1996), 235-252. Zbl0855.43001
  2. [2] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304. Zbl0223.60021
  3. [3] D. L. Burkholder, R. F. Gundy and M. L. Silverstein, A maximal characterization of the class H p , Trans. Amer. Math. Soc. 157 (1971), 137-153. Zbl0223.30048
  4. [4] J. L. Doob, Stochastic Processes, Wiley Publ. Math. Statist., Wiley, New York, 1953. 
  5. [5] J. L. Doob, Semimartingales and subharmonic functions, Trans. Amer. Math. Soc. 77 (1954), 86-121. Zbl0059.12205
  6. [6] H. Helson, Conjugate series in several variables, Pacific J. Math. 9 (1959), 513-523. Zbl0088.05002
  7. [7] K. E. Petersen, Brownian Motion, Hardy Spaces and Bounded Mean Oscillation, London Math. Soc. Lecture Note Ser. 28, Cambridge Univ. Press, 1977. Zbl0363.60004
  8. [8] A. Zygmund, Trigonometric Series, 2nd ed., 2 vols., Cambridge Univ. Press, 1959. Zbl0085.05601
  9. [8] A. Zygmund, Trigonometric Series, 2nd ed., 2 vols., Cambridge Univ. Press, 1959. Zbl0085.05601

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