Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals

Nakhlé Asmar; Stephen Montgomery-Smith

Colloquium Mathematicae (1996)

  • Volume: 70, Issue: 2, page 235-252
  • ISSN: 0010-1354

Abstract

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Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on L p ( G ) , 1 ≤ p < ∞, as was done by Helson [7]. We will show how to use Hahn’s Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in L p ( G ) . This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series.

How to cite

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Asmar, Nakhlé, and Montgomery-Smith, Stephen. "Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals." Colloquium Mathematicae 70.2 (1996): 235-252. <http://eudml.org/doc/210409>.

@article{Asmar1996,
abstract = {Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on $L^p(G)$, 1 ≤ p < ∞, as was done by Helson [7]. We will show how to use Hahn’s Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in $L^p(G)$. This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series.},
author = {Asmar, Nakhlé, Montgomery-Smith, Stephen},
journal = {Colloquium Mathematicae},
keywords = {lexicographic order; martingale theory; Hahn's embedding theorem; probability theory; conjugate functions; measurable order; dual group},
language = {eng},
number = {2},
pages = {235-252},
title = {Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals},
url = {http://eudml.org/doc/210409},
volume = {70},
year = {1996},
}

TY - JOUR
AU - Asmar, Nakhlé
AU - Montgomery-Smith, Stephen
TI - Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals
JO - Colloquium Mathematicae
PY - 1996
VL - 70
IS - 2
SP - 235
EP - 252
AB - Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on $L^p(G)$, 1 ≤ p < ∞, as was done by Helson [7]. We will show how to use Hahn’s Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups with ordered duals. We illustrate our methods by describing a concrete way to construct the conjugate function in $L^p(G)$. This construction is in terms of an unconditionally convergent conjugate series whose individual terms are constructed from specific ergodic Hilbert transforms. We also present a study of the square function associated with the conjugate series.
LA - eng
KW - lexicographic order; martingale theory; Hahn's embedding theorem; probability theory; conjugate functions; measurable order; dual group
UR - http://eudml.org/doc/210409
ER -

References

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  1. [1] N. Asmar and E. Hewitt, Marcel Riesz's Theorem on conjugate Fourier series and its descendants, in: Proc. Analysis Conf., Singapore, 1986, S. T. Choy et al. (eds.), Elsevier, New York, 1988, 1-56. Zbl0646.42011
  2. [2] A. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 157 (1971), 137-153. 
  3. [3] R. R. Coifman and G. Weiss, Transference Methods in Analysis, CBMS Regional Conf. Ser. in Math. 31, Amer. Math. Soc., Providence, R.I., 1977. Zbl0371.43009
  4. [4] M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105-167. 
  5. [5] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, 1960. Zbl0137.02001
  6. [6] D. J. H. Garling, Hardy martingales and the unconditional convergence of martingales, Bull. London Math. Soc. 23 (1991), 190-192. Zbl0746.60046
  7. [7] H. Helson, Conjugate series and a theorem of Paley, Pacific J. Math. 8 (1958), 437-446. Zbl0117.29702
  8. [8] H. Helson, Conjugate series in several variables, ibid. 9 (1959), 513-523. Zbl0088.05002
  9. [9] E. Hewitt and S. Koshi, Orderings in locally compact Abelian groups and the the- orem of F. and M. Riesz, Math. Proc. Cambridge Philos. Soc. 93 (1983), 441-457. Zbl0528.43002
  10. [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, 2nd ed., Grundlehren Math. Wiss. 115, Springer, 1979. Zbl0416.43001
  11. [11] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, 1971. Zbl0232.42007
  12. [12] A. Zygmund, Trigonometric Series, 2nd ed., 2 vols., Cambridge Univ. Press, 1959. Zbl0085.05601

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