# 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

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topAsmar, 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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- [2] A. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 157 (1971), 137-153.
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- [6] D. J. H. Garling, Hardy martingales and the unconditional convergence of martingales, Bull. London Math. Soc. 23 (1991), 190-192. Zbl0746.60046
- [7] H. Helson, Conjugate series and a theorem of Paley, Pacific J. Math. 8 (1958), 437-446. Zbl0117.29702
- [8] H. Helson, Conjugate series in several variables, ibid. 9 (1959), 513-523. Zbl0088.05002
- [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] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, 2nd ed., Grundlehren Math. Wiss. 115, Springer, 1979. Zbl0416.43001
- [11] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, 1971. Zbl0232.42007
- [12] A. Zygmund, Trigonometric Series, 2nd ed., 2 vols., Cambridge Univ. Press, 1959. Zbl0085.05601

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