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 , 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 . 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.
@article{bwmeta1.element.bwnjournal-article-cmv70i2p235bwm,
author = {Nakhl\'e Asmar and Stephen Montgomery-Smith},
title = {Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals},
journal = {Colloquium Mathematicae},
volume = {70},
year = {1996},
pages = {235-252},
zbl = {0855.43001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv70i2p235bwm}
}
Asmar, Nakhlé; Montgomery-Smith, Stephen. Hahn's Embedding Theorem for orders and harmonic analysis on groups with ordered duals. Colloquium Mathematicae, Tome 70 (1996) pp. 235-252. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv70i2p235bwm/
[000] [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. | Zbl 0646.42011
[001] [2] A. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 157 (1971), 137-153.
[002] [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. | Zbl 0371.43009
[003] [4] M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105-167.
[004] [5] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, 1960. | Zbl 0137.02001
[005] [6] D. J. H. Garling, Hardy martingales and the unconditional convergence of martingales, Bull. London Math. Soc. 23 (1991), 190-192. | Zbl 0746.60046
[006] [7] H. Helson, Conjugate series and a theorem of Paley, Pacific J. Math. 8 (1958), 437-446. | Zbl 0117.29702
[007] [8] H. Helson, Conjugate series in several variables, ibid. 9 (1959), 513-523. | Zbl 0088.05002
[008] [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. | Zbl 0528.43002
[009] [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, 2nd ed., Grundlehren Math. Wiss. 115, Springer, 1979. | Zbl 0416.43001
[010] [11] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, 1971. | Zbl 0232.42007
[011] [12] A. Zygmund, Trigonometric Series, 2nd ed., 2 vols., Cambridge Univ. Press, 1959. | Zbl 0085.05601