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## Colloquium Mathematicum

1996 | 70 | 2 | 235-252
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
235-252
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-07-13
Twórcy
autor
• Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, U.S.A.
• Department of Mathematics, University of Missouri-Columbia, Columbia, Missouri 65211, U.S.A.
Bibliografia
• [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.
• [2] A. Calderón, Ergodic theory and translation-invariant operators, Proc. Nat. Acad. Sci. U.S.A. 157 (1971), 137-153.
• [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.
• [4] M. Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105-167.
• [5] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, 1960.
• [6] D. J. H. Garling, Hardy martingales and the unconditional convergence of martingales, Bull. London Math. Soc. 23 (1991), 190-192.
• [7] H. Helson, Conjugate series and a theorem of Paley, Pacific J. Math. 8 (1958), 437-446.
• [8] H. Helson, Conjugate series in several variables, ibid. 9 (1959), 513-523.
• [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.
• [10] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis I, 2nd ed., Grundlehren Math. Wiss. 115, Springer, 1979.
• [11] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton Univ. Press, 1971.
• [12] A. Zygmund, Trigonometric Series, 2nd ed., 2 vols., Cambridge Univ. Press, 1959.
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Bibliografia
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