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1996 | 118 | 2 | 175-184
Tytuł artykułu

Two-parameter Hardy-Littlewood inequalities

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The inequality (*) $(∑_{|n|=1}^{∞} ∑_{|m|=1}^{∞} |nm|^{p-2} |f̂(n,m)|^p)^{1/p} ≤ C_p ∥ƒ∥_{H_p}$ (0 < p ≤ 2) is proved for two-parameter trigonometric-Fourier coefficients and for the two-dimensional classical Hardy space $H_p$ on the bidisc. The inequality (*) is extended to each p if the Fourier coefficients are monotone. For monotone coefficients and for every p, the supremum of the partial sums of the Fourier series is in $L_p$ whenever the left hand side of (*) is finite. From this it follows that under the same condition the two-dimensional trigonometric-Fourier series of an arbitrary function from $H_1$ converges a.e. and also in $L_1$ norm to that function.
Czasopismo
Rocznik
Tom
118
Numer
2
Strony
175-184
Opis fizyczny
Daty
wydano
1996
otrzymano
1995-10-05
Twórcy
autor
  • Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary , weisz@ludens.elte.hu
Bibliografia
  • [1] S. -Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and $H^p$-theory on product domains, Bull. Amer. Math. Soc. 12 (1985), 1-43.
  • [2] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
  • [3] M. I. D'jachenko [M. I. D'yachenko], Multiple trigonometric series with lexicographically monotone coefficients, Anal. Math. 16 (1990), 173-190.
  • [4] M. I. D'jachenko [M. I. D'yachenko], On the convergence of double trigonometric series and Fourier series with monotone coefficients, Math. USSR-Sb. 57 (1987), 57-75.
  • [5] R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 2, Springer, Berlin, 1982.
  • [6] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-194.
  • [7] C. Fefferman and E. M. Stein, Calderón-Zygmund theory for product domains: $H^p$ spaces, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 840-843.
  • [8] R. F. Gundy, Inégalités pour martingales à un et deux indices: L'espace $H_p$, in: Ecole d'Eté de Probabilités de Saint-Flour VIII-1978, Lecture Notes in Math. 774, Springer, Berlin, 1980, 251-331.
  • [9] R. F. Gundy, Maximal function characterization of $H^p$ for the bidisc, in: Lecture Notes in Math. 781, Springer, Berlin, 1982, 51-58.
  • [10] R. F. Gundy and E. M. Stein, $H^p$ theory for the poly-disc, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 1026-1029.
  • [11] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), 3-9.
  • [12] B. Jawerth and A. Torchinsky, A note on real interpolation of Hardy spaces in the polydisk, Proc. Amer. Math. Soc. 96 (1986), 227-232.
  • [13] K.-C. Lin, Interpolation between Hardy spaces on the bidisc, Studia Math. 84 (1986), 89-96.
  • [14] F. Móricz, On double cosine, sine and Walsh series with monotone coefficients, Proc. Amer. Math. Soc. 109 (1990), 417-425.
  • [15] F. Móricz, On the maximum of the rectangular partial sums of double trigonometric series with non-negative coefficients, Anal. Math. 15 (1989), 283-290.
  • [16] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.
  • [17] F. Weisz, Inequalities relative to two-parameter Vilenkin-Fourier coefficients, Studia Math. 99 (1991), 221-233.
  • [18] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [19] F. Weisz, Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series, Studia Math. 117 (1996), 173-194.
  • [20] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.
Typ dokumentu
Bibliografia
Identyfikatory
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bwmeta1.element.bwnjournal-article-smv118i2p175bwm
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