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1997 | 125 | 3 | 231-246
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Hardy type inequalities for two-parameter Vilenkin-Fourier coefficients

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EN
Abstrakty
EN
Our main result is a Hardy type inequality with respect to the two-parameter Vilenkin system (*) $(∑_{k=1}^∞ ∑_{j=1}^∞ |f̂(k,j)|^{p}(kj)^{p-2})^{1/p} ≤ C_p∥f∥_{H^p_{**}}$ (1/2 < p≤2) where f belongs to the Hardy space $H_{**}^p (G_m × G_s)$ defined by means of a maximal function. This inequality is extended to p > 2 if the Vilenkin-Fourier coefficients of f form a monotone sequence. We show that the converse of (*) also holds for all p > 0 under the monotonicity assumption.
Twórcy
autor
  • Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary, simon@ludens.elte.hu
autor
  • Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary, weisz@ludens.elte.hu
Bibliografia
  • [1] J. Brossard, Comparaison des "normes" $L^p$ du processus croissant et de la variable maximale pour les martingales régulières à deux indices. Théorème local correspondant, Ann. Probab. 8 (1980), 1183-1188.
  • [2] J. Brossard, Régularité des martingales à deux indices et inégalités de normes, in: Processus Aléatoires à Deux Indices, Lecture Notes in Math. 863, Springer, Berlin, 1981, 91-121.
  • [3] D. L. Burkholder, Distribution function inequalities for martingales, Ann. Probab. 1 (1973), 19-42.
  • [4] R. Cairoli, Une inégalité pour martingales à indices multiples et ses applications, in: Séminaire de Probabilités V, Lecture Notes in Math. 124, Springer, Berlin, (1970), 1-27.
  • [5] J. A. Chao, Hardy spaces on regular martingales, in: Martingale Theory in Harmonic Analysis and Banach Spaces, Lecture Notes in Math. 939, Springer, Berlin, 1982, 18-28.
  • [6] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
  • [7] S. Fridli and P. Simon, On the Dirichlet kernels and a Hardy space with respect to the Vilenkin system, Acta Math. Hungar. 45 (1985), 223-234.
  • [8] A. M. Garsia, Martingale Inequalities. Seminar Notes on Recent Progress, Math. Lecture Notes Series, Benjamin, New York, 1973.
  • [9] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), 3-9.
  • [10] N. R. Ladhawala, Absolute summability of Walsh-Fourier series, Pacific J. Math. 65 (1976), 103-108.
  • [11] C. Métraux, Quelques inégalités pour martingales à paramètre bidimensionnel, in: Séminaire de Probabilités XII, Lecture Notes in Math. 649, Springer, Berlin, 1978, 170-179.
  • [12] F. Móricz, On double cosine, sine and Walsh series with monotone coefficients, Proc. Amer. Math. Soc. 109 (1990), 417-425.
  • [13] F. Móricz, On Walsh series with coefficients tending monotonically to zero, Acta Math. Acad. Sci. Hungar. 38 (1981), 183-189.
  • [14] F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990.
  • [15] P. Simon, Investigations with respect to the Vilenkin system, Ann. Univ. Sci. Budapest Eötvös Sect. Math. 28 (1985), 87-101.
  • [16] P. Simon and F. Weisz, Hardy-Littlewood type inequalities for Vilenkin-Fourier coefficients, Anal. Math., to appear.
  • [17] N. Y. Vilenkin, On a class of complete orthonormal systems, Izv. Akad. Nauk SSSR Ser. Mat. 11 (1947), 363-400 (in Russian); English transl.: Amer. Math. Soc. Transl. 28 (1963), 1-35.
  • [18] F. Weisz, Hardy spaces and Cesàro means of two-dimensional Fourier series, in: Approximation Theory and Function Series (Budapest, 1995), Bolyai Soc. Math. Stud. 5, Budapest, 1996, 353-367.
  • [19] F. Weisz, Inequalities relative to two-parameter Vilenkin-Fourier coefficients, Studia Math. 99 (1991), 221-233.
  • [20] F. Weisz, Martingale Hardy Spaces and their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [21] F. Weisz, Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series, Studia Math. 117 (1996), 173-194.
  • [22] F. Weisz, Two-parameter Hardy-Littlewood inequalities, ibid. 118 (1996), 175-184.
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Bibliografia
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