PL EN


Preferencje help
Widoczny [Schowaj] Abstrakt
Liczba wyników
Czasopismo
1995-1996 | 117 | 2 | 173-194
Tytuł artykułu

Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The martingale Hardy space $H_p([0,1)^2)$ and the classical Hardy space $H_p(𝕋^2)$ are introduced. We prove that certain means of the partial sums of the two-parameter Walsh-Fourier and trigonometric-Fourier series are uniformly bounded operators from $H_p$ to $L_p$ (0 < p ≤ 1). As a consequence we obtain strong convergence theorems for the partial sums. The classical Hardy-Littlewood inequality is extended to two-parameter Walsh-Fourier and trigonometric-Fourier coefficients. The dual inequalities are also verified and a Marcinkiewicz-Zygmund type inequality is obtained for BMO spaces.
Twórcy
autor
  • Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary, weisz@ludens.elte.hu
Bibliografia
  • [1] D. L. Burkholder, R. F. Gundy and M. L. Silverstein, A maximal function characterization of the class $H^p$, Trans. Amer. Math. Soc. 157 (1971), 137-153.
  • [2] R. R. Coifman, A real variable characterization of $H^p$, Studia Math. 51 (1974), 269-274.
  • [3] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
  • [4] R. E. Edwards, Fourier Series, A Modern Introduction, Vol. 1, Springer, Berlin, 1982.
  • [5] R. E. Edwards, Fourier Series, A Modern Introduction, Vol. 2, Springer, Berlin, 1982.
  • [6] C. Fefferman, N. M. Rivière, and Y. Sagher, Interpolation between $H^p$ spaces: the real method, Trans. Amer. Math. Soc. 191, (1974), 75-81.
  • [7] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-194.
  • [8] N. J. Fine, On the Walsh functions, Trans. Amer. Math. Soc. 65 (1949), 372-414.
  • [9] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, Math. Stud. 116, North-Holland, Amsterdam, 1985.
  • [10] A. M. Garsia, Martingale Inequalities, Seminar Notes on Recent Progress, Math. Lecture Notes Ser., Benjamin, New York, 1973.
  • [11] G. Gát, Investigations of certain operators with respect to the Vilenkin system, Acta Math. Hungar. 61 (1993), 131-149.
  • [12] G. H. Hardy and J. E. Littlewood, Some new properties of Fourier constants, J. London Math. Soc. 6 (1931), 3-9.
  • [13] N. R. Ladhawala, Absolute summability of Walsh-Fourier series, Pacific J. Math. 65 (1976), 103-108.
  • [14] R. H. Latter, A characterization of $H^p(R^n)$ in terms of atoms, Studia Math. 62 (1978), 92-101.
  • [15] J. Neveu, Discrete-Parameter Martingales, North-Holland, 1971.
  • [16] F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Adam Hilger, Bristol, 1990.
  • [17] P. Simon, Strong convergence of certain means with respect to the Walsh-Fourier series, Acta Math. Hungar. 49 (1987), 425-431.
  • [18] W. T. Sledd and D. A. Stegenga, An $H^1$ multiplier theorem, Ark. Mat. 19 (1981), 265-270.
  • [19] B. Smith, A strong convergence theorem for $H^1(𝕋)$, in: Lecture Notes in Math. 995, Springer, Berlin, 1994, 169-173.
  • [20] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
  • [21] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
  • [22] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1986.
  • [23] F. Weisz, Cesàro summability of two-dimensional Walsh-Fourier series, Trans. Amer. Math. Soc., to appear.
  • [24] F. Weisz, Inequalities relative to two-parameter Vilenkin-Fourier coefficients, Studia Math. 99 (1991), 221-233.
  • [25] F. Weisz, Martingale Hardy Spaces and their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [26] F. Weisz, Martingale Hardy spaces for 0 < p ≤ 1, Probab. Theory Related Fields 84 (1990), 361-376.
  • [27] F. Weisz, Strong summability of two-dimensional Walsh-Fourier series, Acta Sci. Math. (Szeged) 60 (1995), 779-803.
  • [28] J. M. Wilson, A simple proof of the atomic decomposition for $H^p(R^n)$, 0 < p ≤ 1, Studia Math. 74 (1982), 25-33.
  • [29] J. M. Wilson, On the atomic decomposition for Hardy spaces, Pacific J. Math. 116 (1985), 201-207.
  • [30] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-smv117i2p173bwm
JavaScript jest wyłączony w Twojej przeglądarce internetowej. Włącz go, a następnie odśwież stronę, aby móc w pełni z niej korzystać.