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1998 | 131 | 3 | 253-270
Tytuł artykułu

Riesz means of Fourier transforms and Fourier series on Hardy spaces

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EN
Abstrakty
EN
Elementary estimates for the Riesz kernel and for its derivative are given. Using these we show that the maximal operator of the Riesz means of a tempered distribution is bounded from $H_p(ℝ)$ to $L_p(ℝ)$ (1/(α+1) < p < ∞) and is of weak type (1,1), where $H_p(ℝ)$ is the classical Hardy space. As a consequence we deduce that the Riesz means of a function $⨍ ∈ L_1(ℝ)$ converge a.e. to ⨍. Moreover, we prove that the Riesz means are uniformly bounded on $H_p(ℝ)$ whenever 1/(α+1) < p < ∞. Thus, in case $⨍ ∈ H_p(ℝ)$, the Riesz means converge to ⨍ in $H_p(ℝ)$ norm (1/(α+1) < p < ∞). The same results are proved for the conjugate Riesz means and for Fourier series of distributions.
Twórcy
autor
  • Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary, weisz@ludens.elte.hu
Bibliografia
  • [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, New York, 1988.
  • [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976.
  • [3] 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.
  • [4] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser, Basel, 1971.
  • [5] C. Fefferman, N. M. Rivière and Y. Sagher, Interpolation between $H^p$ spaces: the real method, Trans. Amer. Math. Soc. 191 (1974), 75-81.
  • [6] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-193.
  • [7] B. S. Kashin and A. A. Saakjan, Orthogonal Series, Transl. Math. Monographs 75, Amer. Math. Soc., 1989.
  • [8] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132.
  • [9] F. Móricz, The maximal Fejér operator on the spaces $H^1$ and $L^1$, in: Approximation Theory and Function Series (Budapest, 1995), Bolyai Soc. Math. Stud. 5, Budapest, 1996, 275-292.
  • [10] N. M. Rivière and Y. Sagher, Interpolation between $L^∞$ and $H^1$, the real method, J. Funct. Anal. 14 (1973), 401-409.
  • [11] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970.
  • [12] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971.
  • [13] F. Weisz, Cesàro summability of one- and two-dimensional trigonometric-Fourier series, Colloq. Math. 74 (1997), 123-133.
  • [14] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
  • [15] F. Weisz, The maximal Fejér operator of Fourier transforms on Hardy spaces, Acta Sci. Math. (Szeged), to appear.
  • [16] N. Wiener, The Fourier Integral and Certain of Its Applications, Dover, New York, 1959.
  • [17] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv131i3p253bwm
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