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## Studia Mathematica

1998 | 131 | 3 | 253-270
Tytuł artykułu

### Riesz means of Fourier transforms and Fourier series on Hardy spaces

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
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.
Słowa kluczowe
EN
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
253-270
Opis fizyczny
Daty
wydano
1998-02-05
otrzymano
1997-09-17
Twórcy
autor
• Department of Numerical Analysis, Eötvös L. University, Múzeum krt. 6-8, H-1088 Budapest, Hungary
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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