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## Colloquium Mathematicum

1999 | 82 | 2 | 155-166
Tytuł artykułu

### Fejér means of two-dimensional Fourier transforms on $H_p(ℝ × ℝ)$

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The two-dimensional classical Hardy spaces $H_p(ℝ × ℝ)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_p(ℝ × ℝ)$ to $L_p(ℝ^2)$ (1/2 < p ≤ ∞) and is of weak type $(H^{#}_1 (ℝ × ℝ), L_1(ℝ^2))$ where the Hardy space $H^#_1(ℝ × ℝ)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_1^#(ℝ × ℝ)$ ⊃ $LlogL(ℝ^2)$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_p(ℝ × ℝ)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_p(ℝ × ℝ)$, the Fejér means converge to f in $H_p(ℝ × ℝ)$ norm (1/2 < p < ∞). The same results are proved for the conjugate Fejér means.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
155-166
Opis fizyczny
Daty
wydano
1999
otrzymano
1998-06-17
Twórcy
autor
• Department of Numerical Analysis, Eötvös L. University, Pázmány P. sétány 1/D, H-1117 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] 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.
• [4] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645.
• [5] P. Duren, Theory of $H^p$ Spaces, Academic Press, New York, 1970.
• [6] R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 2, Springer, Berlin, 1982.
• [7] C. Fefferman and E. M. Stein, $H^p$ spaces of several variables, Acta Math. 129 (1972), 137-194.
• [8] R. Fefferman, Calderón-Zygmund theory for product domains: $H^p$ spaces, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 840-843.
• [9] A. P. Frazier, The dual space of $H^p$ of the polydisc for 0<p<1, Duke Math. J. 39 (1972), 369-379.
• [10] R. F. Gundy, Maximal function characterization of $H^p$ for the bidisc, in: Lecture Notes in Math. 781, Springer, Berlin, 1982, 51-58.
• [11] 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.
• [12] K.-C. Lin, Interpolation between Hardy spaces on the bidisc, Studia Math. 84 (1986), 89-96.
• [13] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132.
• [14] F. Móricz, The maximal Fejér operator for Fourier transforms of functions in Hardy spaces, Acta Sci. Math. (Szeged) 62 (1996), 537-555.
• [15] F. Weisz, Cesàro summability of one- and two-dimensional trigonometric-Fourier series, Colloq. Math. 74 (1997), 123-133.
• [16] F. Weisz, Cesàro summability of two-parameter trigonometric-Fourier series, J. Approx. Theory 90 (1997), 30-45.
• [17] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994.
• [18] F. Weisz, Strong summability of two-dimensional trigonometric-Fourier series, Ann. Univ. Sci. Budapest Sect. Comput. 16 (1996), 391-406.
• [19] F. Weisz, The maximal Fejér operator of Fourier transforms, Acta Sci. Math. (Szeged) 64 (1998), 515-525.
• [20] N. Wiener, The Fourier Integral and Certain of its Applications, Dover, New York, 1959.
• [21] J. M. Wilson, On the atomic decomposition for Hardy spaces, Pacific J. Math. 116 (1985), 201-207.
• [22] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959.
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Bibliografia
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