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

Weisz, Ferenc

Fejér means of two-dimensional Fourier transforms on

H_{p} (ℝ \times ℝ)

Weisz, Ferenc

Colloquium Mathematicae, Tome 79 (1999), p. 155-166 / Harvested from The Polish Digital Mathematics Library

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Résumé

The two-dimensional classical Hardy spaces $H_{p} (ℝ \times ℝ)$ are introduced and it is shown that the maximal operator of the Fejér means of a tempered distribution is bounded from $H_{p} (ℝ \times ℝ)$ to $L_{p} (ℝ^{2})$ (1/2 < p ≤ ∞) and is of weak type $(H_{1}^{} (ℝ \times ℝ), L_{1} (ℝ^{2}))$ where the Hardy space $H_{1} (ℝ \times ℝ)$ is defined by the hybrid maximal function. As a consequence we deduce that the Fejér means of a function f ∈ $H_{1}^{(} ℝ \times ℝ)$ ⊃ $L l o g L (ℝ^{2})$ converge to f a.e. Moreover, we prove that the Fejér means are uniformly bounded on $H_{p} (ℝ \times ℝ)$ whenever 1/2 < p < ∞. Thus, in case f ∈ $H_{p} (ℝ \times ℝ)$ , the Fejér means converge to f in $H_{p} (ℝ \times ℝ)$ norm (1/2 < p < ∞). The same results are proved for the conjugate Fejér means.

Publié le : 1999-01-01

Zbl 0949.42011

EUDML-ID : urn:eudml:doc:210754

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     author = {Ferenc Weisz},
     title = {Fej\'er means of two-dimensional Fourier transforms on $H\_p($\mathbb{R}$ $\times$ $\mathbb{R}$)$
            },
     journal = {Colloquium Mathematicae},
     volume = {79},
     year = {1999},
     pages = {155-166},
     zbl = {0949.42011},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-cmv82i2p155bwm}
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Weisz, Ferenc. Fejér means of two-dimensional Fourier transforms on $H_p(ℝ × ℝ)$
            . Colloquium Mathematicae, Tome 79 (1999) pp. 155-166. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-cmv82i2p155bwm/

Bibliographie

[000] [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, New York, 1988. | Zbl 0647.46057

[001] [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. | Zbl 0344.46071

[002] [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. | Zbl 0557.42007

[003] [4] R. R. Coifman and G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. | Zbl 0358.30023

[004] [5] P. Duren, Theory of $H^{p}$ Spaces, Academic Press, New York, 1970. | Zbl 0215.20203

[005] [6] R. E. Edwards, Fourier Series. A Modern Introduction, Vol. 2, Springer, Berlin, 1982.

[006] [7] C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), 137-194. | Zbl 0257.46078

[007] [8] R. Fefferman, Calderón-Zygmund theory for product domains: $H^{p}$ spaces, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), 840-843. | Zbl 0602.42023

[008] [9] A. P. Frazier, The dual space of $H^{p}$ of the polydisc for 0 | Zbl 0237.32005

[009] [10] R. F. Gundy, Maximal function characterization of $H^{p}$ for the bidisc, in: Lecture Notes in Math. 781, Springer, Berlin, 1982, 51-58.

[010] [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. | Zbl 0405.32002

[011] [12] K.-C. Lin, Interpolation between Hardy spaces on the bidisc, Studia Math. 84 (1986), 89-96. | Zbl 0626.46060

[012] [13] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132. | Zbl 65.0266.01

[013] [14] F. Móricz, The maximal Fejér operator for Fourier transforms of functions in Hardy spaces, Acta Sci. Math. (Szeged) 62 (1996), 537-555. | Zbl 0880.47018

[014] [15] F. Weisz, Cesàro summability of one- and two-dimensional trigonometric-Fourier series, Colloq. Math. 74 (1997), 123-133. | Zbl 0891.42006

[015] [16] F. Weisz, Cesàro summability of two-parameter trigonometric-Fourier series, J. Approx. Theory 90 (1997), 30-45. | Zbl 0878.42007

[016] [17] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994. | Zbl 0796.60049

[017] [18] F. Weisz, Strong summability of two-dimensional trigonometric-Fourier series, Ann. Univ. Sci. Budapest Sect. Comput. 16 (1996), 391-406. | Zbl 0891.42005

[018] [19] F. Weisz, The maximal Fejér operator of Fourier transforms, Acta Sci. Math. (Szeged) 64 (1998), 515-525. | Zbl 0948.42003

[019] [20] N. Wiener, The Fourier Integral and Certain of its Applications, Dover, New York, 1959. | Zbl 0081.32102

[020] [21] J. M. Wilson, On the atomic decomposition for Hardy spaces, Pacific J. Math. 116 (1985), 201-207. | Zbl 0563.42012

[021] [22] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959. | Zbl 0085.05601