On the maximal Fejér operator for double Fourier series of functions in Hardy spaces

Móricz, Ferenc

Móricz, Ferenc

Studia Mathematica, Tome 113 (1995), p. 89-100 / Harvested from The Polish Digital Mathematics Library

Résumé

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1, 0)} (^{2})$ , $H^{(0, 1)} (^{2})$ , or $H^{(1, 1)} (^{2})$ . We prove that the maximal Fejér operator is bounded from $H^{(1, 0)} (^{2})$ or $H^{(0, 1)} (^{2})$ into weak- $L^{1} (^{2})$ , and also bounded from $H^{(1, 1)} (^{2})$ into $L^{1} (^{2})$ . These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1} l o g^{+} L (^{2})$ , $L^{1} {(l o g^{+} L)}^{2} (^{2})$ , and $L^{μ} (^{2})$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures.

Publié le : 1995-01-01

Zbl 0842.47020

EUDML-ID : urn:eudml:doc:216222

@article{bwmeta1.element.bwnjournal-article-smv116i1p89bwm,
     author = {Ferenc M\'oricz},
     title = {On the maximal Fej\'er operator for double Fourier series of functions in Hardy spaces},
     journal = {Studia Mathematica},
     volume = {113},
     year = {1995},
     pages = {89-100},
     zbl = {0842.47020},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv116i1p89bwm}
}

Móricz, Ferenc. On the maximal Fejér operator for double Fourier series of functions in Hardy spaces. Studia Mathematica, Tome 113 (1995) pp. 89-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv116i1p89bwm/

Bibliographie

[00000] [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988. | Zbl 0647.46057

[00001] [2] R. Fefferman, Some recent developments in Fourier analysis and $H^{p}$ theory on product domains, II, in: Function Spaces and Applications, Proc. Conf. Lund 1986, Lecture Notes in Math. 1302, Springer, Berlin, 1988, 44-51.

[00002] [3] A. M. Garsia, Martingale Inequalities, Benjamin, New York, 1973.

[00003] [4] D. V. Giang and F. Móricz, Hardy spaces on the plane and double Fourier transforms, J. Fourier Anal. Appl., submitted. | Zbl 1055.42503

[00004] [5] B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217-234. | Zbl 61.0255.01

[00005] [6] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, ibid. 32 (1939), 112-132. | Zbl 65.0266.01

[00006] [7] F. Móricz, The maximal Fejér operator is bounded from $H^{1} ()$ into $L^{1} ()$ , Analysis, submitted. | Zbl 0927.47022

[00007] [8] F. Móricz, F. Schipp and W. R. Wade, Cesàro summability of double Walsh-Fourier series, Trans. Amer. Math. Soc. 329 (1992), 131-140. | Zbl 0795.42016

[00008] [9] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1959. | Zbl 0085.05601