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

Ferenc Móricz

Studia Mathematica (1995)

  • Volume: 116, Issue: 1, page 89-100
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Móricz, Ferenc. "On the maximal Fejér operator for double Fourier series of functions in Hardy spaces." Studia Mathematica 116.1 (1995): 89-100. <http://eudml.org/doc/216222>.

@article{Móricz1995,
abstract = {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 $L^\{1\} log^\{+\} L(^2)$, $L^1(log^\{+\}L)^2(^2)$, 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.},
author = {Móricz, Ferenc},
journal = {Studia Mathematica},
keywords = {Fejér (or first arithmetic) means; double Fourier series; Hardy spaces; maximal Fejér operator; maximal conjugate Fejér operators},
language = {eng},
number = {1},
pages = {89-100},
title = {On the maximal Fejér operator for double Fourier series of functions in Hardy spaces},
url = {http://eudml.org/doc/216222},
volume = {116},
year = {1995},
}

TY - JOUR
AU - Móricz, Ferenc
TI - On the maximal Fejér operator for double Fourier series of functions in Hardy spaces
JO - Studia Mathematica
PY - 1995
VL - 116
IS - 1
SP - 89
EP - 100
AB - 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 $L^{1} log^{+} L(^2)$, $L^1(log^{+}L)^2(^2)$, 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.
LA - eng
KW - Fejér (or first arithmetic) means; double Fourier series; Hardy spaces; maximal Fejér operator; maximal conjugate Fejér operators
UR - http://eudml.org/doc/216222
ER -

References

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  1. [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, New York, 1988. Zbl0647.46057
  2. [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. 
  3. [3] A. M. Garsia, Martingale Inequalities, Benjamin, New York, 1973. 
  4. [4] D. V. Giang and F. Móricz, Hardy spaces on the plane and double Fourier transforms, J. Fourier Anal. Appl., submitted. Zbl1055.42503
  5. [5] B. Jessen, J. Marcinkiewicz and A. Zygmund, Note on the differentiability of multiple integrals, Fund. Math. 25 (1935), 217-234. Zbl61.0255.01
  6. [6] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, ibid. 32 (1939), 112-132. Zbl65.0266.01
  7. [7] F. Móricz, The maximal Fejér operator is bounded from H 1 ( ) into L 1 ( ) , Analysis, submitted. Zbl0927.47022
  8. [8] F. Móricz, F. Schipp and W. R. Wade, Cesàro summability of double Walsh-Fourier series, Trans. Amer. Math. Soc. 329 (1992), 131-140. Zbl0795.42016
  9. [9] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, 1959. Zbl0085.05601

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