Riesz means of Fourier transforms and Fourier series on Hardy spaces

Ferenc Weisz

Studia Mathematica (1998)

  • Volume: 131, Issue: 3, page 253-270
  • ISSN: 0039-3223

Abstract

top
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.

How to cite

top

Weisz, Ferenc. "Riesz means of Fourier transforms and Fourier series on Hardy spaces." Studia Mathematica 131.3 (1998): 253-270. <http://eudml.org/doc/216579>.

@article{Weisz1998,
abstract = {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.},
author = {Weisz, Ferenc},
journal = {Studia Mathematica},
keywords = {Hardy spaces; p-atom; atomic decomposition; interpolation; Fourier transforms; Riesz means; atom; Fourier transform},
language = {eng},
number = {3},
pages = {253-270},
title = {Riesz means of Fourier transforms and Fourier series on Hardy spaces},
url = {http://eudml.org/doc/216579},
volume = {131},
year = {1998},
}

TY - JOUR
AU - Weisz, Ferenc
TI - Riesz means of Fourier transforms and Fourier series on Hardy spaces
JO - Studia Mathematica
PY - 1998
VL - 131
IS - 3
SP - 253
EP - 270
AB - 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.
LA - eng
KW - Hardy spaces; p-atom; atomic decomposition; interpolation; Fourier transforms; Riesz means; atom; Fourier transform
UR - http://eudml.org/doc/216579
ER -

References

top
  1. [1] C. Bennett and R. Sharpley, Interpolation of Operators, Pure Appl. Math. 129, Academic Press, New York, 1988. Zbl0647.46057
  2. [2] J. Bergh and J. Löfström, Interpolation Spaces. An Introduction, Springer, Berlin, 1976. Zbl0344.46071
  3. [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. Zbl0223.30048
  4. [4] P. L. Butzer and R. J. Nessel, Fourier Analysis and Approximation, Birkhäuser, Basel, 1971. 
  5. [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. Zbl0285.41006
  6. [6] C. Fefferman and E. M. Stein, H p spaces of several variables, Acta Math. 129 (1972), 137-193. Zbl0257.46078
  7. [7] B. S. Kashin and A. A. Saakjan, Orthogonal Series, Transl. Math. Monographs 75, Amer. Math. Soc., 1989. 
  8. [8] J. Marcinkiewicz and A. Zygmund, On the summability of double Fourier series, Fund. Math. 32 (1939), 122-132. Zbl65.0266.01
  9. [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. Zbl0907.47022
  10. [10] N. M. Rivière and Y. Sagher, Interpolation between L ∞ and H 1 , the real method, J. Funct. Anal. 14 (1973), 401-409. Zbl0295.46056
  11. [11] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, N.J., 1970. Zbl0207.13501
  12. [12] E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, N.J., 1971. Zbl0232.42007
  13. [13] F. Weisz, Cesàro summability of one- and two-dimensional trigonometric-Fourier series, Colloq. Math. 74 (1997), 123-133. Zbl0891.42006
  14. [14] F. Weisz, Martingale Hardy Spaces and Their Applications in Fourier-Analysis, Lecture Notes in Math. 1568, Springer, Berlin, 1994. Zbl0796.60049
  15. [15] F. Weisz, The maximal Fejér operator of Fourier transforms on Hardy spaces, Acta Sci. Math. (Szeged), to appear. Zbl0948.42003
  16. [16] N. Wiener, The Fourier Integral and Certain of Its Applications, Dover, New York, 1959. Zbl0081.32102
  17. [17] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, London, 1959. Zbl0085.05601

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.