On the characterization of Hardy-Besov spaces on the dyadic group and its applications

Jun Tateoka

Studia Mathematica (1994)

  • Volume: 110, Issue: 2, page 127-148
  • ISSN: 0039-3223

Abstract

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C. Watari [12] obtained a simple characterization of Lipschitz classes L i p ( p ) α ( W ) ( 1 ≥ p ≥ ∞ , α > 0 ) on the dyadic group using the L p -modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces B p , q α on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces B p , q α by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the type of the Maz’ya inequality, a weak type estimate for maximal Cesàro means and a sufficient condition of absolute convergence of Walsh-Fourier series.

How to cite

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Tateoka, Jun. "On the characterization of Hardy-Besov spaces on the dyadic group and its applications." Studia Mathematica 110.2 (1994): 127-148. <http://eudml.org/doc/216105>.

@article{Tateoka1994,
abstract = {C. Watari [12] obtained a simple characterization of Lipschitz classes $Lip^\{(p)\}α(W) (1 ≥ p ≥ ∞, α > 0)$ on the dyadic group using the $L^p$-modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces $B^α_\{p,q\}$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces $B^α_\{p,q\}$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the type of the Maz’ya inequality, a weak type estimate for maximal Cesàro means and a sufficient condition of absolute convergence of Walsh-Fourier series.},
author = {Tateoka, Jun},
journal = {Studia Mathematica},
keywords = {dyadic group; Besov spaces; Dirichlet kernels; best approximations; weak- type inequality; maximal Césaro means},
language = {eng},
number = {2},
pages = {127-148},
title = {On the characterization of Hardy-Besov spaces on the dyadic group and its applications},
url = {http://eudml.org/doc/216105},
volume = {110},
year = {1994},
}

TY - JOUR
AU - Tateoka, Jun
TI - On the characterization of Hardy-Besov spaces on the dyadic group and its applications
JO - Studia Mathematica
PY - 1994
VL - 110
IS - 2
SP - 127
EP - 148
AB - C. Watari [12] obtained a simple characterization of Lipschitz classes $Lip^{(p)}α(W) (1 ≥ p ≥ ∞, α > 0)$ on the dyadic group using the $L^p$-modulus of continuity and the best approximation by Walsh polynomials. Onneweer and Weiyi [4] characterized homogeneous Besov spaces $B^α_{p,q}$ on locally compact Vilenkin groups, but there are still some gaps to be filled up. Our purpose is to give the characterization of Besov spaces $B^α_{p,q}$ by oscillations, atoms and others on the dyadic groups. As applications, we show a strong capacity inequality of the type of the Maz’ya inequality, a weak type estimate for maximal Cesàro means and a sufficient condition of absolute convergence of Walsh-Fourier series.
LA - eng
KW - dyadic group; Besov spaces; Dirichlet kernels; best approximations; weak- type inequality; maximal Césaro means
UR - http://eudml.org/doc/216105
ER -

References

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  1. [1] R. A. DeVore and R. C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc. 293 (1984). Zbl0529.42005
  2. [2] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799. Zbl0551.46018
  3. [3] V. G. Maz'ya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston, 1985. 
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  7. [7] E. M. Stein, M. H. Taibleson and G. Weiss, Weak type estimates for maximal operators on certain H p classes, Rend. Circ. Mat. Palermo Suppl. 1 (1981), 81-97. Zbl0503.42018
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  9. [9] M. H. Taibleson, Fourier Analysis on Local Fields, Princeton Univ. Press, 1975. 
  10. [10] J. Tateoka, The modulus of continuity and the best approximation over the dyadic group, Acta Math. Hungar. 59 (1992), 115-120. Zbl0774.41026
  11. [11] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983. 
  12. [12] C. Watari, Best approximation by Walsh polynomials, Tôhoku Math. J. 15 (1963), 1-5. Zbl0111.26502
  13. [13] C. Watari, Mean convergence of Walsh Fourier series, ibid. 16 (1964), 183-188. Zbl0146.08901
  14. [14] C. Watari and Y. Okuyama, Approximation property of functions and absolute convergence of Fourier series, Tôhoku Math. J. 27 (1975), 129-134. Zbl0313.42011
  15. [15] S. Yano, On approximation by Walsh functions, Proc. Amer. Math. Soc. 2 (1951), 962-967. Zbl0044.07102
  16. [16] S. Yano, Cesàro summation of Walsh Fourier series, Real Analysis Seminar 1991, 113-163 (in Japanese). 

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