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

Studia Mathematica (1994)

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

## Access Full Article

top## Abstract

top## How to cite

topTateoka, 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

top- [1] R. A. DeVore and R. C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc. 293 (1984). Zbl0529.42005
- [2] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799. Zbl0551.46018
- [3] V. G. Maz'ya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston, 1985.
- [4] C. W. Onneweer and S. Weiyi, Homogeneous Besov spaces on locally compact Vilenkin groups, Studia Math. 93 (1989), 17-39.
- [5] F. Schipp, W. R. Wade and P. Simon, Walsh Series: An Introduction to Dyadic Harmonic Analysis, Hilger, Bristol, 1990. Zbl0727.42017
- [6] A. Seeger, A note on Triebel-Lizorkin spaces, in: Approximation and Function Spaces, Banach Center Publ. 22, PWN, 1989, 391-400.
- [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
- [8] È. A. Storoženko, V. G. Krotov and P. Oswald, Direct and converse theorems of Jackson type in ${L}^{p}$ spaces, 0<p<1, Math. USSR-Sb. 27 (1975), 355-374.
- [9] M. H. Taibleson, Fourier Analysis on Local Fields, Princeton Univ. Press, 1975.
- [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] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
- [12] C. Watari, Best approximation by Walsh polynomials, Tôhoku Math. J. 15 (1963), 1-5. Zbl0111.26502
- [13] C. Watari, Mean convergence of Walsh Fourier series, ibid. 16 (1964), 183-188. Zbl0146.08901
- [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] S. Yano, On approximation by Walsh functions, Proc. Amer. Math. Soc. 2 (1951), 962-967. Zbl0044.07102
- [16] S. Yano, Cesàro summation of Walsh Fourier series, Real Analysis Seminar 1991, 113-163 (in Japanese).

## NotesEmbed ?

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