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1994 | 110 | 2 | 127-148
Tytuł artykułu

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

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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.
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  • Department of Mathematics, Akita University, Tegata, Akita 010, Japan
Bibliografia
  • [1] R. A. DeVore and R. C. Sharpley, Maximal functions measuring smoothness, Mem. Amer. Math. Soc. 293 (1984).
  • [2] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799.
  • [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.
  • [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.
  • [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.
  • [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.
  • [13] C. Watari, Mean convergence of Walsh Fourier series, ibid. 16 (1964), 183-188.
  • [14] C. Watari and Y. Okuyama, Approximation property of functions and absolute convergence of Fourier series, Tôhoku Math. J. 27 (1975), 129-134.
  • [15] S. Yano, On approximation by Walsh functions, Proc. Amer. Math. Soc. 2 (1951), 962-967.
  • [16] S. Yano, Cesàro summation of Walsh Fourier series, Real Analysis Seminar 1991, 113-163 (in Japanese).
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Bibliografia
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bwmeta1.element.bwnjournal-article-smv110i2p127bwm
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