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1995 | 113 | 3 | 199-210
Tytuł artykułu

Denseness of the spaces $Φ_V$ of Lizorkin type in the mixed $L^{p̅}(ℝ^n)$-spaces

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
The spaces Φ_V(ℝ^{n}) are defined to consist of Schwartz test functions φ such that the Fourier transform φ̂ and all its derivatives vanish on a given closed set V ⊂ ℝ^{n}. Under the only assumption that m(V) = 0 it is shown that Φ_V is dense in $C_0(ℝ^{n})$ and in the space $L^{p̅}(ℝ^n)$ with the mixed norm, for $1/p̅$ in a certain pyramid. The result on the denseness for arbitrary $p̅ = (p_1,..., p_n)$, 1 < p_k < ∞, k = 1,...,n,$ is proved for so-called quasibroken sets V.
Słowa kluczowe
Czasopismo
Rocznik
Tom
113
Numer
3
Strony
199-210
Opis fizyczny
Daty
wydano
1995
otrzymano
1991-07-26
poprawiono
1994-09-07
Twórcy
autor
  • Department of Mathematics and Mechanics, Rostov State University, Bol'shaya Sadovaya 105, 344711 Rostov-na-donu, Russia
Bibliografia
  • [1] A. Benedek and R. Panzone, The spaces $L^P$, with mixed norm, Duke Math. J. 28 (1961), 302-324.
  • [2] O. V. Besov, V. P. Il'in and S. M. Nikol'skiĭ, Integral Representations of Functions and Embedding Theorems, Nauka, Moscow, 1975 (in Russian); English transl.: Scripta Series in Math., Winston and Halsted Press, 1979.
  • [3] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds, Acta Math. 113 (1965), 153-180.
  • [4] S. Helgason, The Radon Transform, Birkhäuser, Boston, 1980.
  • [5] P. I. Lizorkin, Generalized Liouville differentiation and the function spaces $L_p^r(E_n)$. Embedding theorems, Mat. Sb. 60 (1963), 325-353 (in Russian).
  • [6] P. I. Lizorkin, Generalized Liouville differentiation and multipliers method in the embedding theory of spaces of differentiable functions, Trudy Mat. Inst. Steklov. 105 (1969), 89-167 (in Russian).
  • [7] B. Muckenhoupt, R. L. Wheeden and W.-S. Young, $L^2$ multipliers with power weights, Adv. in Math. 49 (1983), 170-216.
  • [8] S. G. Samko, The spaces $L_p,r^α(ℝ^n)$ and hypersingular integrals, Studia Math. 61 (1977), 193-230 (in Russian).
  • [9] S. G. Samko, On test functions vanishing on a given set and on division by functions, Mat. Zametki 21 (1977), 677-689 (in Russian).
  • [10] S. G. Samko, On denseness of the Lizorkin type spaces $Φ_V$ in $L_p(ℝ^n)$, ibid. 31 (1982), 855-865 (in Russian).
  • [11] S. G. Samko, Hypersingular Integrals and their Applications, Rostov University Publ. House, Rostov-on-Don, 1984 (in Russian).
  • [12] S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives and Some of their Applications, Nauka i Tekhnika, Minsk, 1987 (in Russian); English transl.: Gordon and Breach Sci. Publ., 1993.
  • [13] E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, N.J., 1970.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv113i3p199bwm
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