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## Studia Mathematica

1995 | 113 | 3 | 199-210
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### Denseness of the spaces $Φ_V$ of Lizorkin type in the mixed $L^{p̅}(ℝ^n)$-spaces

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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,$is proved for so-called quasibroken sets V. Słowa kluczowe Kategorie tematyczne Czasopismo Rocznik Tom Numer Strony 199-210 Opis fizyczny Daty wydano 1995 otrzymano 1991-07-26 poprawiono 1994-09-07 Twórcy autor • Department of Mathematics and Mechanics, Rostov State University, Bol'shaya Sadovaya 105, 344711 Rostov-na-donu, Russia Bibliografia • [1] A. Benedek and R. Panzone, The spaces$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.
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