Interpolation properties of anisotropic Bq pr (Td) Nikol’skii-Besov spaces and embedding theorems
Keywords:
anisotropic Nikol’skii-Besov spaces, anisotropic Lorentz spaces, embedding, retract, inequality of different metricsAbstract
In this paper we study interpolation properties of anisotropic spaces of Nikol’skii-Besov with respect to the anisotropic interpolation A θq , where 0 < θ = (θ 1,..., θ n) < 1, 1 ≤ q = (q 1,..., q n) ≤ ∞. It shows that in the case −∞ < α = (α 1,..., α n) < ∞, 1 < p = (p 1,..., p n) < ∞ and 1 ≤ q = (q 1,..., q n), r = (r 1,..., r n) ≤ ∞, d = (d 1,..., d n) the Nikol’skii-Besov spaces B pr αq (T d ) are the retract spaces l q α (L pr (T d )), and corresponding interpolation theorem follows from the interpolation properties of anisotropic l q α (A) spaces. The interpolation properties of anisotropic Sobolev spaces W pr α (T d ) with dominant mixed derivative are described as a corollary. In the second part of the work the Nikol’skii inequality of different metrics for trigonometric polinomials with the spectrum from parallelepipeds in the anisotropic Lorentz spaces L qτ (T d ) is proved. On the basis of the inequality and interpolation theorems the embedding theorems for anisotropic Nikol’skii-Besov spaces B pr ατ (T d ) and anisotropic Lorentz spaces L qτ (T d ) are obtained. The relations, which connect the spaces parameters α, p and q, i.e. α = (1/p − 1/q) are the limiting. This relations also can not be improved. The obtained theorems generalize the corresponding results from the works of K.A. Bekmaganbetov and E.D. Nursultanov for the case d = (1,..., 1).
References
[2] Nikol’skij S.M. Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables. // Amer. Math. Soc. Transl. Ser. 2. – V. 80. – 1969. – P. 1 – 38.
[3] Besov O.V. Investigation of a family of function spaces in connection with theorems of imbedding and extension // Amer. Math. Soc. Transl. Ser. 2. – V. 40. – 1964. – P. 85 – 126.
[4] Lizorkin P.I. L rp (Ω) spaces. Extension and imbedding theorems // Soviet Math. Dokl., – V. 3. – 1962. – P. 1053-1057.
[5] Triebel H. Spaces of distributions of Besov type on Euclidean n-space. Duality, interpolation // Ark. Mat. – V. 11, No 1–2. – 1973. – P. 13 – 64.
[6] Nikol’skij S.M. Approximation of functions of several variables and imbedding theorems. – Springer–Verlag, New York – Heidelberg: Grundlehren Math. Wiss. – V. 205. – 1975.
[7] Besov O.V., Il’in V.P., Nikol’skij S.M. Integral representations of functions and imbedding theorems. – V. I, II. – Winston, Washington, DC. Wiley, New York – Toronto, ON – London. – 1979.
[8] Triebel H. Interpolation theory, function spaces, differential operators. North–Holland Libruary. – V. 18. North–Holland, Amsterdam – New York – Oxford, 1978.
[9] Nursultanov E.D. Interpolation theorems for anisotropic function spaces and their applications // Doklady. Mathematics. 2004, vol. 69, No, pp. 16-19.
[10] Bergh J. and Löfström J. Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, vol. 223, Springer-Verlag, Berlin – Hiedelburg – New York 1976.
[11] Bazarkhanov D.B. Various representations and equivalent normalizations of Nicol’skii-Besov and Lizorkin-Tribel spaces of the generalized mixed smoothness. Dokl. Akad. nauk. Vol. 402, No. 3, – pp. 298 – 302.
[12] Bekmaganbetov K.A. and Nursultanov E.D. Embedding theorems for anisotropic Besov spaces B pr αq ([0, 2π) n ) // Izvestiya: Mathematics (2009), 73(4):655. – pp. 3 – 16.
[13] Nursultanov E.D. On the coefficients of multiple Fourier series from L p -spaces. (Russian) // Izvestiya: Mathematics (2000), 64: 1, – pp. 93 – 120.
[14] Asekritova I., Krugljak N., Maligranda L., Nikolova L. and Persson L.-E. Lions-Peetre reiteration formulas for triples and their applications // Studia Math. – 2001, – V. 145, – P. 219 – 254.
[15] Bekmaganbetov K.A. and Nursultanov E.D. Method of multiparameter interpolation and embedding theorems in Besov ⃗ [0, 2π) // Analysis Math. – 1998, – V. 24. – P. 241 – 263. spaces B p ⃗ α
[16] Krepkogorskii V.L. Interpolation in Lizorkin-Triebel and Besov spaces // Mat. sbornik. – 1994, – V. 73, No 4. – P. 63 – 76.
[17] Krepkogorskii V.L. Realization of interpolation Sparr method in the class of spaces of smooth funtions // Matem. Zametki. – 2001, – V. 70, No 4. – P. 581 – 590.
[18] Peetre J. Thoughts on Besov spaces: Lecture notes. – Lund. – 1966.
[19] Bekmaganbetov K.A. Interpolation theorem for l q σ (L pτ ) and L pτ (l q σ ) spaces // Vestnik KazNU. – 2008, V. 56, No 1. – P. 30 – 42.
