Abstract
We study the multipliers of multiple Fourier series for a regular system on anisotropic Lorentz spaces. In particular, the sufficient conditions for a sequence of complex numbers {λk}k∈Zn in order to make it a multiplier of multiple trigonometric Fourier series from Lp[0; 1]n to Lq[0; 1]n , p > q. These conditions include conditions Lizorkin theorem on multipliers.
References
[1] Nikol’skii S. M., Inequalities for entire functions of finite degree and their application in the theory of differentiable functions of several variables. (Russian) Trudy Mat. Inst. Steklov, Izdat. Akad. Nauk SSSR, Moscow, 1951, 38, 244–278. Search in Google Scholar
[2] Zygmund A., Trigonometric Series: Vols. I, II, 2nd ed., with a foreward by Robert Ferrerman, Cambridge University Press, 1968. Search in Google Scholar
[3] Marcinkiewicz, J., Sur les multiplicateurs des series de Fourier, Studia Math., 1939, 8, 78–91. 10.4064/sm-8-1-78-91Search in Google Scholar
[4] Sarybekova L., Tararykova T., Tleukhanova N., On a generalization of the Lizorkin theorem on Fourier multipliers, Math. Inequal. Appl. 13, 2010, 13 (3), 613-624. 10.7153/mia-13-42Search in Google Scholar
[5] D’yachenko M. I., Norms of Dirichlet kernels and of some other trigonometric polynomials in Lp spaces. (Russian) Mat. Sb., 1993, 184 (3), 3–20; translation in Russian Acad. Sci. Sb. Math. 1994, 78 (2), 267–282. Search in Google Scholar
[6] E. D. Nursultanov, On multipliers of Fourier series in the trigonometric system, Mat. Zametki, 1998, 63 (2), 235–247; English transl. in Math. Notes, 1998, 63 (1–2), 205–214. Search in Google Scholar
[7] Lizorkin P. I., Multipliers of Fourier integrals in the spaces Lp, θ, Trudy Mat. Inst. Steklov, 1967, 89, 231–248; English transl. in Proc. Steklov Inst. Math., 1967, 89, 269–290. Search in Google Scholar
[8] Nursultanov E. D., Tleukhanova N. T., On multipliers of multiple Fourier series, Tr. Mat. Inst. Steklova, 1999, 227, 237–242; English transl. in Proc. Steklov Inst.Math., 1999, 227 (4), 231–236. Search in Google Scholar
[9] Nursultanov E. D., Nikol’skii’s Inequality for Different Metrics and Properties of the Sequence of Norms of the Fourier Sums of a Function in the Lorentz Space, Proc. Steklov Inst.Math., 2006, 255, 185–202. 10.1134/S0081543806040158Search in Google Scholar
[10] Nursultanov E. D., Net spaces and inequalities of Hardy- Littlewood type, Sbornik: Mathematics, 1998, 189 (3), 83–102. 10.4213/sm309Search in Google Scholar
[11] Nursultanov E. D., Interpolation theorems for anisotropic function spaces and their applications, Dokl. Akad. Nauk, 2004, 394 (1), 22–25; English transl. in Dokl. Math., 2004, 69 (1), 16–19. Search in Google Scholar
[12] Nursultanov E. D., On the application of interpolation methods in the study of the properties of functions of several variables, Mat. Zametki, 2004, 75 (3), 372–383; English transl. in Math. Notes, 2004, 75 (3–4), 341–351. Search in Google Scholar
[13] Nursultanov E. D., On the coeflcients of multiple Fourier series from Lp-spaces, Izv. Ross. Akad. Nauk Ser. Mat., 2000, 64 (1), 95–122; English transl. in Izv. Math., 2000, 64 (1), 93–120. Search in Google Scholar
©2016 A.Zh. Ydyrys et al.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.