D.B. Bazarkhanov. Linear recovery of pseudodifferential operators on classes of smooth functions on an m-dimensional torus. I ... P. 57-79

We construct a linear method for the recovery of pseudodifferential operators on an $m$-dimensional torus with symbols from particular classes with the use of linear spectral information on the symbol of the operator and on the function (finite sets of their Fourier coefficients). Error bounds are given for the error of recovery in the space $L_r(\mathbb{T}^m)$ of values of these pseudodifferential operators on elements of Nikol'skii-Besov and Lizorkin-Triebel function spaces for a number of relations between $r$ and the parameters of the symbol classes and the function spaces (Theorem 1). A key role in the proof of the bounds is played by the boundedness of the pseudodifferential operators between appropriate Nikol'skii-Besov (Lizorkin-Triebel) function spaces (Theorem 2).

Keywords: pseudodifferential operator on $m$-dimensional torus, class of symbols (of product type), Nikol'skii-Besov/Lizorkin-Triebel function space, recovery of operator, error bounds of recovery.

Received August 09, 2018

Revised November 06, 2018

Accepted November 12, 2018

Funding Agency: This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan (grant no. AP05133257).

Dauren B. Bazarkhanov, Сand. Sci. (Phys.-Math.), Prof., Institute of Mathematics and Math Modeling, Almaty, 050010 Kazakhstan,
e-mail: dauren.mirza@gmail.com

REFERENCES

1.   H$\ddot{\mathrm{o}}$rmander L. The Analysis of Linear Partial Differential Operators III: Pseudodifferential operators. Berlin; Heidelberg: Springer-Verlag, 1985, 525 p. ISBN: 9783540499381. Translated to Russian under the title Analiz lineinykh differentsial'nykh operatorov s chastnymi proizvodnymi. T. 3. Psevdodifferentsial'nye operatory. Moscow: Mir Publ., 1987, 696 p.

2.   Chang S.-Y.A., Fefferman R.  Some recent developments in Fourier analysis and $H^p$-theory on product domains. Bull. Amer Math. Soc.,  1985, vol. 12, pp. 1-43. doi: 10.1090/S0273-0979-1985-15291-7

3.   FeffermanR. Harmonic analysis on product spaces. Ann. Math., 1987, vol. 126, no. 1, pp. 109-130. doi: 10.2307/1971346

4.   Yamazaki M. Boundedness of product type pseudodifferential operators on spaces of Besov type.  Math. Nachr., 1987, vol. 133, no. 1, pp. 297-315. doi: 10.1002/mana.19871330120

5.   Carbery A., Seeger A. $H^p$ and $L^p$ variants of multiparameter Calderon-Zygmund theory. Trans. Amer Math. Soc., 1992, vol. 334, no. 2, pp. 719-747. doi: 10.2307/2154479

6.   Stein E.M.  Harmonic analysis: Real-variable methods, orthogonality, and oscillatory integrals.   Princeton: Princeton Univ. Press, 1993, 716 p.

7.   Bazarkhanov D.B. Wavelet approximation and Fourier widths of classes of periodic functions of several variables. I. Proc. Steklov Inst. Math., 2010, vol. 269, pp. 2-24. doi: 10.1134/S0081543810020021

8.   Bazarkhanov D.B. Wavelet approximation and Fourier widths of classes of periodic functions of several variables. II. Analysis Math., 2012, vol. 38, no. 4, pp. 249-289. doi: 10.1007/s10476-012-0401-3

9.   Coifman R., Meyer Y. Au-dela des operateurs pseudo-differentiels. Asterisque, 1978, vol. 57, pp. 1-185.

10.  Bazarkhanov D.B. $(L_p-L_q)$-boundedness of pseudodifferential operators on the $n$-dimensional torus. Math. Notes, 2017, vol. 102, no. 6, pp. 873-877. doi: 10.1134/S000143461711027X

11.  Triebel H. Theory of function spaces. Birkhauser, 1983. 281 p. ISBN: 3764313811. Translated to Russian under the title Teoriya funktsional'nykh prostranstv. Moscow: Mir Publ., 1986.

12.  Schmeisser H.J., Triebel H. Topics in Fourier analysis and function spaces. Chichester: J. Wiley & Sons, 1987, 300 p. ISBN: 0-471-90895-9

13.  Ruzhansky M., Turunen V.  Pseudo-differential operators and symmetries: background analysis and advanced topics. Basel; Birkhauser: Springer, 2009, 709 p. doi: 10.1007/978-3-7643-8514-9

14.  Nikolski S.M. Approximation of functions of several variables and embedding theorems. New York, Springer-Verlag, 1975. Original Russian text (2nd ed.) published in Nikol'skii S.M.  Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya. 2-e izd.  Moscow: Nauka Publ., 1977, 455 p.

15.  Hardy G.H., Littlewood J.E.,  P$\acute{\mathrm{o}}$lya G. Inequalities. Cambridge: Cambridge University Press, 1934, 340 p. ISBN(2nd ed.): 0-521-05206-8. Translated to Russian under the title Neravenstva. Moscow: Inostr. Lit. Publ., 1948, 456 p.

16.  Stein E.M. and  Weiss G. Introduction to Fourier Analysis on Euclidean Spaces. Princeton: Princeton Univ. Press,  1971, 312 p. ISBN: 9781400883899. Translated to Russian under the title Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh. Moscow: Mir Publ., 1974, 333 p.

17.  Nikol’skii S.M. Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables. In: Azarin V.S. et al (eds.), Thirteen papers on functions of real and complex variables. Providence: American Math. Soc., 1969, 278 p. (pp. 1–38).\\ ISBN: 978-1-4704-3291-1

18.  Besov O.V. Investigation of a family of function spaces in connection with theorems of imbedding and extension. American  Math. Soc., Transl., II, 1964, vol. 40, pp. 85-126.