Damped oscillatory integrals and Weierstrass polynomials

Authors

Azimbay Sadullaev, National University of UzbekistanFollow
Isroil Ikromov, Samarkand Division of Institute of Mathematics, Uzbek Academy of SciencesFollow
Shaxriddin Muranov, Samarkand State UniversityFollow

Abstract

In this paper we consider the Sogge- Stein problem related to the damped oscillatory integrals. We show that in three-dimensional Euclidean spaces minimal exponent, which guarantees optimal decaying of the Fourier transform of the surfaces-carried measures with mitigating factor is bounded by 3/2. A proof of the main theorem is based on Weierstrass type results.

First Page

255

Last Page

268

References

1. Arkhipov G.I., Karatsuba A.A. and Chubarikov V.N. Trigonometric integrals. Izv. Akad. Nauk SSSR Ser. Mat., Vol. 43, Issue 5, 971–1003 (Russian); English translation in Math. USSR-Izv., Vol. 15, 21–239 (1980).

2. Arnol'd V.I., Gusen-zade S.M. and Varchenko A.N. Singularities of differentiable maps. Part I. M.: Nauka, (1982).

3. Erdélyi A. Asymptotic Expansions. Dover Publications Inc., New York, (1956).

4. Fedoryuk M.V. Perevals methods. M.: Nauka, (1977).

5. Hörmander L. Analysis of linear partial differential operators. Part I. Distribution Theory and Fourier Analysis, Second edition, Springer-Verlag, (1989).

6. Ikromov I.A. Damped oscillatory integrals and maximal operators. Mathematical notes, Vol. 78, 833–852 (2005).

7. Ikromov I.A., Müller D. and Kempe M. Damped oscillatory integrals and boundedness of maximal operators associated to mixed homogeneous hypersurfaces. Duke Math.J., Vol. 126, No.3, 471–490(2005).

8. Ikromov I.A. and Muranov Sh.A. On estimates for oscillatory integrals with mitigating factor. Mathematical notes, Vol. 104, No.2, 236–251(2018).

9. Kitmanov A.M. and Sadullaev A. On estimates volume of zeros of holomorph functions depending on complex parameter. Math. sb. (to appear).

10. Malgrange B. Ideals of differentiable functions. Tata Institute, Bombay and Oxford University Press, (1966).

11. Muranov Sh.A. On estimates for oscillatory integrals with damping factor. Uzbek Mathematical Journal, Vol. 4, 112–125(2018).

12. Muranov Sh.A. On estimates for oscillatory integrals with phases depending on parameters. Ufa Mathematical Journal, Vol. 11, No.4, 79–91(2019).

13. Oberlin D.M. Oscillatory integrals with polynomial phase, MATH. SCAND., Vol. 69, 45–56(1991).

14. Popov D.A. Estimates with constant for some classes oscillating integrals. UMN., Vol. 52, No.1(313), 77–148(1997).

15. Sadullaev A. On a criterion of algebraicity of analytic sets. Func. anal. and appl, Vol. 6, No.1, 85–86(1972).

16. Stein E.M. Harmonic Analysis Real-Valued Methods, Orthogonality, and oscillatory Integrals. Princeton, Princeton Univ. Press, (1993).

17. Sogge C.D., Stein E.M. Averages of functions over hypersurfaces in ℝⁿ. Invent. Math., Vol. 82, 543–556(1985).

Recommended Citation

Sadullaev, Azimbay; Ikromov, Isroil; and Muranov, Shaxriddin (2020) "Damped oscillatory integrals and Weierstrass polynomials," Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 3: Iss. 2, Article 15.
DOI: https://doi.org/10.56017/2181-1318.1102