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Ufimsk. Mat. Zh., 2019, Volume 11, Issue 4, Pages 79–91 (Mi ufa486)  

On estimates for oscillatory integrals with phase depending on parameters

Sh. A. Muranov

Samarkand State University, Universitetskii boulevard 15, 140104, Samarkand, Uzbekistan

Abstract: We consider estimates for the Fourier transforms of measures supported on analytic hypersurfaces involving a damping factor. As a damper, we naturally take a power of the Gaussian curvature of the surface. It is known that if the exponent in this power is a sufficiently large positive number, then the Fourier transform of the corresponding measure has an optimal decay. C.D. Sogge and E.M. Stein formulated a problem on a minimal power of the Gaussian curvature ensuring an optimal decay for the Fourier transform. In the paper we resolve the problem by C.D. Sogge and E.M. Stein on an optimal decay for the Fourier transform with a damping factor for a particular class of families of analytic surfaces in the three-dimensional Euclidean space. We note that the power we provide is sharp not only for the families of analytic hypersurfaces but also for a fixed analytic hypersurface. The proof of main result is based on the methods of the theory of analytic functions, more precisely, on the statements like a preparation Weierstrass theorem. As D.M. Oberlin showed, similar statements fail for infinitely differentiable hypersurfaces.

Keywords: oscillating integrals, Fourier transform, dumping factor, maximal operator.

Funding Agency Grant Number
КОНИД при Министерстве ВССО РУзб ОТ-Ф4-69
The work is supported by Committee of Education, Science and Innovative Activity of the Ministry of Higher and Special Education of the Republic of Uzbekistan (grant no. OT-F4-69).


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English version:
Ufa Mathematical Journal, 2019, 11:4, 78–90 (PDF, 377 kB); https://doi.org/10.13108/2019-11-4-78

Bibliographic databases:

UDC: 517.518
MSC: 35D05, 35D10, 35G05
Received: 08.10.2018

Citation: Sh. A. Muranov, “On estimates for oscillatory integrals with phase depending on parameters”, Ufimsk. Mat. Zh., 11:4 (2019), 79–91; Ufa Math. J., 11:4 (2019), 78–90

Citation in format AMSBIB
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\paper On estimates for oscillatory integrals with phase depending on parameters
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\yr 2019
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\issue 4
\pages 79--91
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\vol 11
\issue 4
\pages 78--90
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