Average decay estimates for the Fourier transform of fractal measures
- 발행기관 포항공과대학교 일반대학원
- 지도교수 박종국
- 발행년도 2016
- 학위수여년월 2016. 8
- 학위명 박사
- 학과 및 전공 일반대학원 수학과
- 실제URI http://www.dcollection.net/handler/postech/000002292273
- 본문언어 영어
- 저작권 포항공과대학교 논문은 저작권에 의해 보호받습니다.
초록/요약
In this thesis we study average decay estimates for the Fourier transform of fractal measures when the averages are taken over space curves with non-vanishing torsion. We extend some previously known results to higher dimensions and discuss the sharpness of the estimates. Also, we consider the k-plane Nikodym maximal estimates in variable Lebesgue spaces. We first formulate a problem on the boundedness of the k-plane Nikodym maximal operator. Then we show that a maximal operator estimate in Lebesgue spaces is equivalent to a maximal operator estimate in variable Lebesgue spaces.
more목차
I Introduction
1.1 Average decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Nikodym maximal operator . . . . . . . . . . . . . . . . . . . . 11
II Average decay estimates
2.1 Oscillatory integral operators . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Proof of main results on average decay estimates . . . . . . . . . . . 20
2.3 Upper bounds for δ and lower bounds for κ . . . . . . . . . . . . . . 26
2.4 A multilinear approach . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5 A bilinear approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
IIIThe Nikodym maximal operator
3.1 The necessary condition . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2 Proof of the main result on the Nikodym maximal operator . . . . . 51