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Average decay estimates for the Fourier transform of fractal measures

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초록 moremore
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.
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.
목차 moremore
I Introduction
1.1 Average decay estimates . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The Nikodym maximal operator . . . . . . . . . . . . . . . . . . . . 11
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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