Average decay estimates for the Fourier transform of fractal measures

Abstract

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.