Restrição de Fourier em conjuntos de Salem

Abstract

In this work, we show how the s−energy Is(µ) of a Borel measure µ compactly supported is related to the Hausdorff dimension of supp(µ). Using the distributional Fourier transform of the Riesz kernel, we relate Is(µ) to µ^. In this way, we show that Hausdorff dimension and Fourier transforms of measures are closely linked concepts, which is translated into the Fourier dimension. For the construction of examples, we made a study of surface measures. More precisely, we use weak convergence of measures to calculate the Fourier transform of the surface measure in the sphere. In addition, we use the asymptotic behavior of Bessel’s functions to show that it has a rapid decay. More generally, we study oscillatory integrals and apply the results to obtain the decay of the Fourier transform of the intrinsec measure of a compact regular surface with l non-zero principal curvatures. In addition, we use Hausdorff dimension concept to show that the decay of such a measure is optimal. We approach the restriction conjecture in the sphere and use the Knapp Example to get required range. We have dealt with the Stein-Tomas Theorem and obtained it as a consequence of the Littman Theorem. We use the techniques of Carleson-Sjölin to exhibit the proof of the restriction conjecture in the plane. We finish this dissertation by presenting the Mockenhaupt-Mitsis Theorem, which generalizes the Stein-Tomas Theorem, without the end-point. In addition, we present some consequences of the same observed by Mitsis. We briefly deal with the construction of a measure supported on a Salem set, which satisfies the hypotheses of the Mockenhaupt-Mitsis Theorem.