Fourier transforms of Gibbs measures for the Gauss map

Surface Measures - Maximal Functions and Fourier Transforms

A Remark on Circular Means of Fourier Transforms of Measures

In [7] the optimal decay of circular L means of compactly supported measures of finite energy was given for p ≥ 2, with application to Falconer’s distance problem. The question was then raised in that paper as to whether any non-trivial improvement to these estimates were available for p < 2. We answer this question in the negative. This paper is concerned with the quantity σp(α) defined by Wol...

Oscillatory Integrals and Fourier Transforms of Surface Carried Measures

We suppose that S is a smooth hypersurface in Rn+1 with Gaussian curvature re and surface measure dS, it) is a compactly supported cut-off function, and we let pa be the surface measure with dßa = u>Ka dS. In this paper we consider the case where S is the graph of a suitably convex function, homogeneous of degree d, and estimate the Fourier transform ßa. We also show that if S is convex, with n...

L_1 operator and Gauss map of quadric surfaces

The quadrics are all surfaces that can be expressed as a second degree polynomialin x, y and z. We study the Gauss map G of quadric surfaces in the 3-dimensional Euclidean space R^3 with respect to the so called L_1 operator ( Cheng-Yau operator □) acting on the smooth functions defined on the surfaces. For any smooth functions f defined on the surfaces, L_f=tr(P_1o hessf), where P_1 is t...

Large Scale Renormalisation of Fourier Transforms of Self-similar Measures and Self-similarity of Riesz Measures

We shall show that the oscillations observed by Strichartz JRS92, Str90] in the Fourier transforms of self-similar measures have a large-scale renormali-sation given by a Riesz-measure. Vice versa the Riesz measure itself will be shown to be self-similar around every triadic point.