On Homogeneous Distributions and Non-isotropic Dilations

Abstract

The aim of this thesis is to provide an introduction to the relationship between homogeneous distributions and the Fourier transform. We review the properties of isotropic dilations and demonstrates how they may be extended to non-isotropic dilations (x1, . . . , xd) 7→ (a α1 x1, . . . , aαd xd) with positive exponents αj . In the last chapter we drop the positive exponent condition in the dilation and explore the dilation (x1, x2) 7→ (ax1, a−1x2).