Characterization of $L^p $-Solutions for the Two-Scale Dilation Equations

Abstract

We give a characterization of the existence of compactly supported $L^p $-solutions, $1 \leq p < \infty $, for the two-scale dilation equations. For the $L^2 $-case, the condition reduces to the determination of the spectral radius of a certain matrix in terms of the coefficients, which can be calculated through a finite step algorithm. For the other cases, we implement the characterization by the four-coefficient dilation equation and obtain some simple sufficient conditions for the existence of the solutions. The results are compared with known ones.