Convolution Inequalities for Positive Borel Measures on $\mathbb{R}^d$ and Beurling Density

Abstract

We consider certain convolution inequalities for positive Borel measures in Euclidean space and show how they are related to the notions of upper and lower-Beurling density for these measures. In particular, the upper-Beurling density of a measure μ is shown to be the infimum of the constants C>0 such that μ∗f≤C a.e. on ℝ ^dfor some nonnegative function f with ∫f(x)dx=1, and a similar characterization is obtained for the lower-Beurling density of μ. We also consider convolution inequalities involving several measures and provide applications of these results to systems of windowed exponentials and Gabor systems.