Strict definiteness of integrals via complete monotonicity of derivatives

Abstract

Let k k be a nonnegative integer and let ϕ : ( 0 , ∞ ) → R \phi : (0,\infty ) \rightarrow \mathbb {R} be a C ∞ C^\infty function with ( − ) k ⋅ ϕ ( k ) (-)^k\cdot \phi ^{(k)} completely monotone and not constant. If σ ≠ 0 \sigma \neq 0 is a signed measure on any euclidean space R d \mathbb {R}^d , with vanishing moments up to order k − 1 k-1 , then the integral ∫ R d ∫ R d ϕ ( ‖ x − y ‖ 2 ) d σ ( x ) d σ ( y ) \int _{\mathbb {R}^d} \int _{\mathbb {R}^d} \phi ( \|x-y\|^2 ) \, d\sigma (x) d\sigma (y) is strictly positive whenever it exists. For general d d no larger class of continuous functions ϕ \phi seems to admit the same conclusion. Examples and applications are indicated. A section on ”bilinear integrability” might be of independent interest.