COMPACTNESS OF INTEGRAL OPERATORS AND UNIFORM
Universitat Bielefeld, WOLFHARD HANSEN · Revue Roumaine Mathematiques Pures Appliquees · 2024
Let (E, E, μ) be a measure space and let E+, Eb denote the set of all measurable numerical functions on E which are positive, bounded respectively. Moreover, let G: E ×E → [0,∞] be measurable. We show that the set of all q ∈ E+ for which {G(x, ·)q : x ∈ E} is uniformly integrable coincides with the set of all q ∈ E+ for which the mapping f 7→ G(fq) := R G(·, y)f(y)q(y) dμ(y) is a compact operator on the space Eb (equipped with the sup-norm) provided each of these two sets contains strictly positive functions.