Riesz-Kolmogorov type compactness criteria in function spaces with applications (Q6610406)

The paper presents characterizations of precompactness in various function spaces along with proofs and examples of applications.\N\NSection 1 starts out with the Riesz-Kolmogorov characterization in \(L_p(\mathbb{R}^n)\), \(p\in[1,\infty)\) (Theorem~A). Then follows a characterization in Hilbert spaces (Theorem~1.1). Next follow two characterizations in \(L_2(\mathbb{R}^n)\) (Theorem~C and~D) in terms of the Fourier- and short-time Fourier transform, respectively. The introduction part of Section~1 ends with four characterizations in Hilbert [Banach] spaces with a continuous Parseval frame [continuous frame].\N\NIn the second part of Section 1, Section~1.1, it is indicated how the above results are used to find characterizations in Paley-Wiener spaces, weighted Bargmann-Fock spaces, and in weighted Besov-Sobolev spaces. In Section~1.2, characterizations of compactness of a large class of Toeplitz operators on the usual Bergman space, and of little Hankel operators on the Hardy space (2-norm) follow.\N\NSection 2 contains detailed proofs of the results from Section~1.1, while Section~3 provides proofs of the results in Section~1.2.\N\NFor the entire collection see [Zbl 1531.47001].