Characterizing compact families via the Laplace transform

In 1985, Robert L. Pego characterized compact families in $L^2(\mathbb{R})$ in terms of the Fourier transform. It took nearly 30 years to realize that Pego’s result can be proved in a wider setting of locally compact abelian groups (works of G'orka and Kostrzewa). In the current paper, we argue that the Fourier transform is not the only integral transform that is efficient in characterizing compact families and suggest the Laplace transform as a possible alternative.