Dominated Compactness Theorem in Banach Function Spaces and its Applications

Abstract.

A famous dominated compactness theorem due to Krasnosel’skiĭ states that compactness of a regular linear integral operator in L ^p follows from that of a majorant operator. This theorem is extended to the case of the spaces $L^{p(\cdot)}(\Omega, \mu, \varrho), \mu \Omega < \infty$, with variable exponent p(·), where we also admit power type weights $\varrho$. This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces $L^{p(\cdot)}(\Omega, \mu, \varrho)$ is applied to fractional integral operators over bounded open sets.