The bounded approximation property in Banach function lattices, interpolation of the essential norm, and applications

We prove the bounded approximation property for Banach function lattices with order continuous norm on which the Hardy–Littlewood maximal operator is bounded. This allows us to develop a general framework for the quantitative interpolation of compact operators, understood in terms of the essential norm, encompassing a large class of function spaces including weighted Lebesgue spaces, rearrangement invariant Banach function spaces, and variable exponent Lebesgue spaces. Combined with extrapolation techniques, we show how to transfer compactness (or, more generally, control of the essential norm) from unweighted Lebesgue spaces to the three aforementioned classes of function spaces. Finally, we present several applications in analysis and partial differential equations.