Banach Function Spaces, Extrapolation, and Orlicz Spaces

Abstract

AbstractIn this chapter we begin by providing a definition which is more inclusive than that of a “standard” Banach function space (as traditionally used in the literature; cf. e.g., [14]), and indicate that a significant portion of the classical theory goes through for this more general brand, which we dub Generalized Banach Function Spaces. This is done in §5.1. The relevance of this extension is that a variety of scales of spaces of interest, such as Morrey spaces, block spaces, as well as Beurling algebras and their pre-duals, now fit naturally into this more accommodating label. Most significantly, in §5.2 we develop powerful and versatile extrapolation results serving as portal, allowing us to pass from estimates on Muckenhoupt weighted Lebesgue spaces (for a fixed integrability exponent and arbitrary weights) to estimates on the brand of Generalized Banach Function Spaces introduced earlier, on which the Hardy-Littlewood maximal operator happens to be bounded. Finally, in §5.3 we focus on Orlicz spaces which, in particular, are natural examples of classical Banach function spaces for which the machinery developed so far applies.