The ball Banach fractional Sobolev inequality and its applications

The authors obtain a fractional Sobolev inequality for Sobolev spaces W˙s,X(Rn) for ball Banach function spaces X on Rn with the homogeneity and the non-collapse properties. More precisely, the authors show the existence of a positive constant C such that, for any f∈W˙s,X(Rn)∩Xαα+s, ∥f∥W˙s,X(Rn)≥C∥f∥Xαα+s, where α is the homogeneity index of X, s∈(0,min{−α,1}), and Xαα+s is the αα+s-convexification of X. Moreover, under some mild assumptions, the authors prove that the closure of C∞c(Rn) with respect to ∥⋅∥W˙s,X(Rn) modulo constants is identified with W˙s,X(Rn)∩Xαα+s. When X is a Lebesgue space, these results reduce to the well-known Sobolev embeddings for which the restriction s∈(0,min{−α,1}) is sharp. However, these results also provide new Sobolev embeddings for Morrey spaces, mixed-norm Lebesgue spaces, Lebesgue spaces with power weights, Besov–Triebel–Lizorkin–Bourgain–Morrey spaces, and Lorentz spaces. As in the case of the classical Sobolev inequality, these results have a wide range of applications.