On a generalization of Lp-differentiability

In this paper we connect Calderon and Zygmund's notion of L-p-differentiability (Calderon and Zygmund, Proc Natl Acad Sci USA 46: 1385-1389, 1960) with some recent characterizations of Sobolev spaces via the asymptotics of non-local functionals due to Bourgain, Brezis, and Mironescu (Optimal Control and Partial Differential Equations, pp. 439-455, 2001). We showhowthe results of the former can be generalized to the setting of the latter, while the latter results can be strengthened in the spirit of the former. As a consequence of these results we give several new characterizations of Sobolev spaces, a novel condition for whether a function of bounded variation is in the Sobolev space W-1,W-1, and complete the proof of a characterization of the Sobolev spaces recently claimed in (Leoni and Spector, J Funct Anal 261: 2926-2958, 2011; Leoni and Spector, J Funct Anal 266: 1106-1114, 2014).