L^p-Taylor approximations characterize the Sobolev space W^1,p

In this note, we introduce a variant of Calderón and Zygmund's notion of L^p-differentiability - an L^p-Taylor approximation. Our first result is that functions in the Sobolev space W1,p(RN) possess a first-order L^p-Taylor approximation. This is in analogy with Calderón and Zygmund's result concerning the L^p-differentiability of Sobolev functions. In fact, the main result we announce here is that the first-order L^p-Taylor approximation characterizes the Sobolev space W1,p(RN), and therefore implies L^p-differentiability. Our approach establishes connections between some characterizations of Sobolev spaces due to Swanson using Calderón-Zygmund classes with others due to Bourgain, Brézis, and Mironescu using nonlocal functionals with still others of the author and Mengesha using nonlocal gradients. That any two characterizations of Sobolev spaces are related is not surprising; however, one consequence of our analysis is a simple condition for determining whether a function of bounded variation is in a Sobolev space.