On square functions with independent increments and Sobolev spaces on the line

Abstract

We prove a characterization of some $L^p$-Sobolev spaces involving the quadratic symmetrization of the Calderón commutator kernel, which is related to a square function with differences of difference quotients. An endpoint weak type estimate is established for functions in homogeneous Hardy-Sobolev spaces ${\dot{H}}_{\alpha }^1$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.

To appear in Annali di Matematica Pura ed Applicata