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

## TL;DR

This paper characterizes certain Sobolev spaces using a quadratic symmetrization of the Calderón commutator kernel, introduces endpoint estimates for Hardy-Sobolev spaces, and uses a local square function to analyze pointwise differentiability in the Zygmund class.

## Contribution

It provides a new characterization of Sobolev spaces via a specific square function and establishes endpoint estimates, advancing understanding of differentiability and function space properties.

## Key findings

- Characterization of $L^p$-Sobolev spaces using quadratic symmetrization.
- Endpoint weak type estimate for homogeneous Hardy-Sobolev spaces.
- Square function approach to pointwise differentiability in Zygmund class.

## Abstract

We prove a characterization of some $L^p$-Sobolev spaces involving the quadratic symmetrization of the Calder\'on 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 $\dot H^1_\alpha$. We also use a local version of this square function to characterize pointwise differentiability for functions in the Zygmund class.