A note on characterization of $H^1$ Sobolev spaces by square functions

TL;DR

This work develops a square-function framework for characterizing $H^1(\mathbb{R}^n)$ via $U_\alpha(f)$ with $n/2<\alpha<n$, yielding a precise equivalence between $f \in W^\alpha_{H^1}$ and $U_\alpha(f) \in L^1$ and a corresponding norm equivalence. It extends the theory by introducing higher-order kernels $K^{(k)}$ and $\widetilde{E}_\alpha^{(k)}$, showing $f \in W^\alpha_{H^1}$ iff $\widetilde{E}_\alpha^{(k)}(f) \in L^1$ under $\alpha<\min(2k,n)$, and confirms these results without the Fourier-transform size restrictions present in earlier work (Sajf). The special case $\Phi = |B(0,1)|^{-1}\chi_{B(0,1)}$ links $\widetilde{E}_\alpha^{(2)}$ to repeated ball-averaging, highlighting concrete interpretations of the square-function approach. Overall, the paper provides robust, assumption-free square-function criteria for Hardy-Sobolev spaces with explicit norm control.

Abstract

We establish characterization of $H^1$ Sobolev spaces by certain square functions, improving previous results.