In harmonic analysis, there is a big chunk of literature studying the square function $Sf=\|\{P_jf\}_{j=1}^\infty\|_{l^2}$, where $P_jf=(\psi_j\hat f)\check{}$ and $\{\psi_j\}$ is a partition of unity, each is $1$ on the annulus $\psi\sim2^j$ and supported around the annulus. I am wondering if we can generalise this to $\|\{P_jf\}_{j=1}^\infty\|_{l^p}$ for $p\in[1,\infty]$. I am wondering if there are good references on this?