Any references for generalised square functions?

Question

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 $p\in[1,\infty]$. Are there good references for this?

Answer 1

For $0<p<\infty$, $0<q\le\infty$ and $s\in\mathbb R$, the Triebel-Lizorkin space $F_{pq}^s(\mathbb R^n)$ is the set of all (tempered distributions) $f$ such that $\big\||\{2^{js}\cdot P_jf\}_j|_{\ell^q(\mathbb Z_+)}\big\|_{L^p(\mathbb R^n)}$ is finite. In your case the $L^q$ norm of the "$\ell^p$-square function" is the $F_{qp}^0$ norm.