On Littlewood-Paley functions
On Littlewood-Paley functions
We prove that, for a compactly supported $L^q$ function $\Phi$ with vanishing integral on $\mathbf {R}^n$, the corresponding square function operator $S_\Phi$ is bounded on $L^p$ for $|1/p - 1/2| < \min \{(q-1)/2, 1/2\}$.