A note on rough \(g\)-function operators (Q2735300)

Let \(\psi(x)=\Omega(x')|x|^{1-n}\rho(|x|)\) for \(x\in {\mathbb R}^n\setminus\{0\}\), where \(x'=x/|x|\). For \(f\in {\mathcal S}({\mathbb R}^n)\), the Littlewood-Paley \(g_\psi\)-function is defined by NEWLINE\[NEWLINEg_\psi(f)(x)=\left(\int^\infty_0|\psi_t\ast f(x)|^2 \frac {dt}t\right)^{1/2}.NEWLINE\]NEWLINE The author proves that if \(\Omega\in H^1(S^{n-1})\) and \(\rho\) satisfies some integral Dini condition, then \(g_\psi\) is bounded on \(L^p({\mathbb R}^n)\) for \(1<p<\infty\), which generalizes some known result.