Lipschitz estimates for multilinear commutator of Littlewood-Paley operator (Q2379914)

Let \(g_{\mu}\) be the Littlewood-Paley operator and \({\mathbf b} =(b_1,\dots, b_m)\). The multilinear commutator of the Littlewood-Paley operator is defined by \[ g_{\mu}^{\mathbf{b}}(f)(x)=\Bigg[\int \int_{{\mathbb R}^{n+1}_+} \bigg(\frac{t}{t+|x-y|}\bigg)^{n\mu}\big|F^{\mathbf b}_t(f) (x,y)\big|^2 \frac{dydt}{t^{n+1}}\Bigg]^{1/2}, \] where \[ F^{\mathbf b}_t(f)(x,y)=\int_{{\mathbb R}^{n}} \bigg[\prod_{j=1}^m \big(b_j(x)-b_j(z)\big)\bigg]\psi_t(y-z)f(z)dz \] and \(\psi_t(x)=t^{-n}\psi(x/t)\) for \(t>0\). When \(\mu>3+\frac1n\), \(0<\beta<\frac1{2m}\) and \(b_j\in {\mathrm{Lip}_{\beta}}({\mathbb R}^{n})\), \(j=1,\dots, m\), the author proves that the multilinear commutator \(g_{\mu}^{\mathbf b}\) is bounded from \(L^p({\mathbb R}^{n})\) into \(\dot{F}_p^{m\beta,\infty}({\mathbb {R}}^{n})\) for \(1<p<\infty\) and from \(L^p({\mathbb R}^{n})\) into \(L^q({\mathbb R}^{n})\) for \(1<p<\frac{n}{m\beta}\) and \(\frac1p-\frac1q=\frac{m\beta}{n}\). The boundedness of \(g_{\mu}^{\mathbf{b}}\) on Hardy spaces and Herz-Hardy spaces is also considered.