Let \(S^{n-1}\) denote the unit sphere in \(\mathbb R^n\) \((n\geq 2)\) equipped with normalized Lebesgue measure \(d\sigma\). Let \(\Omega\) be a homogeneous function of degree zero on \(\mathbb R^n\), which satisfies the \(\text{Lip}_\gamma\), \(0<\gamma\leq 1\), condition on \(S^{n-1}\) and \[ \int_{S^{n-1}}\Omega(x')\,d\sigma(x')=0. \]
Write \(\Gamma(x)=\{(y,t)\in \mathbb R^{n+1}_+ : |x-y|<t\}\). For functions \(A, f\) defined on \(\mathbb R^n\), \(\delta\geq 0\), and \(m\in \mathbb N\), the author investigates the boundedness of following two multilinear Marcinkiewicz operators for the extreme cases of \(p\)
\[ \begin{aligned} \mu_\delta^A(f)(x)&=\bigg( \iint_{\Gamma(x)} |F_t^A(f)(x,y)|^2\,\frac{dy\,dt}{t^{n+3}}\bigg)^{1/2},\\ \widetilde{\mu}_\delta^A(f)(x)&=\bigg( \iint_{\Gamma(x)} |\widetilde{F}_t^A(f)(x,y)|^2\,\frac{dy\,dt}{t^{n+3}}\bigg)^{1/2}, \end{aligned} \] where
\[ \begin{aligned} F_t^A(f)(x,y)&=\int_{|y-z|\leq t} \frac{\Omega(y-z)}{|y-z|^{n-1-\delta}} \frac{R_{m+1}(A;x,z)}{|x-z|^m}f(z)\,dz,\\ \widetilde{F}_t^A(f)(x,y)&=\int_{|y-z|\leq t} \frac{\Omega(y-z)}{|y-z|^{n-1-\delta}} \frac{Q_{m+1}(A;x,z)}{|x-z|^m}f(z)\,dz, \end{aligned} \]
and
\[ \begin{aligned} R_{m+1}(A;x,z)&=A(x)-\sum_{|\alpha|\leq m}\frac 1{\alpha !} D^\alpha A(z) (x-z)^\alpha,\\ Q_{m+1}(A;x,z)&=R_m(A;x,z)-\sum_{|\alpha|= m}\frac 1{\alpha !} D^\alpha A(z) (x-z)^\alpha. \end{aligned} \]
The author proves that, if \(0\leq \delta<n\) and \(D^\alpha A \in \text{BMO}\) for all \(\alpha\) with \(|\alpha|=m\), then
\(\mu_\delta^A\) is bounded from \(L^{n/\delta}\) to BMO and bounded from \(H^1\) to weak \(L^{n/(n-\delta)}\);
\(\widetilde{\mu}_\delta^A\) is bounded from \(H^1\) to \(L^{n/(n-\delta)}\).