A sharp endpoint estimate for multilinear Marcinkiewicz integral operator (Q1763081)

Let \(\Omega\) be a homogeneous function of degree zero, satisfying \(\Omega\in \roman{Lip}_\gamma(S^{n-1})\) \((0<\gamma\leq 1)\), \(\int_{S^{n-1}}\Omega(x)\,d\sigma(x)=0\). Let \(\mu_{\lambda}^{A}\) be the multilinear Marcinkiewicz operator defined by \[ \mu_{\lambda}^{A}f(x)=\left[\iint_{\mathbb R_+^{n+1}} \left(\frac{t}{t+| x-y| } \right)^{n\lambda} \left| \int_{| z-y| \leq t}\frac{\Omega(y-z)}{| y-z| ^{n-\rho}} \frac{R_{m+1}(A;x,z)}{| x-z| ^m}f(z)dz\right| ^2\frac{dydt}{t^{n+3}} \right]^{\frac12}, \] where \[ R_{m+1}(A;x,y)=A(x)-\sum_{| \beta| \leq m}\frac{1}{\beta!} D^\beta A(y)(x-y)^\beta,\quad D^\beta\in \roman{BMO}(\mathbb R^n)\text{ for }| \beta| =m. \] The author claims: For any \(0<r<1\), there exists \(C>0\) such that for any \(f\in C_0^\infty(\mathbb R^n)\) and any \(x\in \mathbb R^n\), \[ ([(| \mu_\lambda^A (f)| ^r)^{\#}](x))^{1/r}\leq C\sum_{| \alpha| =m}\| D^\alpha A\| _{\text{BMO}}M^2f(x), \] where \(g^{\#}\) is the sharp function of \(g\), and \(Mg\) is the Hardy-Littlewood maximal function of \(g\). However, there are several mistakes in the proofs. In the proof of Lemma 4, he uses an observation: \(| x-z| \leq 2t\), \(| y-z| \geq | x-z| -t\geq | x-z| -3t\) when \(| x-y| , | y-z| \leq t\). But, \(| x-z| -3t<2t<0\), and so one cannot deduce \(| y-z| \geq\bigl| | x-z| -3t\bigr| \), which he uses several times. Avoiding this, one can prove the estimate \(| \mu_\lambda^A f(x)| \leq C\int_{\mathbb R^n}| R_{m+1}(A;x,z)| | x-z| ^{-m-n}| f(z)| dz\). Nevertheless, from this estimate and the results cited in the references, one cannot deduce the conclusion of Lemma 4. In the proof of Theorem 1, he uses the weak type \((1,1)\) of \(\mu_\lambda\). However, \(\mu_\lambda\) is in general not weak \((1,1)\) for \(n\geq2\) (see, for example, [\textit{M. Sakamoto} and \textit{K. Yabuta}, Stud. Math. 135, No.~2, 103--142 (1999; Zbl 0930.42009)]).