BMO-boundedness of Lusin's area function (Q1264299)

Let \(\Gamma= \{(y, t)\in \mathbb{R}^{n+1}_+, | t|< y\}\) and \(\Gamma(x)= x+\Gamma\), where \(x\in\mathbb{R}^n\). For \(f\in L^1(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)\), the Lusin's area function \(S_T\) is defined by \[ S_T(f)(x)= \Biggl\{\iint_{\Gamma(x)} |\nabla_t u|^2 (y,t) y^{-n+1} dy dt\Biggr\}^{1/2}, \] where \(u= u(y,t)= P_y* f(t)\) and \(P_y\) is the Poisson kernel. The authors in this paper prove that there is a constant \(C= C(n)> 0\) such that \(\| S_T f\|_{\text{BMO}}\leq C\| f\|_{\text{BMO}}\). It is interesting to point out a closely related result by \textit{D. S. Kurtz} [Proc. Am. Math. Soc. 99, 657-666 (1987; Zbl 0658.42022)], where he proved that there is a constant \(C>0\) such that either \(S(f)(x)= \infty\) almost everywhere or \(S(f)(x)< \infty\) almost everywhere and \(\| S(f)\|_{\text{BMO}}\leq C\| f\|_{\text{BMO}}\). Here \(S(f)\) is the Littlewood-Paley function.