Linear and bilinear \(T(b)\) theorems à la Stein (Q2795852)

In this paper, the authors mainly discuss \(T(b)\) theorems à la Stein for linear and bilinear Calderón-Zygmund operators. An infinite differentiable function \(\phi\) is called a normalized bump function of order \(M\in\mathbb N\cup\{0\}\) if \(\phi\) is supported in the unit ball centered at the origin and \(\|\partial^\alpha\phi\|_{L^\infty}\leq 1\) for all multi-indices \(\alpha\) with \(|\alpha|\leq M\). For fixed \(x_0\in\mathbb R^d\) and \(R>0\), denote \(\phi(\frac{x-x_0}{R})\) by \(\phi^{x_0,R}(x)\) for \(x\in\mathbb R^d\). A function \(b\in L^\infty\) is said to be para-accretive if there exists \(c_0>0\) such that, for any cube \(Q\), there exists a cube \(\widetilde{Q}\subset Q\) satisfying NEWLINE\[NEWLINE \frac 1{|Q|}\left|\int_{\widetilde{Q}} b(x)\,dx\right|\geq c_0. NEWLINE\]NEWLINENEWLINENEWLINELetting \(b_0\) and \(b_1\) be para-accretive functions and \(T\) a linear singular integral operator of Calderón-Zygmund type associated to \(b_0\) and \(b_1\), the authors prove that \(T\) can be extended to a bounded linear operator on \(L^2\) if and only if there exists \(M\in\mathbb N\cup\{0\}\) such that, for any normalized bump function \(\phi\) of order \(M\), \(x_0\in\mathbb R^d\) and \(R>0\), \(\| T(b_1\phi^{x_0,R})\|_{L^2}\) and \(\| T^*(b_0\phi^{x_0,R})\|_{L^2}\) can both be controlled by \(R^{\frac d2}\).NEWLINENEWLINESimilarly, letting \(b_0\), \(b_1\) and \(b_2\) be para-accretive functions and \(T\) a bilinear singular integral operator of Calderón-Zygmund type associated to \(b_0\), \(b_1\) and \(b_2\), the authors also prove that \(T\) can be extended to a bounded bilinear operator from \(L^p\times L^q\) into \(L^r\) for all \(p,\,q<\infty\) satisfying \(\frac 1p+\frac 1q=\frac 1r\) if and only if there exists \(M\in\mathbb N\cup\{0\}\) such that, for any normalized bump function \(\phi\) of order \(M\), \(x_0,\,x_1,\,x_2\in\mathbb R^d\) and \(R>0\), \(\| T(b_1\phi^{x_1,R},b_2\phi^{x_2,R})\|_{L^2}\), \(\| T^{*1}(b_0\phi^{x_0,R},b_2\phi^{x_2,R})\|_{L^2}\) and \(\| T^{*2}(b_1\phi^{x_1,R},b_0\phi^{x_0,R})\|_{L^2}\) can all be controlled by \(R^{\frac d2}\).