In this paper the authors consider a systematic treatment of multilinear Calderón-Zygmund operators introduced earlier in the papers of Coifman and Meyer and ofM. Lacey andC. Thiele [Ann. Math. (2) 146, 693-724 (1997;Zbl 0914.46034); ibid. 149, 475-496 (1999;Zbl 0934.42012)]. The first main result reads as follows: Let \(m\)-linear operators be \(T: [{\mathcal S}(\mathbb{R}^n)]^m\to{\mathcal S}'(\mathbb{R}^n)\) for which there is a function \(K\) defined away from the diagonal \(x= y_1=\cdots= y_m\) in \((\mathbb{R}^n)^{m+1}\) satisfying \[|K(y_0,y_1,\dots, y_m)|\leq {c_{n,m}A\over (\sum^m_{k,l=0}|y_k- y_l|)^{nm}}\] and \[|K(y_0,\dots, y_j,\dots,y_m)- K(y_0,\dots, y_j',\dots, y_m)|\leq {c_{n,m}A|y_j- y_j'|^\varepsilon\over (\sum^m_{k,l=0}|y_k- y_l|)^{nm+ \varepsilon}},\] whenever \(0\leq j\leq m\) and \(|y_j- y_j'|\leq{1\over 2}\max_{0\leq k\leq m}|y_j- y_k|\). Let \(q_j\in [1,\infty)\) be given numbers with \(1/q= \sum^m_{j=1} 1/q_j\). Suppose that \(T\) maps \(L^{q_1,1}\times\cdots\times L^{q_m,1}\) into \(L^{q,\infty}\) if \(q> 1\) or \(L^1\) if \(q= 1\). Then for any \(p_j\in [1,\infty]\) such that \(1/m\leq p< \infty\), \(T\) extends to a bounded map from \(L^{p_1}\times\cdots\times L^{p_m}\) into \(L^p\) if all \(p_j> 1\) and into \(L^{p,\infty}\) if some \(p_j= 1\). If some \(p_k= \infty\), \(L^{p_k}\) should be replaced by \(L^\infty_c\). Moreover, \(T\) extends to a bounded map from \(L^\infty\times\cdots\times L^\infty\) to BMO. Next, the authors obtain the version of the multilinear T1 theorem byG. David andJ.-L. Journé [Ann. Math. (2) 120, 371-397 (1984;Zbl 0567.47025)]. It is proved that if \(T(e_{\xi_1},\dots, e_{\xi_m})\) and \(T^{*j}(e_{\xi_1},\dots, e_{\xi_m})\) \((\xi_1,\dots, \xi_m\in \mathbb{R}^n\), \(1\leq j\leq m)\) are bounded subsets of BMO, then \(T\) has a bounded extension from \(L^{q_1}\times\cdots\times L^{q_m}\) into \(L^q\) if \(1< q,q_j<\infty\). Here \(j\)th transpose \(T^{*j}\) of \(T\) is defined via \[\langle T^{*j}(f_1,\dots, f_m), h\rangle= \langle T(f_1,\dots, f_{j-1}, h,f_{j+1},\dots, f_m),f_j\rangle\] for all \(f_1,\dots, f_m\), \(g\) in \({\mathcal S}(\mathbb{R}^n)\). This multilinear Calderón-Zygmund theory is applied to obtain some new continuity results for multilinear translation invariant operators, multlinear pseudodifferential operators, and multilinear multipliers.

Reviewer: Sunggeum Hong (Seoul)

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