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Multilinear Calderón-Zygmund theory. (English) Zbl 1032.42020
In this paper the authors consider a systematic treatment of multilinear Calderón-Zygmund operators introduced earlier in the papers of Coifman and Meyer and of {\it M. Lacey} and {\it C. 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 $T: [{\cal S}(\bbfR^n)]^m\to{\cal S}'(\bbfR^n)$ for which there is a function $K$ defined away from the diagonal $x= y_1=\cdots= y_m$ in $(\bbfR^n)^{m+1}$ satisfying $$|K(y_0,y_1,\dots, y_m)|\le {c_{n,m}A\over (\sum^m_{k,l=0}|y_k- y_l|)^{nm}}$$ and $$|K(y_0,\dots, y_j,\dots,y_m)- K(y_0,\dots, y_j',\dots, y_m)|\le {c_{n,m}A|y_j- y_j'|^\varepsilon\over (\sum^m_{k,l=0}|y_k- y_l|)^{nm+ \varepsilon}},$$ whenever $0\le j\le m$ and $|y_j- y_j'|\le{1\over 2}\max_{0\le k\le m}|y_j- y_k|$. 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 $1/m\le p< \infty$, $T$ 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 by {\it G. David} and {\it J.-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 $T^{*j}(e_{\xi_1},\dots, e_{\xi_m})$ $(\xi_1,\dots, \xi_m\in \bbfR^n$, $1\le j\le m)$ 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 ${\cal S}(\bbfR^n)$. This multilinear Calderón-Zygmund theory is applied to obtain some new continuity results for multilinear translation invariant operators, multlinear pseudodifferential operators, and multilinear multipliers.

MSC:
42B20Singular and oscillatory integrals, several variables
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References:
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