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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 M. Lacey and 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 $T:{\left[𝒮\left({ℝ}^{n}\right)\right]}^{m}\to {𝒮}^{\text{'}}\left({ℝ}^{n}\right)$ for which there is a function $K$ defined away from the diagonal $x={y}_{1}=\cdots ={y}_{m}$ in ${\left({ℝ}^{n}\right)}^{m+1}$ satisfying

$|K\left({y}_{0},{y}_{1},\cdots ,{y}_{m}\right)|\le \frac{{c}_{n,m}A}{\left({\sum }_{k,l=0}^{m}|{y}_{k}-{y}_{l}{|\right)}^{nm}}$

and

$|K\left({y}_{0},\cdots ,{y}_{j},\cdots ,{y}_{m}\right)-K\left({y}_{0},\cdots ,{y}_{j}^{\text{'}},\cdots ,{y}_{m}\right)|\le \frac{{c}_{n,m}A{|{y}_{j}-{y}_{j}^{\text{'}}|}^{\epsilon }}{\left({\sum }_{k,l=0}^{m}|{y}_{k}-{y}_{l}{|\right)}^{nm+\epsilon }},$

whenever $0\le j\le m$ and $|{y}_{j}-{y}_{j}^{\text{'}}|\le \frac{1}{2}{max}_{0\le k\le m}|{y}_{j}-{y}_{k}|$. Let ${q}_{j}\in \left[1,\infty \right)$ be given numbers with $1/q={\sum }_{j=1}^{m}1/{q}_{j}$. Suppose that $T$ maps ${L}^{{q}_{1},1}×\cdots ×{L}^{{q}_{m},1}$ into ${L}^{q,\infty }$ if $q>1$ or ${L}^{1}$ if $q=1$. Then for any ${p}_{j}\in \left[1,\infty \right]$ such that $1/m\le p<\infty$, $T$ extends to a bounded map from ${L}^{{p}_{1}}×\cdots ×{L}^{{p}_{m}}$ 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 ${L}_{c}^{\infty }$. Moreover, $T$ extends to a bounded map from ${L}^{\infty }×\cdots ×{L}^{\infty }$ to BMO. Next, the authors obtain the version of the multilinear T1 theorem by G. David and J.-L. Journé [Ann. Math. (2) 120, 371-397 (1984; Zbl 0567.47025)]. It is proved that if $T\left({e}_{{\xi }_{1}},\cdots ,{e}_{{\xi }_{m}}\right)$ and ${T}^{*j}\left({e}_{{\xi }_{1}},\cdots ,{e}_{{\xi }_{m}}\right)$ $\left({\xi }_{1},\cdots ,{\xi }_{m}\in {ℝ}^{n}$, $1\le j\le m\right)$ are bounded subsets of BMO, then $T$ has a bounded extension from ${L}^{{q}_{1}}×\cdots ×{L}^{{q}_{m}}$ into ${L}^{q}$ if $1. Here $j$th transpose ${T}^{*j}$ of $T$ is defined via

$〈{T}^{*j}\left({f}_{1},\cdots ,{f}_{m}\right),h〉=〈T\left({f}_{1},\cdots ,{f}_{j-1},h,{f}_{j+1},\cdots ,{f}_{m}\right),{f}_{j}〉$

for all ${f}_{1},\cdots ,{f}_{m}$, $g$ in $𝒮\left({ℝ}^{n}\right)$. 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:
 42B20 Singular and oscillatory integrals, several variables