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Polynomial growth estimates for multilinear singular integral operators. (English) Zbl 0645.42017
The authors investigate boundedness of Calderon commutators of the form $$\int L(x,y)f(y)dy$$ where $$L(x,y)=K(x-y)[\int^{1}_{0}a(tx+(1- t)y)dt]^ n$$ and $$a\in L^{\infty}(R^ d)$$ is complex-valued. They want to prove$$\| \int L(x,y)f(y)\|_ 2\leq Cn^ M\| a\|^ n_{\infty}\| f\|_ 2.$$ The key to the proof is an extension of the T1 theorem to $$\delta$$-multilinear singular integral forms. The idea is that an n-linear form allows one to define $$U_ i1$$ for $$1\leq i\leq n$$ and then boundedness of the singular integral form is controlled by a suitable weak boundedness property and $$U_ i1\in BMO$$ for $$1\leq i\leq n$$. They also give a Carleson measure version for kernels that form an $$\epsilon$$-family $$| S_ t(x,y)| \leq Cw_{\epsilon,t}(x-y)| S_ t(x,y)-S_ t(x,z)| \leq C(| y-z| /(t+| x- y|))^{\epsilon}w_{\epsilon,t}(x-y),$$ for $$| y-z| \leq (t+| x-y|),where$$ $$w_{\epsilon,t}(x)=t^{\epsilon}/(t^{d+\epsilon}+| x|^{d+\epsilon}),$$ $$x\in R^ d$$. The relation of singular integral operators and the $$\epsilon$$-family in that the T1 theorem requires one to decompose $$T=\int^{\infty}_{0}Q_ tTP_ tdt/t+\int^{\infty}_{0}P_ tTQ_ tdt/t,$$ where Q is a radial nonnegative function with integral 1, $$P_ tf=\phi_ t*f$$, $$\phi_ t(x)=t^{-d}\phi (x/t),$$ and $$Q_ t=-t\partial P_ t/\partial t$$. Then the $$Q_ tTP_ t$$ form an $$\epsilon$$-family. Given a suitable restricted $$\epsilon$$-family, $$T=\int^{\infty}_{0}T_ tdt/t$$ defines a weakly bounded singular integral operator. The Carleson measure T1 theorem says that an $$\epsilon$$-family $$S_ t$$ is bounded in the sense that $$\int^{\infty}_{0}\| S_ tf\|^ 2_ 2dt/t\leq C\| f\|^ 2_ 2$$ iff $$F(x,t)=S_ t1(x)$$ is a Carleson function $$(1/| B| \int_{B\times [0,r]}| F(x,t)|^ 2dx(dt/t))^{1/2}\leq C,$$ where r is the radius of the ball B. As a consequence of their methods the authors find that for the Calderon commutator $T_ k(a,...,a,f)(x)=\lim_{\epsilon \to 0}\int_{| x-y| >\epsilon}(\frac{A(x)-A(y)}{x-y})^ K\frac{f(y)}{x-y}dy,$ where $$A'=a$$, one has the estimate $$\| T_ k(a,...,a,f)\|_{L^ 2}\leq D_{\delta}(1+k)^{1+\delta}\| a\|^ k_{\infty}\| f\|_ 2$$ for any $$\delta >0$$.
Reviewer: R.Johnson

##### MSC:
 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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