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


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