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A $$T(1)$$-theorem in relation to a semigroup of operators and applications to new paraproducts. (English) Zbl 1281.46028
The famous $$T(1)$$-theorem of G. David and J.-L. Journé [Ann. Math. (2) 120, 371–397 (1984; Zbl 0567.47025)] characterizes the $$L^2({\mathbb R}^n)$$-boundedness of Calderón-Zygmund operators. It claims that for such a linear operator $$T$$, if it satisfies a weak boundedness property, then $$T$$ is bounded on $$L^2({\mathbb R}^n)$$ if and only if $$T(1)$$ and $$T^*(1)$$ belong to the John-Nirenberg space $$BMO$$. The author considers a more general situation. Specifically, let $$(M,d,\mu)$$ be a Riemannian manifold with doubling measure. Let $$L$$ be an operator of order $$m$$ acting on it. Examples of such semigroups include second-order elliptic operators and Laplacian operators on a manifold. Consider the semigroup $$\left(e^{-t L}\right)_{t >0}$$. The corresponding $${ BMO}_L$$ space may be defined as the set of functions $$f$$ such that $\sup_{t>0}\sup_Q\frac{1}{\mu(Q)}\int\limits_Q \left|f-e^{-tL}f\right|d\mu < \infty.$ It is also assumed that $$L(1)=0=L^*(1)$$ and that the manifold $$M$$ satisfies the Poincaré inequality.
The main result of the paper states the following: Let $$T$$ be a linear operator, weakly continuous on $$L^2(M)$$ and admitting “off-diagonal decay relative to cancellation built with the semigroup”. If $$T(1)\in BMO_L$$ and $$T^*(1)\in BMO_{L^*}$$, then $$T$$ admits a bounded extension in $$L^2(M)$$.
The author gives applications of the main result by describing boundedness for a new kind of paraproduct, built on the considered semigroup. A version of the classical $$T(1)$$-theorem for doubling Riemannian manifolds is also proved.

MSC:
 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 47B38 Linear operators on function spaces (general) 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B30 $$H^p$$-spaces
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