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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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