A $$T(1)$$ -Theorem for non-integral operators

Abstract

Let \(X\) be a space of homogeneous type and let \(L\) be a sectorial operator with bounded holomorphic functional calculus on \(L^2(X)\). We assume that the semigroup \(\{e^{-tL}\}_{t>0}\) satisfies Davies–Gaffney estimates. Associated with \(L\) are certain approximations of the identity. We call an operator \(T\) a non-integral operator if compositions involving \(T\) and these approximations satisfy certain weighted norm estimates. The Davies–Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on \(T\) in Calderon–Zygmund theory. In this paper, we show, under the additional assumption that a vertical Littlewood–Paley–Stein square function associated with \(L\) is bounded on \(L^2(X)\), that a non-integral operator \(T\) is bounded on \(L^2(X)\) if and only if \(T(1) \in BMO_L(X)\) and \(T^{*}(1) \in BMO_{L^{*}}(X)\). Here, \(BMO_L(X)\) and \(BMO_{L^{*}}(X)\) denote the recently defined \(BMO(X)\) spaces associated with \(L\) that generalize the space \(BMO(X)\) of John and Nirenberg. Generalizing a recent result due to F. Bernicot, we show a second version of a \(T(1)\)-Theorem under weaker off-diagonal estimates, which gives a positive answer to a question raised by him. As an application, we prove \(L^2(X)\)-boundedness of a paraproduct operator associated with \(L\). We moreover study criterions for a \(T(b)\)-Theorem to be valid.