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 Calderón–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.