2011/07/21 by Dorothee Frey, Frey, Dorothee, Peer Christian Kunstmann +1
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #math.CA #math.FA
paper · pdf · doi:10.48550/arxiv.1107.4347
51 pages
arxiv created 2011/07/21 · arxiv updated 2011/07/22
Let X be a space of homogeneous type and let L be a sectorial operator with bounded holomorphic functional calculus on L2(X). We assume that the semigroup \e-tL\t>0 satisfies Davies-Gaffney estimates. Associated to 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 to L is bounded on L2(X), that a non-integral operator T is bounded on L2(X) if and only if T(1) ∈ BMOL(X) and T∗(1) ∈ BMOL∗(X). Here, BMOL(X) and BMOL∗(X) denote the recently defined BMO(X) spaces associated to 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 L2(X)-boundedness of a paraproduct operator associated to L. We moreover study criterions for a T(b)-Theorem to be valid.