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

作者：Frey Dorothee^*; Kunstmann Peer Christian

来源：Mathematische Annalen, 2013, 357(1): 215-278.

DOI：10.1007/s00208-013-0901-x

Let be a space of homogeneous type and let be a sectorial operator with bounded holomorphic functional calculus on . We assume that the semigroup satisfies Davies-Gaffney estimates. Associated with are certain approximations of the identity. We call an operator a non-integral operator if compositions involving 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 in Caldern-Zygmund theory. In this paper, we show, under the additional assumption that a vertical Littlewood-Paley-Stein square function associated with is bounded on , that a non-integral operator is bounded on if and only if and . Here, and denote the recently defined spaces associated with that generalize the space of John and Nirenberg. Generalizing a recent result due to F. Bernicot, we show a second version of a -Theorem under weaker off-diagonal estimates, which gives a positive answer to a question raised by him. As an application, we prove -boundedness of a paraproduct operator associated with . We moreover study criterions for a -Theorem to be valid.

• 出版日期2013-9