A T(1)-Theorem in relation to a semigroup of operators and applications to new paraproducts

TL;DR

This paper extends the T(1)-theorem framework by introducing BMO spaces associated with semigroups of operators, leading to new boundedness results for paraproducts and a version of the theorem on doubling Riemannian manifolds.

Contribution

It develops a generalized T(1)-theorem replacing BMO with BMO_L linked to semigroups, and applies it to new paraproducts and Riemannian manifolds.

Findings

Established boundedness of new paraproducts built on semigroups.

Extended T(1)-theorem to doubling Riemannian manifolds.

Introduced BMO_L spaces for broader Hardy and BMO theory.

Abstract

In this work, we are interested to develop new directions of the famous T(1)-theorem. More precisely, we develop a general framework where we look for replacing the John-Nirenberg space BMO (in the classical result) by a new BMO_{L}, associated to a semigroup of operators (e^{-tL})_{t>0}. These new spaces BMO_L (including BMO) have recently appeared in numerous works in order to extend the theory of Hardy and BMO space to more general situations. Then we give applications by describing boundedness for a new kind of paraproducts, built on the considered semigroup. In addition we obtain a version of the classical T(1) theorem for doubling Riemannian manifolds.