Local Tb Theorems and applications in PDE

A Tb Theorem is a boundedness criterion for singular integrals, which allows the L2 boundedness of a singular integral operator T to be deduced from sufficiently good behavior of T on some suitable non-degenerate test function b. However, in some PDE applications, including, for example, the solution of the Kato problem for square roots of divergence form elliptic operators, it may be easier to test the operator T locally (say on any given dyadic cube Q), on a test function bQ that depends upon Q, rather than on a single, globally defined b. Or to be more precise, in the applications, it may be easier to find a family of bQ’s for which TbQ is locally well behaved, than it is to find a single b for which Tb is nice globally. In this lecture, we’ll discuss some versions of local Tb theorems, as well as some applications to PDE.