# Local <em>Tb</em> theorems and applications in PDE

### Dorina Mitrea

University of Missouri, Columbia, United States

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## Abstract

A *Tb* theorem is a boundedness criterion for singular integrals, which allows the _L_2 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 *b__Q*’s for which *Tb__Q* is locally well behaved, than it is to find a single *b* for which *Tb* is nice globally. In this lecture, we shall discuss some versions of local *Tb* theorems, as well as some applications to PDE.