T(1) theorem

In mathematics, the T theorem, first proved by, describes when an operator T given by a kernel can be extended to a bounded linear operator on the Hilbert space L². The name T theorem refers to a condition on the distribution T, given by the operator T applied to the function 1.

Statement

Suppose that T is a continuous operator from Schwartz functions on Rⁿ to tempered distributions, so that T is given by a kernel K which is a distribution. Assume that the kernel is standard, which means that off the diagonal it is given by a function satisfying certain conditions.
Then the T theorem states that T can be extended to a bounded operator on the Hilbert space L² if and only if the following conditions are satisfied:

T is of bounded mean oscillation.
T^* is of bounded mean oscillation, where T^* is the adjoint of T.
T is weakly bounded, a weak condition that is easy to verify in practice.