T(1) theorem

In mathematics, the T(1) theorem, first proved by David & Journé (1984), describes when an operator T given by a kernel can be extended to a bounded linear operator on the Hilbert space L2(Rn).

The name T(1) theorem refers to a condition on the distribution T(1), given by the operator T applied to the function 1.

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(1) theorem states that T can be extended to a bounded operator on the Hilbert space L2(Rn) if and only if the following conditions are satisfied: