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T(1) theorem
In mathematics, the T(1) 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(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.
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(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:
- T(1) is of bounded mean oscillation (where T is extended to an operator on bounded smooth functions, such as 1).
- T*(1) 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.
References
This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page.
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