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.

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

• David, Guy; Journé, Jean-Lin (1984), "A boundedness criterion for generalized Calderón-Zygmund operators", Annals of Mathematics, Second Series, 120 (2): 371–397, doi:10.2307/2006946, ISSN 0003-486X, JSTOR 2006946, MR 0763911
• Grafakos, Loukas (2009), Modern Fourier analysis, Graduate Texts in Mathematics, vol. 250 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-0-387-09434-2, ISBN 978-0-387-09433-5, MR 2463316