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In mathematics, the '''T(1) theorem''', first proved by {{harvtxt|David|Journé|1984}}, describes when an operator ''T'' given by a  kernel can be extended to  a bounded linear operator on the [[Hilbert space]] ''L''<sup>2</sup>('''R'''<sup>''n''</sup>). The name ''T''(1) theorem refers to a condition on the [[Distribution (mathematics)|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 '''R'''<sup>''n''</sup> to [[Distribution (mathematics)|tempered distribution]]s, 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 ''L''<sup>2</sup>('''R'''<sup>''n''</sup>) if and only if the following conditions are satisfied:
*''T''(1) is of [[Bounded mean oscillation|bounded mean oscillation]] (where ''T'' is extended to an operator on bounded smooth functions, such as 1).
*''T''<sup>*</sup>(1) is of bounded mean oscillation, where ''T''<sup>*</sup> is the [[Adjoint|adjoint]] of ''T''.
*''T'' is weakly bounded, a weak condition that is easy to verify in practice.

==References==
*{{Citation | last1=David | first1=Guy |  last2=Journé | first2=Jean-Lin | title=A boundedness criterion for generalized Calderón-Zygmund operators | jstor=2006946 | mr=763911  | year=1984 | journal=[[Annals of Mathematics]] |series=Second Series | issn=0003-486X | volume=120 | issue=2 | pages=371–397 | doi=10.2307/2006946}}
*{{Citation | last1=Grafakos | first1=Loukas | title=Modern Fourier analysis | publisher=[[Physics:Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | edition=2nd | series=Graduate Texts in Mathematics | isbn=978-0-387-09433-5 | doi=10.1007/978-0-387-09434-2 | mr=2463316  | year=2009 | volume=250}}

{{Functional analysis}}

[[Category:Theorems in functional analysis]]

{{Sourceattribution|T(1) theorem}}