“T(1) theorem”的意思、由来-开放百科全书

词条

T(1) theorem

释义

Statement
References

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 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

{{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=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}}

1 : Theorems in functional analysis

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