This article is about the integral inequality. For the algebraic inequality in 3 variables, see
Schur's inequality.
In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the
operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem).
Here is one version.[1] Let
be two measurable spaces (such as
). Let
be an integral operator with the non-negative Schwartz kernel
,
,
:
![{\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/0178db8271168ff87f5b6f370b2a2cdb19aa4788)
If there exist real functions
and
and numbers
such that
![{\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq \alpha p(x)}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/3f850a189b43e5a8663ea39160fd117b371f362e)
for almost all
and
![{\displaystyle (2)\qquad \int _{X}p(x)K(x,y)\,dx\leq \beta q(y)}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/aa6560ec5433b546f8e735d93134f948443ea393)
for almost all
, then
extends to a continuous operator
with the operator norm
![{\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}\leq {\sqrt {\alpha \beta }}.}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/775701e55707333725e7e94f51cec71afc18a9fd)
Such functions
,
are called the Schur test functions.
In the original version,
is a matrix and
.[2]
Common usage and Young's inequality
[edit]
A common usage of the Schur test is to take
Then we get:
![{\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}^{2}\leq \sup _{x\in X}\int _{Y}|K(x,y)|\,dy\cdot \sup _{y\in Y}\int _{X}|K(x,y)|\,dx.}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/2f11e4ba9b762ca1e686023d259992c018c01155)
This inequality is valid no matter whether the Schwartz kernel
is non-negative or not.
A similar statement about
operator norms is known as Young's inequality for integral operators:[3]
if
![{\displaystyle \sup _{x}{\Big (}\int _{Y}|K(x,y)|^{r}\,dy{\Big )}^{1/r}+\sup _{y}{\Big (}\int _{X}|K(x,y)|^{r}\,dx{\Big )}^{1/r}\leq C,}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/806073b8761de5055cf841c6bbaed6995eace0ec)
where
satisfies
, for some
, then the operator
extends to a continuous operator
, with
Using the Cauchy–Schwarz inequality and inequality (1), we get:
![{\displaystyle {\begin{aligned}|Tf(x)|^{2}=\left|\int _{Y}K(x,y)f(y)\,dy\right|^{2}&\leq \left(\int _{Y}K(x,y)q(y)\,dy\right)\left(\int _{Y}{\frac {K(x,y)f(y)^{2}}{q(y)}}dy\right)\\&\leq \alpha p(x)\int _{Y}{\frac {K(x,y)f(y)^{2}}{q(y)}}\,dy.\end{aligned}}}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/22eb3d50172740dac95fbf1c1836049b6303a9e1)
Integrating the above relation in
, using Fubini's Theorem, and applying inequality (2), we get:
![{\displaystyle \Vert Tf\Vert _{L^{2}}^{2}\leq \alpha \int _{Y}\left(\int _{X}p(x)K(x,y)\,dx\right){\frac {f(y)^{2}}{q(y)}}\,dy\leq \alpha \beta \int _{Y}f(y)^{2}dy=\alpha \beta \Vert f\Vert _{L^{2}}^{2}.}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/0261d85878d65b588d499cb51f5277f2d93f05fc)
It follows that
for any
.
- ^ Paul Richard Halmos and Viakalathur Shankar Sunder, Bounded integral operators on
spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 96., Springer-Verlag, Berlin, 1978. Theorem 5.2.
- ^ I. Schur, Bemerkungen zur Theorie der Beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. reine angew. Math. 140 (1911), 1–28.
- ^ Theorem 0.3.1 in: C. D. Sogge, Fourier integral operators in classical analysis, Cambridge University Press, 1993. ISBN 0-521-43464-5