Schur test


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.
Here is one version. Let be two measurable spaces . Let be an integral operator with the non-negative Schwartz kernel,, :
If there exist real functions and and numbers such that
for almost all and
for almost all, then extends to a continuous operator with the operator norm
Such functions, are called the Schur test functions.
In the original version, is a matrix and.

Common usage and Young's inequality

A common usage of the Schur test is to take Then we get:
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:
if
where satisfies, for some, then the operator extends to a continuous operator, with

Proof

Using the Cauchy–Schwarz inequality and the inequality, we get:
Integrating the above relation in, using Fubini's Theorem, and applying the inequality, we get:
It follows that for any.