Schur test

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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 , , :

If there exist 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 .[2]

Common usage and Young's inequality[edit]

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:[3]

if

where satisfies , for some , then the operator extends to a continuous operator , with

Proof[edit]

Using the Cauchy–Schwarz inequality and the inequality (1), we get:

Integrating the above relation in , using Fubini's Theorem, and applying the inequality (2), we get:

It follows that for any .

See also[edit]

References[edit]

  1. ^ 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.
  2. ^ I. Schur, Bemerkungen zur Theorie der Beschränkten Bilinearformen mit unendlich vielen Veränderlichen, J. reine angew. Math. 140 (1911), 1–28.
  3. ^ Theorem 0.3.1 in: C. D. Sogge, Fourier integral operators in classical analysis, Cambridge University Press, 1993. ISBN 0-521-43464-5