Schur test

In Mathematical Analysis,the Schur Test (named after German mathematician Issai Schur)is the name for the bound on the $L^2\; o\; L^2$ operator normof an integral operator in terms of its Schwartz kernel(see Schwartz kernel theorem).

The following result of Issai Schuris described in [Paul Richard Halmos and Viakalathur Shankar Sunder (1978)."Bounded integral operators on $L^\{2\}$ spaces",Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 96.Springer-Verlag, Berlin, 1978.Theorem 5.2.] . Let "X" and "Y" be two measurable spaces (such as $mathbb\{R\}^n$, or see Measurable space), and let "T" be an integral operator with the non-negative Schwartz kernel $K(x,y)$,$xin\; X$, $yin\; Y$:

:$T\; f(x)=int\_Y\; K(x,y)f(y),dy.$

If there exist functions $p(x)>0$ and $q(x)>0$and numbers $alpha>0$, such that

:$(1)qquadint\_Y\; K(x,y)q(y),dylealpha\; p(x)$for almost all "x", and:for almost all "y",then "T" is continuous from $L^2(Y)$ to $L^2(X)$,with the norm bounded by

:

Such functions$p(x)$,$q(x)$are called the Schur test functions.

The original result appeared in [I. Schur (1911)."Bemerkungen zur Theorie der Beschränkten Bilinearformenmit unendlich vielen Veränderlichen",J. reine angew. Math. 140 (1911), 1-28.] for "T" a matrix and with .

Usage

The most common usage is to take$p(x)=q(x)=1.$Then we get:

:$Vert\; TVert^2\_\{L^2\; o\; L^2\}lesup\_\{xin\; X\}int\_Y|K(x,y)|dycdotsup\_\{yin\; Y\}int\_X|K(x,y)|dx.$This inequality is sometimes called the Schur inequalityor Young's inequality.It is valid no matter whether the Schwartz kernel $K(x,y)$is non-negative or not.

Proof

Using the Cauchy-Schwarz inequality andthe inequality (1),we get:

:$$
TF(x)|^2=left|int_Y K(x,y)f(y),dy ight|^2leint_Y K(x,y)q(y),dyint_Y frac{K(x,y)f(y)^2}{q(y)} dylealpha p(x)int_Y frac{K(x,y)f(y)^2}{q(y)}dy.

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

:

It follows thatfor any $fin\; L^2(Y)$.

References