Schur test

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

The following result of Issai Schur is described in [1] (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), x\in X, y\in 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 α > 0, β > 0 such that

(1) \qquad \int_Y K(x,y)q(y)\,dy\le\alpha p(x)

for almost all x, and

(2) \qquad \int_X p(x)K(x,y)\,dx\le\beta q(y)

for almost all y, then T is continuous from L2(Y) to L2(X), with the norm bounded by

\Vert T\Vert_{L^2\to L^2} \le\sqrt{\alpha\beta}.

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

[edit] Usage

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

\Vert T\Vert^2_{L^2\to L^2}\le \sup_{x\in X}\int_Y|K(x,y)|dy \cdot \sup_{y\in Y}\int_X|K(x,y)|dx.

This inequality is sometimes called the Schur inequality or Young's inequality. It is valid no matter whether the Schwartz kernel K(x,y) is non-negative or not.

[edit] Proof

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

|Tf(x)|^2=\left|\int_Y K(x,y)f(y)\,dy\right|^2 \le\int_Y K(x,y)q(y)\,dy \int_Y \frac{K(x,y)f(y)^2}{q(y)} dy \le\alpha p(x)\int_Y \frac{K(x,y)f(y)^2}{q(y)}dy.

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

\Vert T f\Vert_{L^2}^2 \le \alpha \int_Y \Big[\int_X p(x)K(x,y)\,dx\Big] \frac{f(y)^2}{q(y)}dy \le\alpha\beta \int_Y f(y)^2 dy =\Vert f\Vert_{L^2}^2.

It follows that \Vert T f\Vert_{L^2}\le\sqrt{\alpha\beta}\Vert f\Vert_{L^2} for any f\in L^2(Y).


[edit] References

[1] Paul Richard Halmos and Viakalathur Shankar Sunder (1978). Bounded integral operators on L2 spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete (Results in Mathematics and Related Areas), vol. 96. Springer-Verlag, Berlin, 1978.

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