Schur’s Test

(Schur’s Test) Let $(X,\mu)$ be a measure space ( $\mu$ a positive measure). Let $K$ be a positive, measurable function on $X\times X$ . Define the operator

\displaystyle Tf(x)

\displaystyle:=\int_{X}K(x,y)f(y)\,d\mu(y),x\in X

If for some $1<p<\infty$ there exists a measurable, strictly positive function $h$ and a constant $M>0$ such that

	$\displaystyle\int_{X}K(x,y)h(y)^{q}\,d\mu(y)\leq Mh(x)^{q}$
	$\displaystyle\int_{X}K(x,y)h(x)^{p}\,d\mu(x)\leq Mh(y)^{p}$

with $p^{-1}+q^{-1}=1$ , then $||T||\leq M$ in $L^{p}(X,d\mu)$ .

Let $f\in L^{p}(X,d\mu)$ . We have

\displaystyle|Tf(x)|

\displaystyle\leq\int_{X}h(y)h(y)^{-1}|f(y)|K(x,y)\,d\mu(y)

hence by Hoelder’s inequality

\displaystyle|Tf(x)|

\displaystyle\leq\left[\int_{X}K(x,y)h(y)^{q}\,d\mu(y)\right]^{\frac{1}{q}}% \left[\int_{X}K(x,y)h(y)^{-p}|f(y)|^{p}\,d\mu(y)\right]^{\frac{1}{p}}

By the first inequality in the assumption we have

\displaystyle|Tf(x)|

\displaystyle\leq M^{\frac{1}{q}}h(x)\left[\int_{X}K(x,y)h(y)^{-p}|f(y)|^{p}\,% d\mu(y)\right]^{\frac{1}{p}}

Evaluating $||Tf||_{p}^{p}$ by Fubini and the second inequality in the assumption we obtain

\displaystyle\int_{X}|Tf(x)|^{p}\,d\mu(x)

\displaystyle\leq M^{p}\int_{X}|f(y)|^{p}\,d\mu(y)

This completes the proof. ∎

A noted special case is Young’s Inequality

(Young)

Let $K\colon\mathbb{R}^{n}\times\mathbb{R}^{n}\to\mathbb{C}$ be Borel-measurable such that there is a constant $C>0$ with

	$\displaystyle\sup_{x\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\|K(x,y)\|\,d\lambda^% {n}(y)\leq C$
	$\displaystyle\sup_{y\in\mathbb{R}^{n}}\int_{\mathbb{R}^{n}}\|K(x,y)\|\,d\lambda^% {n}(x)\leq C$

For $f\in L^{p}(\mathbb{R}^{n})$ ( $1\leq p\leq+\infty$ ) define

\displaystyle T(f)(x)

\displaystyle:=\int_{\mathbb{R}^{n}}K(x,y)f(y)\,d\lambda^{n}(y)

Then $||Tf||_{p}\leq C||f||_{p}$ .

References

(Hedenmalm 2000) H. Hedenmalm, Boris Korenblum, Kehe Zhu Theory of Bergman spaces, Springer Verlag, New York, 2000