Schur’s Test

Theorem 1.

(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 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 $L^{p}(X,d\mu)$.

Proof.

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

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

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

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

A noted special case is Young’s Inequality

Corollary 1.

(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
Title Schur’s Test SchursTest 2013-03-22 19:01:19 2013-03-22 19:01:19 karstenb (16623) karstenb (16623) 6 karstenb (16623) Theorem msc 46G99