In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L 2 → L 2 {\displaystyle L^{2}\to L^{2}} operator norm of an integral operator in terms of its Schwartz kernel (see Schwartz kernel theorem). Here is one version. Let X , Y {\displaystyle X,\,Y} be two measurable spaces (such as R n {\displaystyle \mathbb {R} ^{n}} ). Let T {\displaystyle \,T} be an integral operator with the non-negative Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)} , x ∈ X {\displaystyle x\in X} , y ∈ Y {\displaystyle y\in Y} :
T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y . {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy.}
If there exist real functions p ( x ) > 0 {\displaystyle \,p(x)>0} and q ( y ) > 0 {\displaystyle \,q(y)>0} and numbers α , β > 0 {\displaystyle \,\alpha ,\beta >0} such that
( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) {\displaystyle (1)\qquad \int _{Y}K(x,y)q(y)\,dy\leq \alpha p(x)}
for almost all x {\displaystyle \,x} and
( 2 ) ∫ X p ( x ) K ( x , y ) d x ≤ β q ( y ) {\displaystyle (2)\qquad \int _{X}p(x)K(x,y)\,dx\leq \beta q(y)}
for almost all y {\displaystyle \,y} , then T {\displaystyle \,T} extends to a continuous operator T : L 2 → L 2 {\displaystyle T:L^{2}\to L^{2}} with the operator norm
‖ T ‖ L 2 → L 2 ≤ α β . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}\leq {\sqrt {\alpha \beta }}.}
Such functions p ( x ) {\displaystyle \,p(x)} , q ( y ) {\displaystyle \,q(y)} are called the Schur test functions. In the original version, T {\displaystyle \,T} is a matrix and α = β = 1 {\displaystyle \,\alpha =\beta =1} .
Common usage and Young's inequality A common usage of the Schur test is to take p ( x ) = q ( y ) = 1. {\displaystyle \,p(x)=q(y)=1.} Then we get:
‖ T ‖ L 2 → L 2 2 ≤ sup x ∈ X ∫ Y | K ( x , y ) | d y ⋅ sup y ∈ Y ∫ X | K ( x , y ) | d x . {\displaystyle \Vert T\Vert _{L^{2}\to L^{2}}^{2}\leq \sup _{x\in X}\int _{Y}|K(x,y)|\,dy\cdot \sup _{y\in Y}\int _{X}|K(x,y)|\,dx.}
This inequality is valid no matter whether the Schwartz kernel K ( x , y ) {\displaystyle \,K(x,y)} is non-negative or not. A similar statement about L p → L q {\displaystyle L^{p}\to L^{q}} operator norms is known as Young's inequality for integral operators: if
sup x ( ∫ Y | K ( x , y ) | r d y ) 1 / r + sup y ( ∫ X | K ( x , y ) | r d x ) 1 / r ≤ C , {\displaystyle \sup _{x}{\Big (}\int _{Y}|K(x,y)|^{r}\,dy{\Big )}^{1/r}+\sup _{y}{\Big (}\int _{X}|K(x,y)|^{r}\,dx{\Big )}^{1/r}\leq C,}
where r {\displaystyle r} satisfies 1 r = 1 − ( 1 p − 1 q ) {\displaystyle {\frac {1}{r}}=1-{\Big (}{\frac {1}{p}}-{\frac {1}{q}}{\Big )}} , for some 1 ≤ p ≤ q ≤ ∞ {\displaystyle 1\leq p\leq q\leq \infty } , then the operator T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y {\displaystyle Tf(x)=\int _{Y}K(x,y)f(y)\,dy} extends to a continuous operator T : L p ( Y ) → L q ( X ) {\displaystyle T:L^{p}(Y)\to L^{q}(X)} , with ‖ T ‖ L p → L q ≤ C . {\displaystyle \Vert T\Vert _{L^{p}\to L^{q}}\leq C.}
Proof Using the Cauchy–Schwarz inequality and inequality (1), we get:
| T f ( x ) | 2 = | ∫ Y K ( x , y ) f ( y ) d y | 2 ≤ ( ∫ Y K ( x , y ) q ( y ) d y ) ( ∫ Y K ( x , y ) f ( y ) 2 q ( y ) d y ) ≤ α p ( x ) ∫ Y K ( x , y ) f ( y ) 2 q ( y ) d y . {\displaystyle {\begin{aligned}|Tf(x)|^{2}=\left|\int _{Y}K(x,y)f(y)\,dy\right|^{2}&\leq \left(\int _{Y}K(x,y)q(y)\,dy\right)\left(\int _{Y}{\frac {K(x,y)f(y)^{2}}{q(y)}}dy\right)\\&\leq \alpha p(x)\int _{Y}{\frac {K(x,y)f(y)^{2}}{q(y)}}\,dy.\end{aligned}}}
Integrating the above relation in x {\displaystyle x} , using Fubini's Theorem, and applying inequality (2), we get:
‖ T f ‖ L 2 2 ≤ α ∫ Y ( ∫ X p ( x ) K ( x , y ) d x ) f ( y ) 2 q ( y ) d y ≤ α β ∫ Y f ( y ) 2 d y = α β ‖ f ‖ L 2 2 . {\displaystyle \Vert Tf\Vert _{L^{2}}^{2}\leq \alpha \int _{Y}\left(\int _{X}p(x)K(x,y)\,dx\right){\frac {f(y)^{2}}{q(y)}}\,dy\leq \alpha \beta \int _{Y}f(y)^{2}dy=\alpha \beta \Vert f\Vert _{L^{2}}^{2}.}
It follows that ‖ T f ‖ L 2 ≤ α β ‖ f ‖ L 2 {\displaystyle \Vert Tf\Vert _{L^{2}}\leq {\sqrt {\alpha \beta }}\Vert f\Vert _{L^{2}}} for any f ∈ L 2 ( Y ) {\displaystyle f\in L^{2}(Y)} .
See also Hardy–Littlewood–Sobolev inequality
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