In mathematical analysis, the Schur test, named after German mathematician Issai Schur, is a bound on the L 2 → 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 be two measurable spaces (such as R n ). Let T be an integral operator with the non-negative Schwartz kernel K ( x , y ) , x ∈ X , y ∈ Y :
T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y . If there exist functions p ( x ) > 0 and q ( x ) > 0 and numbers α , β > 0 such that
( 1 ) ∫ Y K ( x , y ) q ( y ) d y ≤ α p ( x ) for almost all x and
( 2 ) ∫ X p ( x ) K ( x , y ) d x ≤ β q ( y ) for almost all y , then T extends to a continuous operator T : L 2 → L 2 with the operator norm
∥ T ∥ L 2 → L 2 ≤ α β . Such functions p ( x ) , q ( x ) are called the Schur test functions.
In the original version, T is a matrix and α = β = 1 .
Common usage and Young's inequality
A common usage of the Schur test is to take p ( x ) = q ( x ) = 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 . This inequality is valid no matter whether the Schwartz kernel K ( x , y ) is non-negative or not.
A similar statement about L p → L q operator norms is known as Young's inequality:
if
sup x ( ∫ Y | K ( x , y ) | r d y ) 1 / r + sup y ( ∫ X | K ( x , y ) | r d x ) 1 / r ≤ C , where r satisfies 1 r = 1 − ( 1 p − 1 q ) , for some 1 ≤ p ≤ q ≤ ∞ , then the operator T f ( x ) = ∫ Y K ( x , y ) f ( y ) d y extends to a continuous operator T : L p ( Y ) → L q ( X ) , with ∥ T ∥ L p → L q ≤ C .
Using the Cauchy–Schwarz inequality and the 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 . Integrating the above relation in x , using Fubini's Theorem, and applying the 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 . It follows that ∥ T f ∥ L 2 ≤ α β ∥ f ∥ L 2 for any f ∈ L 2 ( Y ) .
