Schur’s Test

Theorem 1.

(Schur’s Test) Let (X,μ) be a measure spaceMathworldPlanetmath (μ a positive measure). Let K be a positive, measurable functionMathworldPlanetmath on X×X. Define the operator

Tf(x) :=XK(x,y)f(y)𝑑μ(y),xX

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


with p-1+q-1=1, then ||T||M in Lp(X,dμ).


Let fLp(X,dμ). We have

|Tf(x)| Xh(y)h(y)-1|f(y)|K(x,y)𝑑μ(y)

hence by Hoelder’s inequalityMathworldPlanetmath

|Tf(x)| [XK(x,y)h(y)q𝑑μ(y)]1q[XK(x,y)h(y)-p|f(y)|p𝑑μ(y)]1p

By the first inequality in the assumptionPlanetmathPlanetmath we have

|Tf(x)| M1qh(x)[XK(x,y)h(y)-p|f(y)|p𝑑μ(y)]1p

Evaluating ||Tf||pp by Fubini and the second inequality in the assumption we obtain

X|Tf(x)|p𝑑μ(x) MpX|f(y)|p𝑑μ(y)

This completesPlanetmathPlanetmathPlanetmathPlanetmath the proof. ∎

A noted special case is Young’s Inequality

Corollary 1.


Let K:Rn×RnC be Borel-measurable such that there is a constant C>0 with


For fLp(Rn) (1p+) define

T(f)(x) :=nK(x,y)f(y)𝑑λn(y)

Then ||Tf||pC||f||p.


  • (Hedenmalm 2000) H. Hedenmalm, Boris Korenblum, Kehe Zhu Theory of Bergman spaces, Springer Verlag, New York, 2000
Title Schur’s Test
Canonical name SchursTest
Date of creation 2013-03-22 19:01:19
Last modified on 2013-03-22 19:01:19
Owner karstenb (16623)
Last modified by karstenb (16623)
Numerical id 6
Author karstenb (16623)
Entry type Theorem
Classification msc 46G99