Problem 1.Suppose is a measurable function on with for a.e. , and is a measurable function on that satisfies:

- for almost every , and
- for almost every .
Prove that the integral operator defined by

is bounded on with .

Note as a special case that if for all , and for all , then .

*Proof. * If , note that by Cauchy-Schwarz inequality we have

For any , note that by Fubini’s theorem, we have

So we get for all . So by definition of operator norm, we get

An application of the above problem is the following:

Example 2.The operator defined on

is bounded with .

*Solution. *By previous problem, it suffices to find a function so that

Define Then this is our desired function. To use the change of variable, set to get

Similary, we get

Apply the previous theorem to yield .