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 .