For , define

prove that is a bounded operator.

**Proof by Hardy’s inequality**

Hardy’s inequality: Let and

1.

2.

Split as . Then

Here we used Hardy’s inequality case (1) with .

Here we used Hardy’s inequality case (2) with .

So .

**Proof by Minkowski integral inequality**

Changing variable by

So by Minkowski Integral Inequality

Since , . We are done.

Pay attention to the tricks using here. The proof by Hardy’s inequality is hinted from Prof. Chanillo’s notes. For Minkowski method, see stein’s book.