Now, recopying some handwritten notes:

For the other direction: when

Corollary (of the Proof)For , then

- For all
- For

*Proof:* We will prove assertion 1, first beginning with . Note that , so the first half of the regular square function proof shows that

noting that is the convolution operator associated to the multiplier . So the above says

Now substitute . Note that the argument above also works for . Now, for a unit vector in , then

Now choose to exhibit on the left hand side.

Now we turn to assertion 2, beginning with .

Therefore, the right hand side of assertion 2 is that the right hand side of assertion 1. Next we show the reverse (namely, replace “” from the previous sentence by “”).

*Now, Michael Bateman is lecturing.*

Theorem (Fractional Product Rule, Christ–Weinstein 1991)For and with , , then

*Proof:* To prove this, we first use part 1 of the corollary to write the left hand side as a Littlewood–Paley sum:

For each , we have

This implies

Now we use the estimate to get

Thus,

Using Hölder and then assertion 2 of the corollary, then

Theorem (Fractional chain rule, Weinstein, 1991)Suppose satisfieswhere . If , then

whenever .

RemarkRecall the original chain rule: if exists, then .

Example, then

Here , noting that if then .

*Proof:* Recalling that , then

Let’s analyze :

and

We claim that , the proof of which we break into two cases: and . For the first case, we change “” to “” and prove the estimate without . For the second case, we use the derivative of :

So,

which proves the claim. Now

The first term is bounded above by and the second term can be handled similarly. The third term is bounded above by

(if is Schwartz, then is still Schwartz).

Now, consists of terms like

and terms like

and we claim . Given this claim, we are left with expressions like . Now, using that ,

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