PropositionDefine with being a Calderón–Zygmund kernel. Thenuniformly as . Consequently,

extends from to a bounded operator on .

*Proof:* For the first part, we just need to show that uniformly in . Let’s compute (more honestly we ought to do some mollification near infinity)

So

where and are the balls coming from two displays previous. Expanding each term in the right hand side of the previous display

If and

and so . Thus is Cauchy in and so it converges in . To extend to general (i.e., not Schwartz), we approximate and use uniformly in .

TheoremIf is a Calderón–Zygmund kernel, then

- for .

RemarkThe fact that we are dealing with a convolution operator was essentially to prove -boundedness. Once we know -boundedness for a more general operatorthen the proof the Theorem goes through requiring only [the analog of property (3)]

and

Lemma (Calderón–Zygmund decomposition)Let . Given we can decompose where is supported on disjoint dyadic cubes with and

- a.e.
- .

Moreover, .

*Proof:* Run a stopping time argument on the dyadic cubes with stopping rule

This yields a collection of disjoint “stopping cubes” . Note

Define

and . Note that , and the rest follows as before.

*Proof:* (of Theorem) Let’s denote the operator by . We need to bound , for which we do a Calderón–Zygmund decomposition at height and so bound and . Now

Now let (draw a picture and inscribe some circles and their doubles) and so

It remains to estimate

Pick an , then

Note that

Thus

Consequently,

Altogether, we have shown

but we also know that . Thus Marcinkiewicz implies boundedness on for . For we argue by duality,

where the final inequality follows by applying the first part to instead of .

RemarkTypical interesting Calderón–Zygmund operators are not bounded on or . [In fact, if is bounded on or then is a finite measure.] Some examples includeIf , then is bounded analytic on . If , then is bounded analytic on . So, splitting into analytic in / parts.

For next time

Theorem (Mikhlin multiplier theorem, or possibly Marcinkiewicz, or Hörmander)If obeysThen

is bounded on for .

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