Recall our previous computation

Example (Newton), thenNow,

Thus is then has solution

by Hardy–Littlewood–Sobolev (note ).

DefinitionLet be the linear transformation (defined initially for ) by

Here .

Theorem (Sobolev Embedding)For , , and , then

*Proof:* By duality

We’re done if we know , but this is exactly what HLS says!

If , it is natural to want to replace by . It would suffice to show that form bounded Fourier multipliers on when :

i.e.,

We will do this next. These convolution operators are called the Riesz transforms. E.g., gives , which is the Hilbert transform.

Additionally -boundedness of then

obeys

and so

DefinitionA Calderón–Zygmund (convolution) kernel is a function that obeys

- ;
- for all ;
- for all .

The Riesz transforms obey these axioms. For property (3), it is easiest to use the following lemma.

LemmaIf obeys properties (1), (2), and

then is a Calderón–Zygmund kernel.

Wait do we really need (1) and (2)? *Proof:* The fundamental theorem of calculus gives

Now

Coming up in the next lecture

PropositionIf is a Calderón–Zygmund kernel, then “”. Namely, obeys uniformly in and .

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