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# Sobolev embedding

The properties of Sobolev spaces are briefly discussed above. If is a positive integer then means' that has up to derivatives in The question naturally arises as to the sense in which these weak' derivatives correspond to old-fashioned strong' derivatives. Of course when is not an integer it is a little harder to imagine what these fractional derivatives' are. However the main result is:

Proof. By definition, means and . Suppose first that . The Fourier inversion formula shows that

Now, if then the second integral is finite. Since the first integral is the norm on we see that

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This is all for , but is dense. The estimate (9.2) shows that if in with , then in In fact in since in and in both imply that converges, so

Notice here the precise meaning of When identifying with the corresponding tempered distribution, the values on any set of measure zero are lost'. Thus as functions (9.1) means that each has a representative

We can extend this to higher derivatives by noting that

Proof. First it is enough to show that each defines a continuous linear map

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since then (9.3) follows by composition.

If then means . Since , and

we conclude that and

Applying this result we see

Proof. If , then . Thus the weak derivatives' are continuous. Still we have to check that this means that is itself times continuously differentiable. In fact this again follows from the density of in . The continuity in (9.3) implies that if in , , then in (using its completeness). However as before, so .

In particular we see that

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These functions are not in general Schwartz test functions.

Proof. This follows directly from (9.5) since the left side is contained in

Thus every tempered distribution is a finite sum of derivatives of continuous functions of poynomial growth.

Proof. Essentially by definition any is continuous with respect to one of the norms From the Sobolev embedding theorem we deduce that, with

This is the same as

which shows that , i.e., from Proposition 8.8,

In fact, choose and consider defined by . As in the proof of Proposition 8.14 we conclude that

Thus,17

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To get (9.9) we `commute' the factor to the inside; since I have not done such an argument carefully so far, let me do it as a lemma.

Proof. In fact it is convenient to prove a more general result. Suppose is a polynomial of a degree at most then there exist polynomials of degrees at most such that

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The lemma follows from this by taking .

Furthermore, the identity (9.11) is trivial when , and proceeding by induction we can suppose it is known whenever . Taking ,

Writing the identity for as

we may differentiate with respect to This gives

The first term on the right expands to

We may apply the inductive hypothesis to each of these terms and rewrite the result in the form (9.11); it is only necessary to check the order of the polynomials, and recall that is a polynomial of degree .

Applying Lemma 9.6 to (9.10) gives (9.9), once negative powers of are absorbed into the continuous functions. Then (9.8) follows from (9.9) and Leibniz's formula.

Next: Differential operators. Up: Lecture notes for 18.155, Previous: Fourier inversion   Contents
Richard B. Melrose 2003-02-18