The Fourier transform in spaces of distributions

To prove (iii), we must show that the mapping f - f is surjective. Let h E S be given, and set hl(x) =

J

(27r)

n. Then

(ix) hl (x) = (2r)'

h(-x),

and hl E S by (ii). We now compute its Fourier transform.

hl (.\) = Je1'hi(s)dx

_{= (2Ir)n}

Ji(_x)eitAdx

=

(21r)n

J(x)e_dx.

By (i), hl = h. This shows that the Fourier transform is surjective. The inverse transform, given by (ix), is continuous by (ii). Both the isomorphism S -- S and its inverse are continuous: it is thus a topological isomorphism.

Applying the Fourier isomorphism to formula (iv), which has already been proved, gives (v).

Since f, g E S C L1, 1.6.2 can be applied and (f *g)^ = fg. It is clear

that the product of two functions in S is in S: if f E S and g E S, then

f g E S. It follows that f * 9 E S. This proves the first part of (vii), and the second part follows from (vi) by the Fourier isomorphism. 0

(ii) Multiplication by a polynomial P of degree k is a continuous operation on S.

Since

IIP(z)f(x)II,n,r <- clIf II.n+k.r, where c = c(P), multiplication by a polynomial on S' can be defined by

(Pf,I) = (f,P1).

(iii) S is an algebra: the product of two functions in S is a function in S. That S' is an S-module follows from the formula

(hf,1) = (f, hl), df E S, where 1 and h are fixed elements of S' and S, respectively.

4.3.3 The weak topology on S'

Definition. A sequence In E S' is said to converge weakly to to if (f,1n) converges to (f, to), V f E S.

Proposition. The operations defined in 4.9.2 are continuous in the weak topology on S'.

In particular, if In to weakly, then

Ox1In ^8x1!0.

In other words, the differentiation operator is a continuous operator on S in the topology of weak convergence of sequences.

PROOF. We prove this for differentiation:

/ \

\Or

ln/

Since 4 E S if f E S, the right-hand side converges to

C-

fl,to>. o

4.3.4 Theorem (Laurent Schwartz). Let a mapping FS' : S' - S' be

defined by setting

(f, rs'l) = (1,1).

Then FS' is an isomorphism from S' onto S', mapping weakly convergent sequences to weakly convergent sequences.

Moreover, FS' can be restricted to L' and L2 by means of the inclusions Ll C S', L2 C S'. The restriction of FS' to L' gives the Fourier integral;

the restriction of FS' to L2 gives the Fourier-Plancherel transform.

Finally, the inverse of FS' is given by

YS" (u) = FS' (u), du E S'.

REMARK. If A is a positive measure satisfying 4.3.1(1), YS, (A) is defined even though the integral µ(t) might diverge for every t.

PROOF. Fixing I E S' and setting

'P(f) = (f,1),

we obtain a linear functional on S which, as the composition of continuous map- pings, is itself continuous. Hence there exists ii E S' such that ap(f) = (f,11).

Let

11 = FS'(1).

Since f - f is an isomorphism of S onto S, its transpose FS' is an isomor- phism from S' onto S'. Moreover, by Parseval's relation (cf. 2.6),

(f,u) _ (f, u), ^{df E S, Vu E L1.}

Hence FS, is an extension of the Fourier integral on L'. The same result holds on L2.

Finally, the inversion formula for FS, is proved by transposing the inversion formula on S.

4.3.5 Support of a distribution

Let 1 E S'. We say that l is zero on the open set 0 if l(w) = 0 for any

cp E S(Rn) such that supp (W) C 0. Differentiable partitions of unity can be used to show that there exists a largest open set ci on which I is zero.

The complement of f is called the support of 1.

4.3.6 Sobolev scales of distributions

For a fixed positive real number s, let D(R') be given the H-' norm

defined by

II'PIIH- =sup

JR ^cpfdx, where f E H', IIlIIH° <_ 1.

Since V is dense in H', hPIIH- = 0 implies that io = 0.

Using the notation of Sobolev, we let H-'(R') denote the completion of the space V with respect to the H-' norm.

Theorem (Sobolev). The Fourier transform extends from V to H-' and

realizes an isometric isomorphism from H-' onto L2(Rn, µe), where d148 = (1 + IItII2)-8dt.

PROOF. If f E H8, then f E L2 and the Fourier-Plancherel isomorphism gives

f

^{c fdx =}

_J

ⁱ (t)Tf(t)dt.

R° R^

Hence

IIWIIH = supj'ca(t)v(t), with

J

Iv(t)I2(1 + IItII2)'dt < 1.

By the Cauchy-Schwarz inequality,

L1(t)(1 + IItII2)-°/2(1 + IItIi2)'/2v(t)dt

L

/

^{1 (t)12}

Lf (1 + llt112)8 JJ

whence

r l

[J ,v(t)2

1 /2

(1 + 11t112)'J

Equality occurs when v(t) = ccp(t)(1+lltll2)-', with the constant c determined so that IIvIIH. = 1. 0

4.3.7 Comparison of the two theories

(i) Proposition. For every s > 0, H-'(R) C S'(R°).

PROOF. S(R") C H'(R"). Moreover,

IIflIm.r ? IllIIH" if

r > 8, m> 2

Let 0 E H-'. Then 0 defines a linear functional on H' and

10(/)1 <_ CIIIIIH" <_ CIIfIIm.r Vf E H.

Hence 8 is continuous on H' if H' is given the topology induced by that of S.

Restricting 8 to S gives a continuous linear functional 0 on S and 0 H 01 defines the desired map H-' - S'.

This map is injective: V is dense in H'; a fortiori, so is S; thus a linear func- tional on H' that vanishes on S is identically zero. E3

(ii) Proposition. Let 1 E S' and suppose that l has compact support. Then there exists p such that I E H-p(R").

PROOF. There exists a pair of integers m, r such that Il(f)I < CIIfIIm,r Vf E S(R' ).

Let W E D(R") such that 'p = 1 on the support of 1. Then I(f) = I(f), whence I1(f)I <_ dlVfllm.r. But

IIcoflim,r <_ CI10IIm,rDfIIw20.

Moreover, by the wrollary to the trace theorem, Ill II_{L W _} cll f f l H. if

s > 2 and IIfIIw <_ Hence II(f)I <_

Thus l extends to a continuous linear functional on H'+', whence l E

H-'-r. o

5 Pseudo-differential Operators

The Fourier transform on R" diagonalizes linear differential operators with constant coefficients. This property leads to representation theorems for the solution of the homogeneous equation as a limit of sums of complex expo- nentials, as well as existence theorems for the nonhomogeneous equation.

These theorems, due to Leon Ehrenpreis and Bernard Malgrange, use the Fourier transform in C" as a fundamental tool.

Complex-analytic methods are needed to prove these theorems, which are naturally formulated in the context of Laurent Schwartz's theory of distributions.

To obtain such general results, we would need not only to study locally convex topologies on spaces of distributions and duality between locally convex spaces, but also to prove minimum modulus theorems for holomor- phic functions of several complex variables. All these methods originate in different currents of thought from those we have followed up to now.

We will study differentiable operators with variable rather than constant coefficients, and on bounded open subsets of R" rather than on all of R". In physics, differentiable operators with variable coefficients invariably appear when an inhomogeneous medium is considered.

At first glance, Fourier analysis seems to have no means of obtaining re- sults in this setting. It was thus a striking result when Alberto Calderon, in 1957, introduced an "infinitesimal Fourier transform on the tangent space", which assigns a "symbol" to an operator and thereby embeds differential operators in the wider class of pseudo-differential operators. In this class, one introduces an infinitesimal symbolic calculus which consists of multiply- ing symbols. Calderon's symbolic calculus theorem states that the symbolic calculus corresponds to the composition of operators modulo regularizing operators, i.e. with the gain of one derivative.

The pseudo-inverse of a differential operator can be explicitly constructed in integral form.

This section ends with an application of the pseudo-inverse, in the proof of the elliptic regularity theorem.

Pseudo-differential operators are a basic tool of the theory of partial differential equations. The spectral pseudo-decomposition they effect, and the integral estimates they entail, make up, to some degree, the extension of Sections 1 to 4 of this chapter.