18. Distributions

18. Distributions Letf(x) be a continuous function defined in R" or at least on an open subset of that space. If the function is bounded, we set llfll a = sup, If(x)I and define the support off, written suppfas the closure of the set If(x)l > Oand thus the support is the complement of the interior of the setf(x) = 0. We are particularly interested in the functions which are infinitely differentiable with compact support; we have already seen that many such functions exist, and we call them testjiinctions. Let R be an open subset of Rn;by 9 = 9 ( R ) we designate the class of all testfunctions having support in R and this is evidently a linear space. We think of it as a linear space over real scalars when we are considering only real valued testfunctions, and for complex valued testfunctions we take it as a space over complex scalars. It is convenient to introduce a family of seminorms on 9 as follows:

these seminorms are actually norms and form an increasing sequence. It is natural then to define a locally convex topology on 9 with these seminorms; the resulting topology makes 9 a metric space which unfortunately is not complete. To see this, we suppose that R = R' = the real axis and take a fixed testfunction q ( x ) ; the partial sums of the infinite series

n= 1

2-nq(x

- n)

form a Cauchy sequence for the metric topology determined by the seminorms, but the series does not represent a function with compact support. We therefore reject the topology which we have just defined, but we will find the norms JJrpJJ,useful in any case. Definition: A linear functional T o n 9(R) is a distribution if and only if for every compact subset K of Q, there exist constants C and N such that

for all testfunctions cp with support in K. If the integer N can be chosen independent of the compact K, and N is the smallest such choice, the distribution is said to be of order N. 91

92

11. DISTRIBUTIONS

We consider some important examples of distributions. Let dp be a Radon measure on R; we define the corresponding distribution by the equation

If K is a compact subset of !J and C = !,Jdp(x)I is the total mass of dp on K then IT(rp)l 5 c IIrpllm =

c llrpllo

for all testfunctions supported by K. Thus the functional T corresponding to dp is in fact a distribution and is of order 0. We will make a canonical identification of Radon measures and the distributions which correspond in this way. An important special case is given by the Dirac &distribution; this is the measure which consists of a positive unit mass at the origin, or equivalently, the distribution 6 defined by S ( q ) = rp(0). Letf(x) be a locally integrable function on R; we form the Radon measure f ( x ) dx and pass to the corresponding distribution

f(rp)=

/ rp(x)f(x) dx

*

In the sequel we shall often identify locally integrable functions and the corresponding distribution and shall not speak explicitly of this identification, Thus, we will speak of a distribution which is a polynomial, a Cm-function,or the characteristic function of a set, etc. Let {U,}be a countable family of open subsets of R having compact closures in R and which form a covering of that set: every x in R belongs to at least one U,.We suppose also that the covering is locallyjinire, that is, that every x in R belongs at most to a finite number of the sets of the covering. It is then easy to show that a compact subset of R intersects at most a finite number of the Ui . By induction, we can always construct a further locally finite covering subordinate to the covering { U i };this is a locally finite covering { V i } such that the closure of V i is a compact subset of U,. Iff,(x) is the characteristic function of V i, then for sufficiently small positive E , the regularization offi(x) of order E will be a testfunction rpi(x) in R which is positive on V i and which has its support in U,. We form the infinite series

@(XI = C rpi(x) to obtain a function which is strictly positive in i2 and infinitely differentiable there, since only finitely many terms of the series are nonzero on any compact subset of R. Finally, we form the functions

18. DISTRIBUTIONS

93

and obtain a system of testfunctions satisfying the conditions 0 5 $i(x) 5 1, G i ( x ) = 0 outside U i , and ZI+~~(X) = 1. The system is called a partition of unity subordinate to the covering { U i } .We should remark that there is no difficulty concerning the existence of partitions of unity, since any open subset R of R" has locally finite coverings. We use the partitions of unity first to show that the distributions are determined by their local behavior. More exactly, if two distributions T and S on R have the property that for every x in R there exists a neighborhood U such that T ( q )= S ( q ) for all testfunctions q(x) supported by U,then T = S. The proof consists in passing to a locally finite covering { U i } consisting of neighborhoods on which the distributions coincide and taking a corresponding partition of unity. For any testfunction q(x) we have and

CP(X> =

C $i(X)dX)

T ( q )= C T($i CP) = C S($i CP) = S(V) * i

i

since only finitely many terms in the series are nonzero. Theorem: measure.

T is a distribution of order 0 if and only if T is a Radon

PROOF: We have already seen that the Radon measures are distributions of order 0. On the other hand, T being of order 0, we take a locally finite covering { U , } of R and for each i we consider the continuous function space %'(Bi). This, of course, is the space of continuous functions on the compact, Bi with the usual supremum norm. The linear functional T ( q )is defined on the subspace consisting of testfunctions with support in Ui , and because T is of order 0, it is continuous for the norm of W(Bi).Thus the functional can be extended by continuity to the closure of the testfunctions, and by HahnBanach to a continuous linear functional on the whole continuous function space. The theorem of F. Riesz guarantees that this extension is a measure on U i of finite total mass, whence ~ ( q=)

SP(X)

dpi(X)

for all testfunctions with support in U i . If, now, $i(x) is a partition of unity subordinate to the covering U i , we have, finally, T ( v ) = T ( Z +iq) = =

1T($iq)

c j$i(x)q(x) dpi(x)

= jrp(x) d p ( 4

94

11. DISTRIBUTIONS

where dp(x) is the Radon measure x

y JIi(x) dp,(x).

A distribution Ton R is positive if and only if T(cp) 2 0 for Definition: all testfunctions satisfying q(x) 2 0. If the distribution T is positive, and K a compact, we select a positive testfunction JI which equals + 1 on a rreighborhood of K. Since for any cp(x) supported by K

- IlcplIw

s cp(x)

5 +IlcpIImJI(X) for all points x in R, we obviously have IT(cp)I 5 I T(JI)I llcpll

and therefore T is of order 0. A slight variant of our proof establishes the following result.

Theorem: A distribution T is positive if and only if it is a positive Radon measure. Let R be the positive half-axis in R' and let {Xk} be the sequence {l/k}. It is easy to verify that the form

is a distribution on R which is not of finite order. Moreover, this distribution cannot be extended to a distribution defined on the whole R".

19. Differentiation of Distributions Let T be a distribution on R and xk one of the coordinate functions. We define the derivative of T with respect to xk by the equation

Since the derivative of a testfunction is again a testfunction, the differentiated distribution is a linear functional on 9 ( R ) ; that it is a distribution follows from the inequality

valid for any testfunction cp supported by a compact K on which IT(cp)l 5 Cllcpll,.