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l
Thus, we proved that the following lemma holds true. If a generalized function in 1"(0) vanishes in some nighbourhood of every point of an open set 0, then it also vanishes in the whole set (}. LEMMA.
Suppose f E V'(O). The union of all neighbourhoods where f :::: 0 forms an open set Of which is called the zero set of the generalized function f. By the lemma, f :::: in Of; furthermore Of is the largest open set in which f vanishes. The support of a generalized function f is the complement of Of to 0; the support of f is symbolized as sUPP!, so that supp! = 0 \ Of; Stipp! is a closed set in O. If supp! @ 0, then I is said to be with compact support in O. From the foregoing follow the assertions: (a) If the supports of f E V'(O) and'P E 1'(0) do not have any points zn common, then (I, 'P) :::: 0. (b) To have x E supp I, it is necessary and sufficient that! should not vanish in any neighbourhood of the point x. Let A be a closed set in O. We denote by V' (0, A) the collection of generalized functions in 1"(0), whose supports are contained in A, and with convergence: fk ---+ 0, k ---+ 00 in V'(O, A) if /k ---+ 0, k ---+ 00 in V'(O) and supp!k c A. We write D'(A) = V'(R n , A). A similar meaning is also attributed to other spaces of generalized functions, for example, S'(A), £;(A) and so on (see Sees. 5 and 7 below). The lemma that was proved in this subsection admits of a generalization. In Sec. 1.3 we saw that any generalized function f in V' (0) induces, in each 0' cO, its own local element f E '[)'(O'). The converse is also true: it is possible to "join together" a single generalized function out of any collection of compatible local elements. To be more precise, the following theorem holds.
° I
THEOREM ("of piecewise sewing"). Suppose that for each point yEO there exist a neighbourhood U (y) @ 0 and in it there is specified a generalized function !y, wherein !Yl (x) :::: f!J2(x) if x E U(yt} n U(Y2) :I 0. Then there is a unique generalized function f in V' (0) that coincides with fy in U (y) for all yEO. PROOF. Again, as in the proof of the lemma, we construct, with respect to the cover {U(y), YEO}, a locally finite cover {Ok}, Ok C U(Yk), of the set 0 and the appropriate partition of uni ty {ek}. Set (/, 1 are defined (see Sec. 2.7). For instance, the generalized function P f N = 1,2, ... , are defined by the formula
I
W'
f
) = Pf_1_ ( ]xjN I If'
If'(x) - SN-dx; If') d
Ixl N
x
f
+
Ixl 2, If'o(x) > 0 and
= 1.
If'o(x) dx
Then If'k(X)
= e- k / 2 klf'o(kx) -+ 0,
f
e1/xipdx)
dx
k --+
in
00
f =f ek(~-~)lf'o(Y) >f =
V,
= e:-~kipo(kx) dx 2
dy
1
2
If'o(Y) dy
1,
1
(O): epk ¢:::::::>
8 cx epk(X)
~
xeo
k --+
OJ
00
in
COO (0)
J
~
0,
k --+
00
for all
Q
and
0'
~
O.
From this definition it follows that convergence in V(O) implies convergence III Ceo (0), bu t not vice versa. Suppose a generalized function fin V'(O) has compact support in 0, supp f J{ @ O. Suppose 1] E V(O), 1J(x) ::::: 1 in the neighbourhood of J{ (see Sec. 1.2). We will construct a functional f on C<X>(O) via the rule
=
(j. Ifl) =
(5.1 )
(!, TJep) ,
Clearly, j is a linear functional on Coo (0). Furthermore, since the operation ep ~ 1J1fl is continuous from Coo (0) into V( 0), it follows that is a continuous functional on C<X> (0). The functional f is an extension of the functional from V(O) onto COO(O), since for ep E "D(O)
i
(1, ep) = (f, 'flIP) = ('fIf, IP) = (f, ip) by virtue of the equality (10.2) of Sec. 1.10.
2.
DIFFERENTIATION OF GENERALIZED FUNCTIONS
37
We will show that there is a unique linear and continuous extension of f onto COO(O) (~n particular, the extension (5.1) does not depend on the auxiliary function Let j be another such extension of f. We introduce a sequence of functions {7]k} in V(O) such that 1Jk(X) 1, x E Ok (0 1 @ O 2 @ ... ,0 Uk Ok), so that 7]k --+ 1, k --+ (X) in COO(O). Therefore, for any 'P E Coo(O) we will have T]k'P --+ 'P, k --+ 00 in COO (0). Hence, 7]).
=
(1, Ifl)
=
= (j, k-+oo lim 11kV') = lim (1,1]k'P) = lim (f, 7]k'P) k-+oo k-too -
-
~
= lim (1,7]kY') = (], lim 7]k'P) = (], 'P), k-too k-tlXl
Ifl E
ClXl(O),
=
so that f I· We have thus proved the necessity of the conditions in the following theorem. THEOREM. For a generalized function f in 'V'(O) to have compact support in 0, it is necessary and sufficient that it admit of a linear and continuous extension onto Coo (0). PROOF OF S_UFFICIENCY. Suppose f E 1>'(0) admits of a linear and continuous extension I onto Coo (0). If f did not possess compact support in 1>, then it would be possible to indicate a sequence of functions {'Pk} in V( 0) such that SUPP!Pk C 0 \ Ok (0 1 @ O 2 (§ ... , Uk Ok = 0) and (I, !Pk) = 1. On the other h~nd (0') for all tp E 1>(0). By the theorem of Sec. 1.3 there exist numbers J( = j{(O') and m m(O') such that the following inequality holds:
=
I(f, ep)1
= 1(/, TJtp)1 < KII1J'Pllcm(o')1
tp E V(O),
whence immediately follows the inequality 'P E V(O).
(5.2)
Inequality (5.2) implies the following assertion: any generalized function with compact support in 0 has a finite order in 0 (see Sec. 1.3). We denote by £' the collection of generalized functions with compact support in ~n. It has thus been proved that. £' COO (ffi. n )'.
=
2.6. Generalized functions with point support. Generalized functions whose supports consist of isolated points admit of explicit description. This is given by the following theorem. THEOREM. If the support of a generalized function / E V' consists of a single point x = 0, then it is uniquely representable in the form
f(x) =
L
caoQo(x)
[QI:SN
where N is the order of f, and
CO'
are certain constants.
(6.1 )
38
1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
Ixi :::;
Suppose T/ E V(Ud, 1](x) ::::: 1, 1](~) f and, hence, for any cp E 1)
PROOF.
f :::::
(/,>0)
1/2. Then for any
E:
> 0 we
have
= ('7 (;) f. >0) =
(f,lJG)(>o-SNJ) + (f''7G)SN)
(6.2)
where
ao cp(O) x a
SN(X; cp) = ~ L.....J
o!
lal5 N
is the Taylor polynomial of cp at zero of degree N (see Sec. 1.8). Since 1J(~) (
O-SNJ
':;C
::::: G max aa I$I O. We say that a generalized function fa E V' dependent on the complex parameter Q' is holomorphic (meromorphic) with respect to Q in a domain G if, for any t.p E V, the function (fa, O. Let us rewrite equality (7.1) for ?Ra > 0 and N 1,2, ... in the form (1r v
=
f
lxl a - 1 ,lp) =
1rv (x)l x IO:-
l
[lp(x) - SN-:-dx;1
2
{
OU
+ k'
1
if l/ + k is even if v + k is odd,
we deduce from (7.2) and (7.3) the following theorem.
The generalized function ?Tvlxl o- l , ~(} > 0, admits the meromorphic continuation P (1Tv I x 10 ) from V' onto the whole plane a with simple poles and residues THEOREM.
_2_ 6(2n)(x), (2n)!
(} = -2n, a
= -2n -
I,
In every half-plane
(2n ~(}
2
(p{7r IX\ct-l), I{)) = V
) 8(2n+l)(x),
+1
> - N,
n
= 0,1, ... ,
n=O,l, ... ,
!
if ?Tv (x)
= sgn x.
= 1,2, ... , it admits the representation
N
f
1l"v(x)l x IO -
l
[(X k-+oo
=
+ y))
lim (ojl(x) x g(y), 1]k'P(X + y))
k-+oo
= (OJ f
~~k) .1
+ y))
x g(y)] , TJkip(X
Xj
y)] - I"(x +y)
J
*9
l
+ f * g, 'P)
- (I * g, 'P)
'P) ,
whence follows the first equality (2.4) for OJ. The second one of (2.4) follows from the first one and from the commutativity of a convolution:
OJ (I * g) = OJ (g
* f)
= OJ 9
* 1 = f * OJ g.
From (2.1) and (2.4) there follow the equalities
f
E V'.
Note that the existence of the convolutions 0 01 f * 9 and f * oO:g for 10'1 > 1 is not yet enough for the existence of a convolution f * 9 and for the truth of (2.4). For example,
0' * 1 = 0 * 1 = 1,
but
0 * l' = B * 0 = O.
4.2.6. The operation f --+ f *g is linear on the set of those generalized functions f for which the convolution with 9 exists. This property of a convolution follows directly from the definition of a convolution (1.4) and from the linearity of the operation f --+ f x 9 (see Sec. 3.2.3). In passing we may note that the operation f --+ f *9 is not, generally speaking, continuous from V' into V', as the following example shows:
J(x - k) --+ 0l 4.2.7.
k --+
If the convolution
00
1 *9
supp(f
in
V';
however,
1 * J(x - k) = 1.
exists, then
* g)
C StiPP! + suppg.
(2.5)
Indeed, suppose {1]k} is a sequence of functions taken from V(~2n) that converges to 1 in 1R 2n and ({J E V (JR.n) is such that sUPP ({J n supp! + supp 9 = 0. Since supp(f x g)
= supp f
(2.6)
x supp 9 (see Sec. 3.2.4), we conclude that
supp[J(x) x g(y)] nSUpp[1]k(X; y) 0 the set TR is bounded in 1R 2n . Then the convolution f *g exists in V'(A + B) and may be represented as THEOREM.
(!*g,cp)
= (!(x) x g(y),e(x)7](Y)CP(x +y)),
(3.1)
where ~ and TJ are any functions in Coo that are equal to 1 in AE and BE and are equal to 0 outside A 2 E and B 2 E respectively (c is any positive number). Here the operation f --+ f * 9 is continuous from V'(A) into V'(A + B).
Let cp E V(UR) and let converges to 1 in IR 2n. Since PROOF.
{m-l be a sequence of functions in V(~2n)
that
supp(! x g) = supp! x suppg C A x B (see Sec. 3.2.4), it follows that supp{ [I(x)
x g(y)] ip(x + y)}
C
[(x, y): x E A, y E B
Ix + yl < R] = TR·
=
And since T R is a bounded set, there is a number N N(R) such that 77k(X; y) in the neighbourhood of TR for all k 2:: N. For this reason,
=1
(f * g, cp) = lim (/(x) x g(y), TJk(X; y) 0 is bounded
in }R2n; furthermore, the function
vanishes in a neighbourhood of TR. This completes the proof of the representation (3.1). From (2.5) it follows that supp(f * g) C A
+B
I
so that the operation 1 -+ f *g carries D'(A) into V'(A + B). Its continuity follows from the continuity of the direct product f x g with respect to 1 (see Sec. 3.2.3) and from the representation (3.1): if !k --+ 0, k --+ 00 in V'(A), then
(!k * g, cp)
= (/d x ) x 9(Y),~(X)1J(Y)CP(x + y))
--+ 0
1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
58
!J
Figure 20
for all r.p ED, that is, fk is complete.
* 9 --+ 0,
k --+
00
in D'(A
+ B).
The proof of the theorem 0
Note that. t.he continuity of the convolution f * 9 relative to the collection of f and 9 may not occur, as the following simple example illustrates:
6(x
+ k)
--+ 0,
k --+
+00,
6(x - k) --+ 0,
k --+
+00.
However,
We note here an important special case of the theorem just proved. If f E V' and 9 E £', then the convolution in the form
(/ * g, ',0) = (f(x)
f *9
x g(y), 1J(Y)r.p(x
exists and can be represented
+ y)),
(3.3)
r.p E V,
where 1] is any test function taken from D that is equal to 1 in a neighbourhood of the support of g.
Indeed, in this case the boundedness condition of the set TR is fulfilled for all R> 0 (Fig. 20): if suppg C UR', then T R = [(x,y): x ERn, y E suppg,
Ix+ yl S R]
C UR+Rf
X
URI.
Similarly, if f E V' and gl, ... ,9m E £' 1 then there exists a convolution
f * 91 *
... * 9m (see Sec. 4.1) that is associative and commutative (see Sec. 4·2.1, 4.2.8) and the formula (3.3) is generalized thus:
(J
* 91 * ... * 9m, rp) = (f(x)
x gl(Y) x ... x 9m(Z),
1JI(y) .. . 1Jm(z)r.p(x + Y + ... + z)), But if f E COO and 9 E £1, then the convolution mula (3.3) takes on the form
f *9
9 is
the extension of 9 onto COO
= coo(I~n)
(3.4)
E Coo! and the for-
(f * g)(x) = (g(y), f(x - y)), where
r.p E 1).
(see Sec. 2.5).
(3.5)
4. THE CONVOLUTION OF GENERALIZED FUNCTIONS
59
True enough, as in the proof of the lemma of Sec. 3.1, it is established that the function
(g(y) , f(x - y))
= (g(y), 1J(y)f(x -
Y))
E
Cr.xJ.
Then, from the representation (3.3) we have, for all tpEV,
U * 9. OJ ...• (en) x) > 0]
is acute (convex and open) if and only if the vectors e1, ... , en form a basis in ~n. Then
In particular, the positive quadrant ~+.
= [x:
Xl
> 0, ... , X n > 0],
(IR+')*
= IRt..
60
1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
(b) The future light cone in JRn+l.
v+
= [x:
(xo,x): Xo
> Ixl],
(V+)*
= V+.
(c) The origin of coordinates {O}, {O}* = }Rn. Note, however, that the cone lR~ x ~n-l [x: Xl > 0] is not acute: (d) The cone Pn C ~n2 of positive (Hermitian) n x n matrices X = (x pq ), P~ = P;t is the cone of nonnegative matrices. This follows from the assertion that, for X E Pn , it is necessary and sufficient that for all S E p;t, S #- 0,
=
(X , 3)
= Tr(X2) = L
Xpq~qp > O.
p,q
LEMMA
1. The following statements are equivalent:
(1) the cone r is acute; (2) the cone ch r does not contain an integral straight line; (3) int 0; (4) for any C J @ int r* there exists a number u u(C J ) > 0 such that
r· :/;
=
(~ , x) ~ u I~ II x I,
~ EC
J ,
x E ch r;
(4.1 )
(5) for any e E pr int r* the set
Be [x: 0 < (e, x) :::; 1, x E ch is bounded in
~n
rJ
(Fig. 22).
r contains an integral straight line x = xO+te, (lei = 1), then it also contains the straight line x = te, -00 < t < 00.
PROOF. (1) 4' (2). If the cone ch -00
< t < 00
Consequently, any plane of support for ch f must contain that straight line, but this contradicts (1). (2) --+ (3). Ifintf* = 0, then, since f* is a convex cone with vertex at 0, it lies in some (n - I)-dimensional plane (e,x) = 0 (lei = 1). For this reason, ±e E f** = ch r. But then t.he integral straight line y = te, ~OO < t < 00 too lies in chr, but this contradicts (2). (3) --+ (4). Since all points of the cone C' different from 0 are interior points relative to f"', it follows that (~, x) > 0 for all E C' and x E ch f. From this fact and also from the continuity and the homogeneity of the form (~, x) follows the existence of a number rr > 0 for which the inequality (4.1) holds true. (4) --+ (5). Let us take an arbitrary e E printf*. Then, by applying the inequality (4.1)' (e, x) ~ ulxl, x E ch r, we conclude that the set Be is bounded: Ixl :::; (e~x) ~ ~. (5) --+ (1). If for some e E pr int f'" the set Be is bounded, then the plane (e, x) = 0 cannot have any other points in common with ch f, with the exception ofO. 0
e
LEMMA
2. Let
r
be a convex cone. Then
r :::: r + r.
is obvious. Let x E r + r so that x = y + z, where y E f and z E f. Then for all A E (0,1) we have x = At + (1- A)l~>" E r and for this reason r + fer, thus completing the proof of the lemma. 0 PROOF. The inclusion
r c r +r
The indicator of the cone
r
is the function
Jlr(~)
= - xEpr inf (~,x). I'
THE CONVOLUTION OF GENERALIZED FUNCTIONS
4.
61
From the definition of the indicator it follows that /-lr(~) is a convex (see Sec. 0.2) and, hence, continuous (see, for example, Vladimirov [105, Chapter II]) and homogeneous first-degree function defined on the whole of lR n . Besides,
JLr«) < JLchr(~), j.tr«) = -Ll(~, ar*), and j.tr(~)
> 0 for a. Let the point 6 E [* realto r*, 8«0, r*) = I~o ize the distance from ell. Then, since is a convex cone (Fig. 23), it follows that
(b)
~
r·;
eo, 6) = O.
From the inequality (a) it follows that r*'" f' and therefore
=
a 2: Jlr(eo) = -
6 - eo
.
mf jeo, x)
xEprf'
E
Figure 23
2: - ( eo, l~61 -- eo0 I) .
e
Now the latter is equivalent, by virtue of (b), to the inequality 16 - eol ::; a. Thus, the point eo = 6 + (eo - ed is represented in the form of a sum of two terms 6 E r* and E U a that is, E r* + U a. This completes the proof of the inverse inclusion of (4.3) and also the equality (4.2). The proof of Lemma 3 is complete. 0
eo - e1
eo
62
I. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
Figure 24
Suppose r is a closed convex acute cone. Set C = iut r* (via Lemma 1, C =j:. 0). The smooth (n - 1}-dimensional surface S without an edge is said to be C -like if each straight line x = Xo + te, -00 < t < 00, e E pr f, intersects it in a unique point; in other words: for any x E 5 the cone r + x intersects S in a unique point x (Fig. 24). Thus, the C-like surface S cuts ffi.n into two infinite regions S+ and S-: S+ lies above 5 and S_ lies below S; S+ U S- u S = ~n. At every point x of the surface 5, the normal OJ; is contained in the cone f* + x (Fig. 24). EXAMPLE.
The surface S in ~n+l, which surface is specified by the equation
Ivr f(x)1 < a < 1,
Xo = f(x),
xE
f E
jRn,
c1,
is V+ -like (space-like). LEMMA
4. If S is a C-like surface, than (4.4)
5+=3+f.
Suppose Xo E S+. The straight line x = Xo + te, ItI < 00, e E pr r, intersects 3 at some point Xl = Xo - i l e, il 2: 0 (Fig. 24) so that Xo = Xl + t l e, x I E 3, tIe E r, and the inelusion S + C S + r is proved. Clearly, the inelusion s+r C S+ is true; together with the inverse inclusion it leads to the equality (4.4), D thus completing the proof of the lemma. PROOF.
LEMMA
R' (R)
>0
5. Let S be a C-like surface. Then for any R
>0
there is a number
such that the set
TR =
[(x, y): x
E 5, y E f,
Ix + yl
~
R]
is contained in the ball URJ C jR2n.
Since 5 is a C-like surface. it follows that any point xES that can be represented as ~ - y, y E f, I~I ~ R, is of the form x = ~ - eT, e E pr [, where the number T T(e,~) is uniquely determined bye and { and constitutes a continuous function of the argument (e,~) on the compact e E prf, I~I ~ R. Hence the set [(y,O: y eT(e,O, e E prr, I~I::; R] is bounded and so also is the set TR. The lemma is proved. D PROOF.
=
=
We will say that a C-like surface S is a strictly C -like surface if, under the conditions of Lemma 5,
R'(R) ~ a(1
+ Rt,
v ~
1,
a
> O.
(4.5)
4. THE CONVOLUTION OF GENERALIZED FUNCTIONS
63
v o
Figure 25
= 0,
EXAMPLE. The plane (e, x) virtue of Lemma 1).
e E pre, is strictly C-like with v = 1 (by
4.5. Convolution algebras V'(f+) and V'(f). We will say that a set A is bounded on the side of the cone r if A C r + Ii, where J( is a certain compact (Fig. 25). It is clear that the sets bounded on the side of the cone {OJ are compacts in lR n . Suppose r is a closed cone in lR n . The collection of generalized functions in V' whose supports are bounded on the side of the cone r will be denoted by V' (r +). We define convergence in V'(f+) in the following manner: fk 4- 0, k 4- 00 in V' (f+), if !k --+ 0, k 4- 00 in V', and supp fk C r + j{, where the compact l{ does not depend on k. 2 Set 1J' ({O} +) £'; £' is the space of generalized functions with compact supports (compare Sec. 2.5). Let f be a closed convex acute cone, C = int f*, SaC-like surface, and S+ the region lying above S (see Sec. 4.4). If f E V'(f+) and 9 E V'(5+)1 then the convolution f * 9 exists in V' and can be represented as
=
(f *9,'P) = (f(x) x g(y),€(x)1](Y) 3;
2)(Tnf,
~ V2
= -21f f.
Indeed, using the formula (4.2) of Sec. 2.4 and (2.4), we obtain, for n AVn
~
3,
= A CXI~-l * f) = Alxl~-2 * f
=-(n -
2)(TnJ
* f = -(n -
2)(Tnf.
We proceed in similar fashion in the case of n = 2 as well. If f = p(x) is a function with compact support integrable on ~nJ n > 3, then the corresponding Newtonian potential Vn is called the volume potential. In this case, Vn is a locally integrable function in lR n and is given by the integral
=
Vn(x)
f Ix -
p(y) dy
yln-2
.~
(9.1)
in accordance with formula (1.1) for the convolution of a function p(x) with compact support integrable in lR n with the function Ixl- n +2 locally integrable in jRn. ve5s) be a simple layer and a double layer on Let f fla n and f == a piecewise-smooth surface S C jR n, n > 3, with surface densi ties fl and v (see Sees. 1.7 and 2.3). The corresponding Newtonian potentials
:n(
=
(0)
Vn
1
= Ixl n -
(1) _ _
Vn
-
2
* Jib's
1
I
~
Ixln-2 * on (vJ s )
are, respectively, the surface potentials of a simple layer and a double layer with densities J.L and v. If S is a bounded surface, then the surface potentials V~O) and VJ1) are locally integrable functions in lR n and can be represented by the integrals V:(O)(x) n
f _! ({)
= Ix _Ji(Y) dS, yln-2 y s
(1)
Vn (x) -
1/
y) any
1
Ix _ yln-2 dSy .
(9.2)
s For the sake of definiteness, let us prove the representation (9.2) for the potential V~l). Using the representation (3.3) and the definition of a double layer (see
4, THE CONVOLUTION OF GENERALIZED FUNCTIONS
69
Sec. 2.3), for all t.p E D we obtain a chain of equalities (the function TJ E V and 1J(x) _ 1 in a neighbourhood of S):
(V~l), 3, be bounded by a piecewise-smooth boundary S and let the function /-l E C2 (G) n C 1 (G). Then it can be represented in the form of a sum of three Newtonian potentials via Green's formula (n is an outer normal to S):
1 - (n - 2)o-n
{J Ix~u(y) G
J[Ix -
d yln-2 y - s
1 8u(y) yln-2 an
() 8
any Ix -
-
u y
=
{U(X), 0,
1
yln-2
~,
x E x ~ G.
] dS } y
(9.3)
Indeed, assuming that the function u(x) has been continued by zero for x ~ (; and taking advantage of the formula (3.7 /) and (3.10) of Sec. 2.3, we conclude that u=J*u==-
1
(n - 2)O"n 1
-,----,-----,---...,.-~
(n - 2)O"nlxln-2 1
= - (n _ 2)o-nl x ln-2
~
1
Ixl n -
2
*u
* ~u
au 65 - ana (u6 s )] . * [ .6.c1 u - an
70
1.
GENERALIZED FUNCTIONS AND THEIR PROPERTIES
Whence, using (9.1) and (9.3), we convince ourselves that the representation (9.3) holds true. 0 In particular, if the function u(x) is harmonic in the region G, then the representation (9.3) transforms into the Green's formula for harmonic functions: 1
f [Ix - vl 1
(n - 2)un
n- 2
ou(y) _ u(y) ~
5
on
1
any Ix - vl n - 2
] dS
y
= {u(x), 0,
x E G, x
rt G.
(9.4)
Formulas similar to (9.3) and (9A) occur in the case of n = 2 as well. In this case the fundamental solution - (n-2)u:lxl n 2 must be replaced by 2~ In Ixl· Green's formula (9A) expresses the values of the harmonic function in the domain in terms of its values and the values of its normal derivative on the boundary of that domain. In that sense, it is similar to Cauchy's formula for analytic functions. REMARK.
4.9.3.
A convolution equation has the form
(9.5) where a and f are specified generalized functions in V' and u is an unknown generalized function in V'. Convolution equations involve all linear partial differential equations with constant coefficients:
a(x)
I:
=
aa oa6 (x),
latS;m a
L
*u =
aaaau(x);
lal~m
linear difference equations:
a
*u =
Laau(X - x a ); a
linear integral equations of the first kind: a E
.ctOCl
a
*u =
!
u(y)a(x - y) dy;
linear integral equations of the second kind: a = 6 + IC
a *u
= u(x) +
K. E £toc,
I
!
u(y)K.(x - y) dy;
linear integra-differential equations; and so forth. The fundamental solution of the convolution operator a* is a generalized function f E V' that satisfies the equation (9,5) for f = 6, a
*£ =
6.
(9.6)
4. THE CONVOLUTION OF GENERALIZED FUNCTIONS
71
Generally speaking, the fundamental solution E is not unique; it is determined up to the summand £0, which is an arbitrary solution in V' of the homogeneous equation a * Eo = O. Indeed, a
* (E + Eo)
= a *E+ a
* Eo
=
o.
(1) The function En(x) defined in Sec. 2.3.8 is a fundamental solution of the Laplace operator: !!J.En = O. (2) The formula O(x) + C yields the general form of the fundamental solution in V' of the operator d~ = 0'* (see Sec. 2.2 and Sec. 2.3.3). EXAMPLES.
Let the fundamental solution E of the operator a* in V' exist. We denote by A( a, E) the collection of those generalized functions f taken from V' for which the convolutions E * f and a * £ * f exist in V'. The following theorem holds. Suppose I E A(a, E). Then the solution u of the equation (9.5) exists and can be expressed by the formula THEOREM.
(9.7)
u=E*f. The solution of (9.5) is unique in the class A(a, E).
=
The generalized function u £ * f satisfies (9.5) since, by virtue of the commutativity and the associativity of a convolution (see Sec. 4.2.8) (the convolutions £ * j, a * E * f and a * £ = 0 exist): PROOF.
a *u = a Uniqueness: if a u
* (£ * 1)
*u =
= a
*£ *I
= (a
* £) * f
= 0 * I = f.
0 and u E A(a, f), then
= u * 0 = u * (a * E) = u * a * £ = (u * a) * £ = 0 * E = 0,
which is what we set out to prove. The proof is complete.
o
We can give the solution u = £ * I, (9.7), the following physical interpretation. Let us represent the source I(x) in the form of a "sum" of point sources f(~)c5(x -~) (see Sec. 4.2.2), REMARK.
The fundamental solution E(x) is the disturbance due to the point source o(x). Whence, by virtue of the linearity and translational invariance of the convolution operator a* (see Sec. 4.8) it follows that each point source I(~)o(x -~) generates a disturbance f(~)£(x - ~). It is therefore natural to expect that the "sum" (superposition) of these disturbances
will yield a total disturbance due to the source I, that is, the solution u of the equation (9.5). This nonrigorous reasoning is brought into shape by the theorem proved above.
1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
72
4.9.4. Equations in convolution algebras. Let A be a convolution algebra, for example V'(r+L 'D/(r) (see Sec. 4.5). Let us consider the equation (9.5) in the algebra A, that is, we will assume that a E A and f E A; the solution u will also be sought in A. In the algebra A, the above theorem takes the following form: if the fundamental soLution £ of the operator a* exists in A, then the soLution u of equation (9.5) is unique in A, exists for any 1 taken from A, and can be expressed by the formula u = [ * f. The fundamental solution [; of the operator a* in the algebra A is conveniently denoted as a-I so that, by (9.6),
(9.8) In other words, a-I is the inverse element of a in the algebra A. The following proposition is very useful when constructing fundamental solutions in the A algebra: if all and a2"l exist in A, then
(9.9) Indeed, (a1 * a2)
* (all * a 21) =
(a2
* ad * (all * a 21)
= a2 * ((a 1 * all) * a2" 1) = a2 * (0' * a2"l) = a2 * a2"l = O'. Formula (9.9) forms the basis of operational calculus. 4.9.5. Fractional differentiation and integration. Denote by V~ the algebra V'(~~).
We introduce the generalized function fa., taken from real parameter a, -00 < a < 00, via the formula
a> a
V~
I
that depends on a
0,
< O.
Let us verify that fa. Indeed, if Q'
* ff3 = fo:+f3'
(9.10)
> 0 and fJ > 0, then (see Sec. 4.1)
!
x
fa. * 1(3
_
-
O(x) r(a)r(fJ)
o a f3 = O(x)x + -
1
r(a)r(f3)
=
Y
a.-I
!
(x - y)
dy
1
r:x-l(1-t)f3- 1 dt
o
O(X)Xa.+ f3 -1
r(a)f(f3) B(a,fJ) O(X)Xa.+ f3 -1 ---:....-.:---- = 101+13'
r(a + f3)
j3-1
4. THE CONVOLUTION OF GENERALIZED FUNCTIONS
Now if Q'
::;
0 or {3 ::; 0, then, by choosing integers m fa
* ffJ
= f~~~
* f~~n
(m+n)
= (!Ot+m
> -(}
and n
> -{3,
73
we obtain
* ffJ+n)(m+ n )
F
= f a+fJ+m+n = J a+f3 which is what we set out to prove.
J
0
Let us consider the convolution operator 1a* in the algebra V~. Since 10 = 0' 6, it follows from (9.10) that the fundamental solution 1;;1 of the operator fa* exists and is equal to I-a: 1;1 = I-a. Furthermore, for integer n < 0, In = 6(-n), and for this reason fn * U = 6(n) * u u(n), which means the operator fn* is the operator of n-fold differentiation. Finally, for integer n > 0,
=
=
* u)(n) = f-n * (In * u) = (1-n * In) * U = 6 * u = u, which is to say that In *u is an antiderivative of order n of the generalized function (In
u (see Sec. 2.2). By virtue of what has been said, the operator ja* is termed the operator of fractional differentiation of order -(}' for (}' < 0 and the operator of fractional integration of order (}' for (} > 0 (it is also called the Riemann-Liouville operator). EXAMPLE.
Let
f
E V~. Then
J x
81/ 2 I
= 8(11 2 * f) = _1 .!!/ ft dx
f(y) dy .
.Jx - y
o
4.9.6. Heaviside's operational calculus is nothing but analysis in the convolution algebra V~. To illustrate, let us calculate in the algebra V~ the fundamental solution £(t) of the differential operator d) dm dm- 1 P ( dt = dtm + al dt m - 1 + ... + am, where form
aj
are constants. In the V~ algebra the corresponding equation takes the
*[: = 0, P(6)(t) = o(m)(t) + Ul0(m-l)(t) + ... + umo(t).
P(o)
If, in the V~ algebra3 , we factor P(6), P(6)
= *II (6' -
Aj 6)k j
,
j
and take advantage of (9.9), we obtain
p-l (o)(t)
=Eft) = [*J} (0' -
Ajo)k;]
-1
= *J} (0' -
AjW k;,
(9.11)
eAt
(9.12)
But it is easy to verify that
*(6' - A6)-k = *[(6' _ A6)-lt =
*8(t)t
k
1 -
(k-l)!
'
74
1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
whence, by continuing the equalities (9.11), we derive
£(t)
=*
n O(t)t . (k j
-
J
k;-l
I)!
e
Ajt
.
(9.13)
The convolution (9.13) admits of explicit calculation. By decomposing the righthand side of (9.11) into partial fractions in the V'-t algebra, we obtain £(t) =
* II(JI -
AjJ)-k,
j
AjJ)-k j + ... + Ci,t
= L[Cj,kj * (JI -
* (6 ' -
Aj 6)-1],
j
whence, using (9.12), we finally derive tkj-1 _ j
£(t) = O(t) ~ [ Cj,kj (k J
]
1)1 + ... + Cj ,1 e
Ajt
(9.14)
.
We thus have the following rule for finding the fundamental solution of the operator p{ft): substitute p for set up the polynomial P(pL decompose the expression ~ into partial fractions:
it,
P;P)
= U(p - Aj)-k = ~[Cj,kj(P j
)..j)-k j
+ ... + Cj,l'(p _
)..j)-l] ,
J
J
and with each partial fraction (p - A)-k associate the right-hand side of formula (9.12). As a result, we obtain the formula (9.14). EXAMPLE.
Find E if E" + w2 £
We have p2
1
+w2
=~ ( 2Wl
P-
= J.
1. _ +1.) lW
P
lW
foot
O(t~
(eiwt _ e-iwt)
= O(t) sin wi = £(t). W
2Wl
5. Tempered Generalized Functions 5.1. The space 8 of test (rapidly decreasing) functions. We refer to the space of test functions S = S(lR n ) all functions infinitely differentiable in lR n that decrease together with all their derivatives, as Ixl ---1 00, faster than any power of Ixl-t. We introduce in S a countable number of norms via the formula 8 1 :J ... .
(1.2)
Each imbedding
8 p +1 C 8 p ,
p
= 0,1, ... ,
is continuous, by (1.1). We will now prove that this imbedding is totally cantin uous (compact), that is, it is possible, from each infinite bounded set in Sp+l, to choose a sequence that converges in Sp. Indeed, let M be an infinite set bounded in Sp+l, Ilcpllp+l < C, 'P EM. From this, for all If' E M and lal < p, we obtain
a~. aOlcp(x) < C,
j = 1, ...• n;
J
(1 Suppose Rk, k
+ IxI2y/2aal.p(x) -t OJ
Ixl-+ 00.
= 1.2, ... , is an increasing sequence of positive numbers such that (1.3)
By the Ascoli-Arzehi lemma there is a sequence {If'jl)} offundions in M that converges in CP(U R I ); furthermore, by the same lemma there is a subsequence {tp)2)} of the functions {cpJl)} that converges in CP(UR-J, and so on. It remains to k
remark that by virtue of (1.3) the diagonal sequence { R, [0:1 < p. [xl < R + 1, k ~ N 1.
(1.4)
Let N l be a number such that TJk(X) = 1, Finally, from Theorem II of Sec. 1.2 it follows the existence of a number N ~ N l such that for all k ~ N, Ixl ~ R + 1, and 10:1 S p, the following inequality holds true:
(1 + I x I 2 )p/2I aa cp(x) - 8 a CPl/k(X)1 < E.
(1.5)
1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
76
Now, using (1.4) and (1.5) for k
Ilcp -
~
N, we obtain
'Pl/k7]kllp == suPx(l + Ix]2y/2I aa [cp(x) - Ipl/k"1k(X)] !al:Sp
R+l lal:Sp
I
[18 alp (x)1 + /3R+l 1~I:Sp
~ 2e + C;
sup / wl/dy) Ixl>R+l IlJlsp
< 2£ + Cpe + Cpe / ~
[(1 + jx -
Wl!k (y) (1
y12y/2
+ IvI P ]
la lJ kll'Pkllkl
k=1,2,....
(2.2)
The sequence of functions
epk(X)
1/Jdx)
k=1,2, ... ,
= YkII'Pkllk'
tends to 0 in S because for k
~
P IIlf'k
1 lip Vk. o
The resulting contradiction proves the theorem.
From the Schwartz theorem we have just proved there follow a number of corollaries. COROLLARY 1. Every tempered generalized function has a finite order (compare Sec. 1.8), that is to say, it admits of an extension as a continuous linear functional from some (least) conjugate space S:n; then, for I, the inequality (2.1) takes the form
1(/, 'P)I where
IIfll-m
:s Ilfll-mll'Pllm,
is the norm of the functional
f
(2.3)
in S:n and m is the order of f·
Thus, the following relations hold true:
Sb
c
S~
c S~
C ... ,
S' =
US;. p~O
They are duals of (1.2) and (1.6).
(2.4)
1. GENERALIZED FUNCTIONS AND THEIR PROPERTIES
78
Also note that every imbedding
s; C s; +1 ,
P
= 0, 1, ... ,
is totally continuous (see Sec. 5.1); in particular, every (weakly) convergent sequence of functionals taken from S; converges in norm in S;+1' COROLLARY 2. Every (weakly) convergent sequence of tempered generalized
functions converges weakly in some space S; and, hence, converges in norm in
S;+1'
This follows from the Schwartz theorem since every (weakly) convergent sequence of functionals taken from S' is a weakly bounded set in S'; it also follows 0 from the remark referring to Corollary 1. COROLLARY
3. The space of tempered generalized functions is complete.
This follows from the completeness of the conjugate spaces
S; and from Corol-
lary 2.
[]
5.3. Examples of tempered generalized functions and elementary operations in S'. A function f (x) is called a tempered function in ~n, if, for some m> 0
f
A tempered function Sec. 1.6,
If(x)l(1
+ lxj)-m dx < 00.
f defines a regular functional f in S' via the formula (6.1) of (I, rp) =
!
I(x)rp(x) dx,
'P E S.
Not every locally integrable function defines a tempered generalized function, for example, eX rt. S'. On the other hand, not every locally integrable function taken from S' is tempered. For example, the function (cos eX)' = _eX sin eX is not a tempered function, yet it defines a generalized function from 5' via the formula
((coseX)','P) = -
!
rp E S.
cosexrp'(x)dx,
However, there can be no such unpleasantness as regards nonnegative functions (and even measures), as we shall now see. A measure Jl specified on ~n (see Sec. 1.7) is said to be a tempered measure if for some m > 0
f (1 + f
Ixl)-mJl(dx)
0 su.ch that
(4.2) PROOF.
Taking into account the equalities (3.2) of Sec. 5.3 and (3.3), we obtain,
for all t.p E S,
(8 Cl F[fL t.p) = (_l)IClI (F[J], 8Cl rp) = (-1 )Ia l (I, F[aa"k), k ~ 00, the special1-sequence in S (in V). By the convolution f 9 of generalized functions f and 9 taken from 5' we call the limit
*"
f"* 9 = lirn(ekJ) * 9 in S', if this limit exists for any special I-sequence {'7k} in S and does not depend on it. Then f *9 E S', If I 9 exists, then 9 f also exists and they are equal
"*
*
(5.4)
f*g=g*f.
The convolution example shows:
f
*" 9
is more general than the convolution
9 =~,
F(f]
~,
f·g =
f
*9
= 2 sin e,
e
as the following
F[g] = 1.
The convolution I * $i~ ( does not exist (see Kaminski [54]), however, the convolution 1 6i~ e exists and is equal to 11". (Prove this fact without using formula (5.6).) Let I, 9 E S' and the convolution 1"* 9 exist in S'. Then there exists F[/] , F[g] in S' and the formula of the Fourier transform of the convolution
*
F[J holds. Let
F(fJ "*
I, 9
E 8' and the product
* g] =
(5.5)
F(f] . F[g]
f .9
exist in 5', Then there exists the convolution F[g] in 5' and the formula of the Fourier transform of the product I
F[f , g) = (211")n F[f)
*" F[g]
(5.6)
holds. See the details and the proofs in Kaminski [54], Hirata, Ogata [45), Dierolf, Voigt
[16].
6.6. Examples. 6.6.1.
F
[e-
Q "']
= ~ e-£',
n= 1.
"'I' 0,
True enough, the function e-.:f~X2 is integrable on ~1 and therefore (0F [e- a2x2 ]
=
f e-a2x2+i~x = ~ f f ( dx
1 (2 = o-e-~
(6.1)
> 0)
e-.,.2+i!.,. du
.!..{.)2 e- "'+2'" du
~ = 0-1 e-4;2
/
e-( 2
de·
SC=eJ(2a)
In the last integral, the line of integration may be shifted onto the real axis and therefore 00
1 (2 = -e-~
/
0-
-li-:s e- q 2 du = e - ~'" , 0-
-00
o
6, THE FOURIER TRANSFORM OF TEMPERED GENERALIZED FUNCTIONS
6.6.2.
97
A multi-dimensional analogue of formula (6.1) is
F
nj2
=
[e-(AJ;,J;)]
1
(6.2)
e-}CA- €,{),
1r
vldet A where A is a real positive-definite matrix. To obtain (6.2) with the aid of a nonsingular real linear transformation x let us reduce the quadratic form (Ax, x) to a sum of squares
(Ax, x)
= By,
= (ABy, By) = (B T ARYl Y) = IYI 2 ,
Note that
= BBT
A-I
det AI det BI 2 = 1.
I
From this, using the formula (6.1), we obtain F
[e-CAX,J;)]
= f e-(Ax,x)+iC{.x) dx = Idet BI =
1
e-IYI2+1(BT {,y)
vldetA =
f e-(ABy,BY)+I(~,By) f II f
n/2
dy T
1 vldet A 1 0,
B(Q', v; 13, /1) =
~(3
> 0,
(6.37)
and 3r(a + (3)
f 7I"v(y)lyla-lrr~(l-
< 1. In (6.38)
(6.38) we set
y)11 - yl.B- 1 dy
1
= 2rrrv(a)r~({3)fv+~(1- 0" - (3).
(6.39)
The function B(a, v; 13, /1) is called the beta function of the characters rr v and 1T"1l of the field~. (Below, we shall define the integers v and /1 = 0,1 modulo 2; v, /1 E F2, so 1I"1I(x)rr~(x) = 1T"1I+~(x).) Equalities (6.37)-(6.39) can be meromorphic continued with respect to a and (3 to all complex pairs (0",13) E ([:2 (in this case, by virtue of Sec. 6.6.17, the poles are defined uniquely). As a result, we obtain the following equalities
P(1r v lxl a P(rrlllxl a -
1 1
)
) .
P(1r ll lxl.B- 1 )
* P(1T"ttlxl,B-l)
= P(7l"v+~lxlo:+.B-2),
(6.37')
= B(a, v; 13, /1)P(rrv+~lxla+,B-1).
(6.38')
Let us prove equality (6.39). To this end, we apply the formula of the Fourier transform of the convolution (see (5.5) of Sec. 6.5) to equality (6.38) and make use of equality (6.33). As a result, we obtain i V r v (a)P (rr v I~ 1- a) 'ilL r JL (j3)P ( rr IL I~ 1- 13 )
= iV+J.lB(a, v; (3, /1)f v+JL(a
+ j3)p( rrv+J.lI~I-o:-I3),
and this, by virtue of (6.37) and (6.36), imply equality (6.39) in the domain ~a > 0, 3rj3 > 0, W(a +;3) < 1. By meromorphic continuation on (0", (3) the relations (6.37)(6.39) are valid for all (a, {3) E «:2 except poles. 0 By virtue of (6.39) we can represent the beta function B on the variety a + {3 +, = 1, v + /1 + T/ = 0,0", {3" E C, v, jJ" E F 2 , in the following form symmetric with respect to the transposition of the arguments (Q',v), ((3,/1), (,,1]): 1
B(a,v;{3,/-l;"T/) = 211"f v (a)fJ.l((3)r 1J (,).
°
(6.40)
The equation v + jJ + T/ = in F 2 has only four solutions: 000, 110, 101, OIl. Therefore, there exists only fOUf beta functions (of the field IR). For v = P := T/ = 0, the beta function 1
Bo(a,,B,,) =B(a,O;(3,O;"O:= 2rrro(a)fo((3)I'o(,L
0'+,6'+,= 1, (6.41)
6. THE FOURIER TRANSFORM OF TEMPERED GENERALIZED FUNCTIONS
109
defines the crossing-symmetric amplitude of Veneziano in the quantum field theory and in the string theory (see Vladirnirov, Volovich, Zelenov [123], Green, Schwartz, Witten [41]). Let us note one more interesting representation of the beta function B o: 1 Bo(Q, p, ')') = 211" [B(Q, (3) + B(Q, ')'} + B(,8, /,)], (6.42) O'+f3+')'==l t where B(O',,8) is the Euler beta function,
B( Q, (3)
= f(Q)f(,8) f(et+,B),
6.7. The Mellin transform. Let IR* denote the multiplicative group of the field JR, JR'" IR \ {a} and d*x be the Haar measure on JR*, We denote
=M
=
p~
Ap(X) = max(lxl- P , IxIP),
O.
Let us introduce the countable- normed space S (IR.. . ) of Coo -funct ions
0 and
(j'
= a takes the form
J J =~ J 00
00
=~ 211"
X-iT
-00
'IjJ(t)e itT In x dt dr
-00
00
e- iT In x F[1JI]( r) dr
211'
-00
= p- 1 [P[1/;]J (In x)
= 1/;(ln x) = /f'(x),
by virtue of formula (1.3) of Sec. 6.1 of the inversion of the Fourier transform. Since
by virtue of (7.4) the estimate
IMo[/f'](a)lxl-al :::; C 2 (0')lrl- 2 x-0-
x > 0,
holds, the contour of integration (0" - ioo 0" + ioo) in integral (7.5) can be shifted (0' - ioo, cr + ioo) -7 (-ioo , ioo) by any shift 0' for any x > O. I
6. THE FOURIER TRANSFORM OF TEMPERED GENERALIZED FUNCTIONS
III
Similarly formula (7.5) can be proved for odd functions as well as for the arbitrary functions t.p = 'Peven + I{}odd taken from S(m. *), where 1 'Peven(X) = "2 [cp(x) + cp( -x)], are even and odd components of the function cpo In order to obtain formula (7.6), let us multiply equality (7.5) by 1/J(x) and integrate over all x. As a result, we obtain (7.6)
f
rno - 1, respectively) .
Consider the case f(x) = 0, Ixl < a. The fact that the functions M v [J]( 0:), V 0 1, are holomorphic in the half-plane (]" < 1 - rna follows from definit.ion (7.6) and from the inclusion 1]1l"vlxla-1 E Sm (lR'"), by virtue of the estimate PROOF.
=
I
11](x)ll'v(x)1
Ixl a - I S CAja_ll(x) < CAmo(X).
112
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
Inequalities (7.7) follow from inequalities (7.1) by virtue of the estimates I
Mil [J](a) I ~
=
IIIII-m 11111l"lIl x la- 1 m 1I
1I111-m
suPx Am(X) 1(11 Xa - 1 )(k)1 Ob O~kJ~m
< C:n (O")II/I1-m In - rnl m :S Cm(O")lIfll-m(1 + ITl m ). (Here, the number b < a is such that 1J(x) 0 for Ixl < b.) Now let us prove the inversion formula (7.8). Let {/k, k --+ oo} be a sequence of functions fk E S (~"') converging to f in S' (~.), and such that fk (x) = 0, Ixi < b, k = 1,2, .... (Such a sequence always exists, compare Sec. 5.6.2). Then Ilf - !k Ilmo+l --+ 0, k --+ 00 (compare Sec. 5.2). Applying equality (7.6) to
a, can be considered similarly. Theorem I' is proved. 0 In the case of an arbitrary generalized function the form of the sum
f(x) = lo(x) + foo(x),
fa,
f
E S' (l~"), we represent it in
100 E S'(IR·),
(7.9)
where la(x) - 0, Ixl < a and foo - 0, Ixl > b. (One can always make this, by setting fa = (1 -11)!, 100 = 111, where TJ E V, 17(x) 1 Ixl < a.) By the Mellin transform of the generalized function f E S' (IR "') we call four functions
Mv[IJJ](a) =
~ (fJJ(x), 1rv(x)lxla-1) ,
V
= 0,1,
fJ = 0,00.
lfthe order of 1 E S'(IR*) is mo, then the functions Mv[fJJ](a) satisfies estimate (7.7) and the inversion formula If' E S (IR• ),
(7.10)
holds; moreover, the integrals in (7.10) does not depend on 0"0 < 1 - rna and U oo > rna - 1, and the right-hand side on (7.10) does not depend on the representation of f as the sum (7.9). In particular, if 1 is a generalized function taken from S'(IR*) with a compact support in R*, then, setting fa = I, foo = 0, we obtain that its Mellin transform
7. FOURIER SERIES OF PERIODIC GENERALIZED FUNCTIONS
113
1
o
-2T 4
x
Figure 27
Mv(J](Q), V = OJ 1, (7.6) is an entire function satisfying estimate (7.7), and the inversion formula (7.10) takes the form (!, rp)
=
:i L
J
O'+ioo
rp E S(lFP!).
Mv[f](a)Mv[ 0,
since Q' * cp E V if a E £' and cp E V (see Sec. 4.2.7 and Sec. 4.6). Thus, condition (1.4) is fulfilled. Conversely, suppose the condition (1.4) is fulfilled, so that if a E V, then f * (]I * Q* is a continuous positive definite function and therefore, by (1.1), (f * Q * 0:* )(0) 2: O. Now taking advantage of formula (6.3) of Sec. 4.6, we have, for all a E V. 1
(/( -y), 0:
* 0:*)
°=
= (I, * a*)( -y)) =([ * * 0:*)(0) > ° (0:
0:
so that I(-x»> and therefore I» 0. If I »0, then I 1* . Indeed, from what has been proved, for all a E V,
(I * 0: * a*)"'
=
r
* (a * a*)
=
o
I * (a * a*).
If in the last equation we let a -+ J in £1 [and then a* -+ J in £' and from formula (5.1) of Sec. 4.5 it follows that 0: * a* -+ J in £' as well] and use the continuity property of a convolution (see Sec. 4.3) I we obtain I = 1*. 0 For what follows we will need the following lemma. LEMMA.
For every integer p ~
°
2
w E C p;
there is a function w(x) with the properties: w(x) = 0,
Ixl > 1;
F[w](~) > (1 + I~;rn+l PROOF.
Let X E V, X(x) = 0 for
l'(x)
(15)
Ixl > 1/2 and
1 J e-i(x,Od~ [1 ] = (2rr)n (1 + 1~12)p+n+l = (1 + 1~12y+n+l . p-l
Let us verify that the function w = I'(X * X*) has the properties (1.5). Since l' E C 2p and X * X* E Coo, it follows that w E C 2 P. Furthermore, by virtue of Sec. 4.2.7,
8. POSITIVE DEFINITE GENERALIZED FUNCTIONS
123
suppw C supp X + supp x· C U1 / 2 + U1 / 2 == U1 • Finally, using the formula of the Fourier transform of a convolution (see Sec. 6.5), we have
F{w](~) = Fb(x
* x*)]
- F [F- 1 [
-
1
(1 + reI2)p+n+l
(2..)n (1+ 1
f
- (21r)n
] F- 1 [F[ }F{ .]]]
X
X
ll~ F)'+n+1 * IFlxJl'
IFx1l 2 (y) dy (1 + I~ _ yI2)p+n+l
and therefore
A . - (1 + 1€1 2 )p+n+l
>
o
The proof of the lemma is complete.
8.2. The Bochner-Schwartz theorem. Suppose / E V' and / in the ball U3 = [x: Ixl < 3], / has a finite order m ~ 0 (see Sec. 1.3),
1(/, '1')1 < I 0, in the form
= (1 -
/(x)
Ll)m fo(x)
where fo (x) is a continuous positive definite function.
This is a consequence of the following chain of equalities:
f
= Flit) = F
= (1 where the measure v finite on JRn.
[(1 + I€IT (1 +
~12)m]
~)m F[v]
= fL(1 + 1~12) -m > 0, for sufficiently
8.3. Examples. 8.3.1. Let the polynomial P(~)
2::
large m, may be made 0
O. Then
P( -i8)6
» O.
In particular, 0 » o. 8.3.2. If f» 0 and gEE', g» 0, then / *g» O. Indeed, the measure F[g] > 0, F[g] E OM (see Sec. 6.4) and therefore the o measure F[J * g] F[J]F[g] ~ 0 is tempered.
=
126
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
8.3.3. If f» 0 and F[g] E £', 9 »0, then gf »0. Indeed, 9 E OM, gl E S' and F- 1 [gf] = F- 1 [g]
* F- 1U]
is a nonnegative measure taken from S' and, hence, tempered (see Sec. 5.3). 0 8.3.4. e-(Ax,x»> 0, where A is a real positive definite matrix (see Sec. 6.6.2). 8.3.5.
1
~»
0, n
= 3 (see Sec. 6.6.7).
=
8.3.6. lI"J(x) ± iP~ »0, n 1 (see Sec. 6.6.10). 8.3.7. For f E V!r to be positive definite, it is necessary and sufficient that its Fourier coefficients Ck (I) be nonnegative. Then for all'P E coo n V!r the following inequality holds:
(I, 'P ® 'P·)T
~ O.
(3.1)
This follows from the theorem of Sec. 7.2, by which theorem
F- 1 [f] =
L cdf)8(€ -
kw),
(3.2)
Ikl?:o
and from the Bochner-Schwartz theorem. To prove the inequality (3.1), let us take advantage of the machinary developed in Sec. 7. Using (3.1) of Sec. 7.3 and (3.3) of Sec. 7.3, we have the chain of equalities (/,
0 is chosen so that it is positive for z iy, y > O. Let (} > O. Then
=
" )_!oo~D:_l -Y{d c 1 L[Ia: ]( 1, Y r ((}) e ~ - yO' f ((} ) o
/00
a-I
u
e
-ud _ 1 u - yO'
0
so that the functions L(Ja](z) and (-iz)-a, which are holomorphic in the upper half-plane, coincide on the line z = iYI Y > O. By virtue of the principle of analytic
9. THE LAPLACE TRANSFORM OF TEMPERED GENERALIZED FUNCTIONS
continuation, (3.1) holds for Q' > O. But if Q < 0, then a + m m. Therefore, fer f~r:;~ and, by what has been proved,
131
> 0 for some integer
=
L[fa]
= L[f~~~J = (-iz)m L[Ja+m] . )ffl( -zz . )-a-m = ( -zz
= (-'lZ. )-er .
9.3.2.
L[O({)sinw{] = L[O({) cosw~]
2
w
=
W
-
2' Z
(3.2)
-lZ 2 2'
-z
These follows from the equations [see 9.3.1 for (}
9.3.3.
w
= IJ
Let us prove the equation
v>
-1/2.
(3.3)
By 9.3.1 we have the equations
Therefore, using the formula for the Laplace transform of a convolution, we have
But
I (1 1
=
O({){2V f2(v + 1/2)
e-i{v
v 2) v-l/2 dv 4 2
-1
8(eh/1r
= f(v + 1/2) where u Sec. 6.6.
= (v + 1)/2, and
(e)V 2" JII(O,
(3.3) is proved. Here we made use of formula (6.31) of
132
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
9.3.4.
We now prove the formula {
sine
=
f
(3.4)
lo(e - t)Jo(t) dt.
o Since the right and left members of (3.4) are odd, it suffices to prove the formula for ~ > O. It is therefore sufficient to prove the convolution equation Osin~
= (Olo) * (Bla),
which is equivalent, by virtue of 9.3.2 and 9.3.3, to the trivial equality 1 1 1 y> O. 1 - z2 - ";1 - z2 ";1 - z2 ' 9.3.5. algebra
S+.
D Let us find the fundamental solution £ of the operator ({)]o)* in the By 9.3.3, we have
L[Ola]
= ..;1 1-
f: 0,
z2
y
> O.
Consequently,
L[£]
1-
z2
= VI - z2 = ";1~=~ z2
whence
t'(~) == O(€)Jo(€) + [O(~)Jo(~)]" == B(~)Ja(~) + 6f(~)JO(O + 26(~)J~(~) + O(~)J~'(~)
== _ O(~):~(e) + o'(e), that is
t'(e) = B(e) J 1 (~)
e
9.3.6.
Let f E
+ c5f(~).
.cloc n'Dr (see Sec. 7), n = 1.
(3.5)
Then
f !(~)eiil€ d~. T
L[O f]
= 1 - 1e . T 1Z
(3.6)
a
Indeed,
f f
00
L[B/Hz)
=
l(e)e iZ {
d~
o
T
00
=::
eiz(t+T)
+ T) dt +
f(t
o
f
eiz €1(0
d~
a T
=eizT L[O f] +
f
eiz { !(e) de,
a whence follows (3.6).
o
10, CAUCHY KERNEL AND TRANSFORMS OF CAUCHY-BOCHNER AND HILBERT 133
10. The Cauchy Kernel and the TransforIns of Cauchy-Bochner and Hilbert 10.1. The space 1£50 We denote by functions g(O with finite norm
£;
the Hilbert space consisting of all
We denote by 1l s the collection of all (generalized) functions I(x) that are Fourier transforms of functions in 1.:;, I = F[g], with norm (1.1 ) The parameter s can assume any real values. Clearly, 1£0 = £2 = £~ and
IlglI(o) = IIgll = (271")-n/21Ifll
=
11/110
by virtue of the Parseval-Steklov equation (see Sec. 6.6.3). From the definition of the space 1l s we find that for I E 1l s it is necessary that the function I be representable as
I (x) =
(1 - L\) m !l (x),
= 1+
m
h
E 1.: 2 ,
[- ~],
if s
m = 0,
£;.
c
< 5,
1l s
c
1l SiC 8',
s
,
s
> 0; (1.2)
< O.
The space 1l s is the Hilbert space isomorphic to
8
if
And
where inclusion is to be understood as embedding together with the appropriate topology, 1I/IIsi < lillis, f E ll s . Let us now prove that S is dense in 1l s . By virtue of (1.1) it suffices to prove that 1) is dense in 1.:;. Let 9 E 1); and c > O. Then
But V is dense in £2 (see Sec. 1.2, Corollary 1 to Theorem II). Therefore there is a function 1/J1 in 7J such that 111/J -1Plll < c. Putting 91 (~) =
we obtain
Ilg - gillts)
=
?h (~)( 1 + 1~12) -s/2
E 7J
f Ig(~) - gd~)12(1 + 1~12r d~
= 111P -1P111 2 < c 2 , o
which is what we affirmed. Let us now prove that -=l
1l s C Co
if
l
is an integer,
l
<s-
n/2.
This assertion is a simple special case of the Sobolev imbedding theorem (Sobolev [97]). REMARK.
134
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
To prove this, note that if f E 1£,,, then for all
10'1
~
I
< s - n/2
{a p - 1 [lJ E I: 1
by the Cauchy-Bunyakovsky inequality
11~ s + n/2.
(1.11)
Indeed, since f E 1£s and 9 E 1ls for all s, it follows, in particular, that f E [,2 and 9 E £2. We put j F- 1[f] and 9 == p-l[g]. Using the definition (1.1) of a norm in 1£5' the formula of the Fourier transform of a convolution (see Sec. 6.5), the Fubini theorem, the Cauchy-Bunyakovsky inequalities, and
=
138
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
for all
5 ~
Il/gII; =
0 and p> s
+ n/2,
f IF-l[Jg](~)12
= 1(1
(1
we obtain the inequality (1.11):
+ 1~12r d~
+ 1~12r 1 j(~')(1 + 1~'12)P/2 g(~ - e) de (1 + leI 2)p/2
2
d~
~ !IJ(()1 2 (1 + 1e'1 2 y dE' ! (1 + 1~12V !Ig(~ - e)12(1 + 1~'12)-P de dE.
! < Ilfll~ ! + 11]1 ! (1 + 1e'1 r- d~' de = ! + 1~'12)P-S IIfll llgll =
2
11/11; Ig( 1]) 1 (1 + Ie' + 171 2 )"' (1 + 1e'1 2 ) -p dry de' 2
2
19(1]) 1 (1
)"'
2
(1
dry
2
p
2 s'
p
o We set
Sb = U£:s = F['D~2] 8>0
as the inductive limit (union) of the spaces £~8' s = 0, 1, .... For f to belong to S', it is necessary and sufficient that it be representable as
f(x) = x fo(x), Q
fa E V~2'
(1.12)
Sufficiency is obvious and necessity follows from the representation
=
where 9 is a tempered continuous function in IRn (see Sec. 5.4), that is, fa F- 1 [g] E 1i s for some 5, whence it follows the required representation (1.12). 0 Let the generalized function fa from S' be continuously dependent, in S' on the parameter u on the compact f{, that is, (fl1l Ip) E C(K) for any Ip E S, and let p, be a finite measure on IC We introduce the generalized function fK foJl(du) taken from S' by means of the equation J
cp E S. It is easy to see that
(1.13)
(cf. definition of Sec. 2.7).
10.2. The Cauchy kernel Kc(z). Let C be a connected open cone in jRn with vertex at 0 and let C* be the conjugate cone C (see Sec. 4.4). The function Kc(z)
=
!
C·
ei(;;.O
d~ =L[Oc·] = F[Oc. e-(Y'{)]
(2.1)
10. CAllCHY KERNEL AND TRANSFORMS OF CAUCHY-BOCHNER AND HILBERT 139
is termed the Cauchy kernel of a tubular region T e ; here, Be. (~) is the characteristic function of the cone C*. If the cone C is not acute, then by virtue of Lemma 1 of Sec. 4.4, mesC* = 0 and, hence, Ke(z) O. Furthermore, since C* = (ch C)*, it follows that Kc(z) _ KchC(z). Therefore, without restricting generality, we may regard the cone C as acute and convex. By what has been proved (see Sec. 9.1)' the kernel Kc(z) is a holomorphic function in T C ; and, moreover, the integral in (2.1) converges uniformly with respect to z in any tubular region T K , K @ C (K is a compact). We will now show that the kernel Ke{z) can be represented by the Figure 28 integral drr
I
(z,O")" ,
z E TC .
(2 2)
.
prC·
Indeed, since (y,O') > 0 for all y E C, rr E pr C*, it follows that the denominator of the integrand on the right of (2.2) is equal to [(x, 0") + i(y, (T)r and does not vanish in T C , and, consequently, the right-hand side of (2.2) is a holomorphic function in T C . Since the kernel Kc (z) is also a holomorphic function in T C , it suffices to prove (2.2) on the manifold z iy, y E C. But when x 0 the formula (2.2) follows readily from (2.1):
=
Ke(iy)
=
I
II
=
00
e-(Y'{)
d~ =
C·
I
e- p (y,a)pn-l
dpdO"
pr c· 0
00
=
I
prC·
drr (y,o-)n
e
-u
u
n-l
du
= z'n r ( n )
a
I
dO" (iy, (T)n .
prC·
From the representation (2.2) it follows that the kernel Kc(z) and also the kernel K-c are holomorphic in the domain
D=cn\
U
[z:(z,O")=O].
aEpr c·
It is easy to see that the domain D contains the tubular domains T C and T- C and also the real points of t.he cones C and -C. The kernels K c and K- c satisfy the relations K-c(z) = (-1)nKc(z) = Kc(z) = Kc(-z), Kc(.\z) = .\-nKc(z)
A E ([1 \ {OJ,
z ETC U T- e .
(2.3)
140
2. [NTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
Let us now prove the estimate
18 a IC c (z)1 :S
z E T C UT- C ,
MoA-n-1al(y),
(2.4)
where A(y) = t1.(y, -Be U Be) is the distance from y to the boundary of the cone -CUC:
=
~(y)
inf (0", y),
YEC
aEpr c·
(see Sec. 0.2 and Fig. 28). Indeed, using the representation (2.2), we have, for z E T C , the estimate (2.4):
f
a
18 Kc(z)1 ~ M
Q
l(Tal dfJ
!(z, fJ)ln+1a!
prC·
< Ma
sup (y, fJ)-n-la l aEpr
c·
= MaD.. -n-!QI(y).
The estimate (2.4) for z E T- c follows from (2.4) that was proved for z E T C and from the properties (2.3). More rigorous reasoning yields the estimate
18 ° K c(z)l:s
M~~-n+l-lal(y)[lxI2
+ ~2(y)]-1/2!
(2.4')
z ETc UT- c .
> 0, lIa Kc(x + iy)118 ~ Ks,a (1 + ~ -s (y)] ~ -n/2- la l (y), y E -cue.
We now prove the estimate for all s a
(2.5)
Indeed, by (2.1) and (2.2) for Y E C we have the estimate (2.5):
1l8 a Kc(x + iY)II; = I F - 1 [8 a K c (x + iy)] II~s)
= 11{-i~)aec.(~)e-(Y'()II:s)
f :s f
=
e- 2 (y,O (1
r
+ lel 2 l~al2 d~
c·
co
+ p2)s pn-l+2 Ia l
(1
o
f
pr
f e-2p~(Y)(1 +
e- 2p (y,u) dO" dp
c·
co
~ ~n
p2)spn-l+2 Ia l dp
o
f
2
00
-
fJ
n
- 2n+1+2Iol~n+2Ial(y) ~ K;,a [1
+~
-8
o
e
-u
[1 + 4~2(y) u ]
s
u
n-1+2Ial d U
(y)] 2 ~ -n- 2 Ial (y).
(Here, (Tn is the surface area of a unit sphere in ~n) see Sec. 0.6.) The case y E -C can be considered with the use of the relation (2.3). 0 The kernel Kc{z) assumes a boundary value equal to (±It F[t9±c·L respectively,
(2.6)
10, CAUCHY KERNEL AND TRANSFORMS OF CAUCHY-BOCHNER AND HILBERT 141
as
y
--+ 0, Y E ±C in norm in 1£3 for arbitrary s
< -n/2.
Indeed by what has been proved, K c (x + iy) E 1£ 3 for y E - C U C and for any s, while the generalized functions F[B±co ] E 1l s for all s < -n/2 (since ()±c o E for s < -n/2). Therefore, when s < -n/2 and y E C, Y --+ 0, we have I
.c;
IIKc(x + iy) - F[Bco]ll: = IIOcoe-(Y'O - ()coll~s)
=
!
1] 2 (1
[e-(Y.r.) -
+ 1~12r d~ --+ O.
Co
But if y E -C, Y --+ 0, then
Kc(x+iy) = (-ltKc(-x-iy) --+ (-ltKc(-x) = (-l)nF[O_c
o ]
and the formula (2.6) is proved, D From the formulas (2.3) and (2.6) we have the following relations for the boundary values of the kernels Kc{z) and K-c(z):
= Kc(x) = Ke(-x), x E ~n, } n = (-1) K c (x), x E CU (-C); 1 ~Kc(x) = 2"F[B c . + B- c ·]
K-c(x) K-c(x)
(2,'7)
1
= 2" [Kc(x) +Kc(-x)],
(2.8) <J'Kc(x) = ;iF[Oeo - O-c·] 1
= 2i[Kc(x)-Kc(-x)]. From this, taking into account the trivial equalities
(B c • - (Lc.)2
= (Bc. + (L c .)2 = Bco + B_co,
(B c • - O-c· )(Oc o + B-c·) = Bc· - O-c·, and making use of (1.9) for the convolution, we obtain the following relations between the generalized functions ~Kc (x) and r;sKe (x): -r:sKc
* ~J(c = ?RICe * ffiK c = ~ (2rr)n~Kc r;sK c
* ffiK c
I
1
= 2(211" tr;sKc,
(2.9) (2.10)
Let us now calculate the real and imaginary parts of the kernel Ke (x). To do this, we introduce, for k = 0, 1, . , " the generalized functions J(k)[(x,u)]
p(k)
1 (x,O") that operates on the test functions
O. We have
=
Kv+(iyo, O)
=
J J
e-Yo{o
v+
J J 00
de
=
e- Yo {
0
1€1' pepp d3. Fn
Then the transformation ~pq -+ J~pq equal to
P.1 ... An)n =
>'pA'I
carries Pn onto itself, and its Jacobian is
(det Yo)n = (det y)n. Consequently,
ICP n(iYa)
= J(p" (iY) = (det'Y)-n /
e- Tr3 d2.
Pn
The last constant has been computed (see, for example, Bochner (6])
f
e- n-s dB
=
1r n (n-l)/2l! ...
(n - 1)!.
F..
Therefore Kp..{iY) = i
n2
1I"n(n-l)/21!· ..
(n - I)! [det(iY)] -0,
YE
Pn.
Extending this relation to all Z E TFn, we obtain the formula (2.18). 10.3. The Cauchy-Bochner transform. Suppose
f(z)
= (2~)n (f(x'), Kc(z - x')),
0
f E ll. s . The function
z E T C U T- c ,
(3.1)
is called the Cau.chy-Bochner transform (integral). It is assumed here that the cone C is convex and acute. Since Kc(x + iy) E 1£3 far all sand y E -C U C (see Sec. 10.2), it follows that by (1.10) the right-hand side of (3.1) may be rewritten in the form of a convolution:
f(z)
= (2~)n f(x') * Kc(x' + iy)
I
Z
E
r C U T- c .
(3.2)
10. CAUCHY KERNEL AND TRANSFORMS OF CAUCHY-BOCHNER AND HILBERT 145
When n = 1, C = (0,00) and I E (3.1) turns into the classical Cauchy integral: EXAMPLE.
J
£2!
the Cauchy-Bochner integral
00
I(z) =
~ 27rl
f(x') dx'. X' -
Z
-00
The function I(z) is holomorphic in T C U T- c and I
80: f(z)
1 = (21l")n (l(x')J 80:Kc(z -
Xl)),
10·I( z) I :'0 ~';'i: 11/11. [1 + ~. (y)] ~ -n/2- 1• I(y),
(3.3) (3.4)
where the numbers Klsl,o: are the same as in the estimate (2.5). The holomorphy of the function I(z) in T C U T- c and the differentiation formula (3.3) follow directly from the facts that the Cauchy kernel Kc(z) is a holomorphic function in T C U T- c and Kc(x + iy) E ll s for all sand y E -0 U 0 (see Sec. 10.2). The estimate (3.4) follows from (3.3) and also from the lemma of Sec. 10.1 and from (2.5):
180:/(z)l:s =
(2~)nll/llsI18aKc(z-x')II_s Ilfll.,
(21r)n
Ila
u
Kc(x + iy)
1 0 there is a number M(£) such that LEMMA.
Ilf(x + iy)lls < M(c)e(a+€)!yl [1 + ~ -'Y(y)] , y E C,
(5.2)
for certain 5, a ~ 0 and, ~ 0 (all dependent only on f). Then f(z) is the Laplace transform of the function 9 in £;,(C* + era), where Sl == s if, == 0, s' < s - , if , > 0; here the following estimates hold true:
llgll(s) < 2 inf M(E), 0 (see Fig. 23). This inequality also holds, in continuity, in a sufficiently small
o
10 CAUCHY KERNEL AND TRANSFORMS OF CAUCHY-BOCHNER AND HILBERT 149
neighbourhood
Ie -
~ol
< 6. Therefore, putting
y
t > 0, the inequality
J
e 2t (a+x)
Ig(~)12(1 + 1~12r de
:::;
= tyo in (5.6), we obtain, for all !g(e)1 2e- 2(Y'0(1 + 1€1 2 r de
/ 1(-(01 O. Let 1=0. Passing to the limit in (5.7) as t we obtain
J Ig(~)12(1 + 1~12r
de
~
(5.7)
+0 and using the Fatou lemma,
:S 4M 2 (c),
c
> 0,
C"+Oo
whence follows the inequality (5.3). Now let I > O. Take into account the inequality ((1,~) :S I~ Ldivide the inequality (5.7) through by t l - 26 , where 6 is an arbitrary number 0 < 6 < 1, integrate the resulting inequality with respect to t on (0,1), and take advantage of the Fubini theorem. Assuming 0 < c < 1, we then obtain the inequality
f
1
Ig(~)12(1 + 1€1 2 f
c"+Oa
J
t2h+O)-le-2tIEI
dtd~
0
Now I taking into account the estimate
J 1
t2h+o)-le-2tlel dt
J 1
> min (1, 1€1-
»)
2 h+ O
o
u 2-y+le- 2u du
0
-2
~ 2(~ + 1) (1 + lel 2 ) -1-
0 ,
we derive from (5.8) the estimate
J
2(a+2)
19(~)12(1 + 1~12r-1-6 d~:S e 6
b + 1)M 2(c)[1 + ~--Y((1)]2,
ce+O a
=
whence it follows that 9 E £;,(C" + Va) for all s' S - "I - 6 < 8 - "y and the estimate (5.3') holds true. Finally, from (5.4) it follows that f = £[g]. The proof of the lemma is complete. 0
150
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
For the function f(z) to belong to the class H~s)(C), it is necessary and sufficient that its spectral junction g(~) belong to the class .L:;(C'" + Ua). Here, the following equalities hold: THEOREM.
Ilfll~s) =
Ilflis = Ilgll(s)I
(5.9)
where f+(x) is the boundary value in 1£s of the function f(z) as y ~ 0, y E C, and f+ = F[g].
Let f E His)(C). Then from the lemma [for 'Y = 0 and M(c) independent of e] it follows that f(z) £[g], where 9 E £;(C· + Ua ). SUFFICIENCY. Let f(z) = £[g], where 9 E £;(C'" + Ua ). By what has been proved (see Sec. 9.1), the function f(z) is holomorphic in T intC •• T C and it is given by the integral PROOF. NECESSITY.
=
=
f
f(z) =
g(~)ei(z,O d~ = F[g(~)e-(Y'O],
z E Te.
(5.10)
c·+u o Let us prove that f E His) (C). Using the relations of the norms in the spaces H~s)(c), 1£3 and we obtain from (5.10)
£;,
Ilfll~srl
= sup e- 2a1Y 'lIf(x + iY)II; yEC
= sup e-2alyl yEC
Ilg(~)e-(Y'O 11
(3)
J Ig(012 e-2(y,~) (1 + 1~12r d~
= sup e-2alYI yEC
2
c·+O a
That is, (5.11)
We now prove that the function f(z) assumes, when y ~ 0, y E C, a (unique) boundary value in 1l s equal to f+ = F[g]. This follows from the limiting relation
Ilf(x
+ iy)
- P[glll; = IIL[g](x + iy) - F[g](x)lI;
=
f
Ig(e)[2
[e-(Y'O -
If (1 + 1~12r d~ -} 0,
y
~
0,
Y E C.
c·+u a Thus, complete.
IIgll(s)
= 11/+118'
which together with (5.11) yields (5.9). The proof is 0
COROLLARY 1. The spaces
H~s\C) and £;(C'" +Ua ) are (linearly) isomorphic
and isometric, and the isomorphism is realized via the Laplace transformation 9 ~
L[g] = f. 2. Any function f(z) in H~s)(C) has, for y ~ 0, Y E C, a (unique) boundary value f + (x) in 1I. s and the correspondence f ~ f + is isometTic. COROLLARY
The theorem on the existence of boundary values in V cp was proved by a different method by Tillmann [103] and Luszczki and Zieleiny [70] (n = I). REMARK.
10 CAUCHY KERNEL AND TRANSFORMS OF CAUCHY-BOCHNER AND HILBERT 151
3 (an analogue of Liouville's theorem). It the cone C is not acute and f E Has) (C), then f(z) - o. COROLLARY
True enough, by Lemma 1 of Sec. 4.4, mes C* = O. In that case, as follows from the proof of the Lemma, the function g(~) = 0 almost everywhere in lR n so that f(z) L[g] _ O. 0
=
10.6. The generalized Cauchy-Bochner representation. Here we con-
tinue the investigation started in Subsection 10.5 when a
= O.
I. For a function f(z) to belong to H(s)(C), it is necessary and sufficient that it possess the generalized Cauchy-Bochner integral representation THEOREM
z E TC z E T- c ,
(6.1)
where f+(x) is a boundary value in 1i s of the function f(z) as y ---1- 0, Y E C.
Let f E H(8)(C). By the theorem of Sec. 10.5,1(11) is the Laplace transform of the function 9 in £ ~ (C*) so that PROOF. NECESSITY.
f{z) = F[g(~)e-(Y'OOc· (~)],
z E TC,
z E T- c .
o= F[g(~)e-(Y'OO_c. (~)],
From this fact, using the definition of the kernel Kc{z) [see (2.1)] and using (1.7) and (1.9) for the convolution, we obtain the representation (6.1):
f(z)
= (2~)n P[g] * Kc = (2~)n (f+{x'), Kc(z -
0=
(2~)n F[g] * L
c=
x')),
i;~?: (f+(x'), Kc(z -
x')),
Z
E T- c .
=
Here we made use of one of the equalities of (2.3): IC-c{z) (-1)nKc(z), and also used the relation f+ P[g]. SUFFICIENCY. Suppose f(z) has the representation (6.1). Then f E H(s)(C) (see Sec. 10.3). Theorem I is proved. 0
=
=
=
For s 0, Theorem I becomes the Bochner theorem [6]; for n 1, Theorem I was obtained by Beltrami and Wohlers [4]; for arbitrary nand s see Vladimirov [101]. REMARK.
THEOREM
II. The following statements are equivalent:
(1) f+ is a boundary value in 1f:; of some function taken from H(:;)(C); (2) f + belongs to 'Ji s and satisfies the relations 2
s.Rf+ 0, Y E C. 11.1.2.
J
Pc(x, y) dx = 1,
y E
C
(1.5)
follows from the Parseval-Steklov equation applied to (2.1) of Sec. 10.2:
J
IKc(x
+ iY)1 2
dx = (27r)n
Je~2(y,O d~
= (27rt ]{c(2iy),
y E C.
c· 11.1.3. Pc (x, y)
s + 11./2,
(1.13)
so that Pc(x, y) E 1£s for all sand y E C. This follows from the inequalities (1.11) and (2.5) of Sec. 10.2 and from the estimate (1.12):
118:Pc(x,y)lls
= (21r)n~c(2iY) IlaQKc(x + iy)Kc(x + iy)IIs
:l:,. L (;)8~ + :::,.c L (;) 118~Kc(x
1
:0; (2
Kc(x
iy)8a-P Kc(x
~
1
:0; (2
p -,
s
+ iY)II,
~
:0; (;;;-:,.
+ iy)
W-~Kc(X+ iy)ll.
L (;) K,,~Kp.a-~ [1 + ~ -'(y)] jJ
x
[1 + il-P(y)]il-n-lal(y)lyln.
o
11 POISSON KERNEL AND POISSON TRANSFORM
155
11.2. The Poisson transform and Poisson representation. Let for some s, -00 < S < 00. We call the convolution [see (1.10) of Sec. 10.1]
F(x, y) = f(x)
f
E 1l s
* Pc(x, y)
= (f(x / ), Pc(x - x', y)),
(2.1)
the Poisson transform (or integral). By virtue of Subsec. 11.1.10, the Poisson integral exists for every y E C and is continuous operation from 1l s to 1l s . If Poisson integral: EXAMPLE.
f E
£,2
= 1l o , then the Poisson integral becomes the classical
:F(x,y)
=
I
f(x')Pc(x-x',y)dx'.
The following is a partial list of the properties of the Poisson integral. 11.2.1.
:F(x, y) E Coo (T c ).
(2.2)
This follows from (1.4) and from (1.13). 11.2.2.
(2.3)
Y E C.
This follows from (1.8) by virtue of the following manipulations:
11:F(x,y)lI; = IIFx-l[.:F(X,y)]II~$) = IIF[f]F;l[Pc(x,y)]II~s) < IIfll;· 11.2.3.
(generalized Poisson representation). For f(x) to belong to H(s)(C), it is necessary and sufficient that it be uniquely represented as the Poisson integral THEOREM
f(z) = (X(x'), Pc{x - x', y)),
z ETc,
(2.4)
where X E 1i s and suppF-l[X] C C"'; here, X = f+ where f+(x) is the boundary value in 1l s of the function f(z) as y --t 0, Y E C.
Since f E H($)(C), it follows, by the theorem of Sec. 10.5, that there is a function 9 E £;(C*) such that f+ = F[gJ E 1I. s and PROOF. NECESSITY.
f(z)
=F
[g(~)e-(Y'€)] (x),
z E Te .
(2.5)
From this, using (1.10), we obtain for the function f(z) the generalized Poisson representation (2.4):
f(z) = F [g(~)Fx [Pc(x, y)](e)] = f+(x)
* Pc(x, y)
= F[g](x) * Pc(x, y)
= (f+(x
l ),
Pc(x - x', y)) ,
z E TC.
The generalized Poisson representation (2.4) is unique since, by (1.8), F x- 1 [Pc(x, y)] (e)
¥ 0,
eE IR
n
,
y E C.
Suppose a generalized function X is such that 9 = p-l [X] E £;(C*). Then by the theorem of Sec. 10.5 the function f(z) defined by (2.5) belongs to H($)(C) and, by what has been proved, can be represented by the integral (2.4) wi th X = F [g] = f +. This completes the proof of the theorem. 0 SUFFICIENCY.
156
2 INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
COROLLARY
1. Under the hypothesis of the theorem, we get
?Rf(z) = (?Rf+(x'),P(x - x/,y)), ~f(z) = (~f+(x'), P(x - x',
(2.6)
y)).
COROLLARY 2. If f(x) is a real generalized function in ?is and supp F(J] C -C* U C*, then the function
u(x,y)
= (f(x'),Pc(x -
x/,y))
(2.7)
is a real part of some function of the class H(s)(C) and assumes, in the sense of ?is as y -t 0, Y E C, the value of f(x).
Indeed, putting
f +(x) =
F [() C· (~) F - 1 [I] (~) J(x) ,
we obtain
so that
u(x, y)
= 25R (f+(x'), Pc(x -
x', y)) .
o
From this and from the theorem follow the required assertions.
The function Kc(z+iy') belongs to the class H(s)(C) for all y' E C and s [see estimate (2.5) of Sec. 10 in which ~(y + y') 2: ~(y'), Y E C]. Suppose C I is an arbitrary (convex open) subcone of the cone C, C/ C C. Applying (2.4) to the function Kc (z + iy') of the class H( s) (G/), we obtain EXAMPLE.
Kc(z
+ iy')
= f KC(x '
(x, y) E
+ iy')PCI(X - x',y) dx ' ,
Tc ,'
(2.8)
I
y E C.
From this, using the Cauchy-Bunyakowsky inequality and (1.5), we obtain the following inequality: 2
2
IKc(z + iy')1 < fIKc(x' + iy')1 pcl(x - x', y) dx' f PCI(X - x', y) dx' = f PCI(X -
2
Xl,
y)[Kc(x ' + i yl)1 dx ' .
(2.9)
In terms of the Poisson kernel (1.1), the inequality (2.9) takes the form ') P cx,y+y (
ICc ( /)P cx,y (I ')d' O. By the properties (1) and (2) there exists a number c > 0 such that Iw(x) I :S 1 - E, Ixl > O. From this fact. taking into account property (3). we obtain
1=
;~I1J, [ j' yEC
"
dX] ,
Ixl> 0, ... , (y, en) > 0]
the limiting relation (3.1) admits of extension to a more general class of functions I;'(s), namely: if 0, Q' > a and f3 2: a (that depend solely on f). Then f(z) can, jor > 0: + n/2, be represented in the form
J
j(z)
= lO(z)!o(z), 10 E H~8)(C), S < -(3 - n(o - 1/2), lifo 11(8) ~ J{s,o inf M(£) inf [1 + ~ -fJ- n (O-1/2)(a)]. a 0 0 [see
[(en,xf+ (e n ,y)2] [(en, x)2
+ (J'2IyI2]
~ (o-IYI)2n-2[(el,x)2+ ... +(e n ,x)2+ a2IYI 2 J,
zET C
(2.5)
.
Since the vectors el, ... ,en are linearly independent, there is a number b > that (el' x)2 + ... + (en, X)2 ~ b21x1 2.
a such
From this, continuing the estimates (2.5), we obtain
Il(x
~ (aIYI)2n-2[b2IxI2 + u21yI 2L
+ iy)1 2
z E Te .
Taking into account the estimate thus obtained and the estimate (2.2), we have, for all z E T C I
Ih(x
+ iy)1
2
= If(x + iY)1 2 Il (x + iy)I- 26 < M 2(E)e 2(a+€)IYI (I + Ix + iyI2)O[1 + ~_)3(y)]2 (ulyl) 26(n-l) [b21x12 + u21y12] Ii
cr.
Therefore
J [1 + Ixl++ ~2(y)] ° J[1 + ~2(y)(1 + 1~12)]cr d~ + .
> n,
2
[lxj2
~2(y)]O
(1
but then 2ex
+ ~-fj-n(O-1/2)(y)]2,
dx
1~12)1i
< n(2 a + n/2, f(z) can be represented as (2.7), and the function h E H~s)(C)) s < -(3 - n(cS - 1/2), and satisfies the estimate PROOF.
(2.8) with some ](s,o. The estimate (2.8) is what signifies that the operation f --+ fa is continuous from H~cr,{3)(C) to H~s)(C). (2) --+ (3). Suppose j(z) can be represented as (2.7). Assuming is > 00 ~ n/2 to be integer and using the theorem of Sec. 10.5 and property 9.2.2 of Sec. 9.2, we conclude that the spectral function 9 of the function j can be represented as (2.9) that is, 9 E S'(C* + U a). (3) --+ (1). Let f = L[gL where 9 E S'(C* + Ua). Then j(z) is a holomorphic function in Tint C" = T C and can be represented in the form (see Sec. 9.1) f(z)
= (g(~)) 1J(~)ei(z,O),
where 11 E Coo; 11(~) = 1, ~ E (C* + U a y:/2; TJ(O = 0, ~ rt (C* + Ua)e; 18aTJ(~)1 < ccr(c); c is an arbitrary number, 0 < 6 :S 1. Since 9 E S', it follows, by the Schwartz theorem (see Sec. 5.2), that it is of finite order m. Furthermore, by what was proved in Sec. 9.1, 1J(€}e i (z,O E S for all z ETc. Hence, for all z E C the
r
12. ALGEBRAS OF HOLOMORPHIC FUNCTIONS
163
following estimates hold true:
If(z) I ~ Ilgll_mll7](~)ei(z,O 11m
= IIgll-m sup€ (1 + 1€1 2 )m/2I aQ [7J(~)ei(z,O] I Icrl::;m
~ IIgll-m
sup€
1001~m
(1 + 1€1 2 ) m/2
(Q) eJ3::;cr f3 L
:s K:n(E)lIgll-m(1 + Iz]2)m/2 < ]{:n (£) Ilgll-m (1 + Iz12) m/2
(Y,€l
sup (1 €E(C· +u.. ). sup 6EC·,
IzJ3llacr-J31](~) I
+ 1€1 2)m/2 e _(y,O
(1 + 16 + 61 2 ) m/2 e-(y,~J)-(Y,b)
161::;a+~
~ ]{~ (£) Ilgll_me(a+~)IYI (1 + [ZI2) m/2 sup (1
+ 161 2 ) m/2 e- (y,€d
€l EC·
:s K~ (£) Ilgll_me(a+~)IYI (1 + Iz 1 m/2 sup (1 + p2)m/2 2)
p>O
~ K;:' (elllgll-m e(a+, )1.1 (1 + Iz12) m/2 ~~~ [1 + a~;Yl
e-D.(y)p
r/
2 e -t,
that is
If(z)1
:s f{m(£)lIgll_me(l1+~)[YI
(1 + IzI 2 )m/2 [1 + ~ -m(y)] ,
Thus, the function f( z) satisfies the conditions of the lemma with ex = f3 = m and M(£) = Km(£)llgll-m. In this case, when 6 > m + n/2, it can be represented as (2.7), where fo E H~s)(C) for s < -m - n(J - 1/2) < 0, and it satisfies the estimate Ilfoll~S)
< ]{~ ollgll-m inf ,
f{m(E)
O 0 such that ~(y)
=
inf (0", y)
aEpr
c·
> xlyL
YE
ct.
From this and from the inequality (1.1) it follows the inequality (4.6) for E = 0, {3' = j3 and for certain a' > a and M'(C') 2: M. Conversely, if f(z) is holomorphic in T C and, for arbitrary C' @ C and E > 0, satisfies the est.imate (4.6), t.hen, taking into account that ~/(y) ~ lyL where Ll'(y) is the distance from y to 8e', we obtain f E H a+€ (G'), whence, by (4.3), it. follows that f E Ha(C). 0
168
2 INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
12.5. The Schwartz representation. Suppose an acute (convex open) cone C is such that the Cauchy kernel Kc(z) -lOin the tube T C :::: ~n + ie. Such cones C will be called regular 3 • For example, the cones ~+ and V+ are regular (see Sec. 13.5 below). If a cone C is regular, then its Cauchy kernel Kc(z) is a divisor of the unity in the algebra H(G), i.e., K C1 (z) E H(G). LEMMA.
Since Kc (z) =j:. 0 in T C , to prove the lemma it is sufficient to establish the following estimate (see Sec. 12.4): for any cone G' @ G there exist nonnegative numbers p, 0", and f3 such that PROOF.
However, this estimat.e follows immediately from representation (2.2) in Sec. 10.2 of the kernel K.c IKc(z)1 = f(n)jzl-n
lf [ rC·
y
X
(p 0-) I
du
+ ~ (q, 0-)
]"
P=~'
q=~,
if we note that the function dt7
is positive and continuous on the compact
hence, it is bounded from below by a positive number previous equality imply the estimate
Setting
Q'
IT
= 0-( G'). This and the
= nand f3 = 0, we obtain the statement of the lemma.
D
The Schwartz kernel of the region T C , where G is a regular cone, relative to the point zO = xO + iyo E T C is the function
S (z' zO) c,
= 2K c (z)Kc( -;0)
(2rr) nK c(z _ zO)
_ p .(x O 0) c ,y,
(5.1)
We note some properties of the Schwartz kernel. 12.5.1.
Sc(z; z)
= Pc(x, y),
(5.2)
This property follows from (5.1) when ZO = z, from the definition of the Poisson kernel (1.1) of Sec. 11.1, and from the property (2.3) of Sec. 10.2 of the Cauchy kernel. 3For n = I, 2, 3 all acute cones are regular; for n ~ 4 it is not the case (Danilov [14]); homogeneous cones of positivity are regular (Rothaus [86]).
12, ALGEBRAS OF HOLOMORPHIC FUNCTIONS
12.5.2.
f
Se(Z - x'; ZO - x') dx '
169
= 1,
(5.3)
This property follows from the Parseval-Steklov equation applied to (2.1) of Sec. 10.2,
f
Kc(z - x/)Ke( -zo
+ x') dx' =
f
Kc(z - x')Ke(zO - x') dx '
= (2 7r t
f
ei(z-zo,{)
d~
c· and from the property (1. 5), Sec. ILl, of the Poisson kernel. 12.5.3.
ISc(z; z ) I ~ 0
IICc(z - -I 'Pe(x, y) + 1 -I + zO) Ke(z - zO) Kc(2iy)
[ Kc(2iyO)
z E Te,
D
°
]
0
1 Pc(x, Y ),
(5.4)
zO E T C .
This property follows from the definitions of the Schwartz and the Poisson kernels and from the estimate 21abJ < lal 2 + Ib1 2 . D EXAMPLE
SlIRn
+
1 (see (2.16) of Sec. 10.2 and (1.2) of Sec. 11.1).
(z; zO) = Sn(z; zo)
=
(:i; 1f'
n
(~_ Zl
1 ) ...
ZOl
(~_ Zn
1 ) _ Pn(x o, yO).
zan
In particular, for n = 1, C = (0,00),
8, (2; 2°) = : EXAMPLE
G- I:aOI2) ,
2 (see (2.17) of Sec. 10.2 and (1.3) of Sec. 11.1).
r(ni 1) Sl/+(z; zo) =
_ ] (n+1)/2 [ -(z - zO)2
(
1l'(n+3)/2 (_z2)(n+1)/2 [_ (zO)2]
+1)/2 -
Pv+(xo, yO).
n
Let the boundary value f+ (x) of a function f(x) of the class H (C) (see Sec. 12.2) satisfy the condition (5.5) for some s and for all zO E T e . Then the generalized function (5.5) is the boundary val ue in S/ of the function f (z )ICe (z - zO) of the class H (C) and therefore the support of its inverse Fourier transform is contained in the cone C·. By Theorem II of Sec. 10.6, the function f(z)Ke(z - zO) belongs to the class H(s)(C) and its boundary value in 1l s is equal to f+(x)ICe(x - zO) since Ke(x + iy) E OM for all y E C [see Sec. 5.3 and Sec. 10.2, estimate (2.4)]. Applying Theorem I of Sec. 10.6
170
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
to the function f(z)Kc(z - z°), we obtain
f(z)Kc(z - zO)
= (2~)n (f+(x')Kc(x' Z
E T
C
O
z ET
,
C
z°),Kc(z - x')),
(5.6)
.
Putting zO = z in (5.6) and taking into account (5.2) for the function f(z), we derive the generalized Poisson representation
f(z)
= (f+(x'),Pc(x -
x',y)),
z E Te.
(5.7)
Then, interchanging z and zO in (5.6), we obtain
whence, passing to the complex conjugate, we derive
Subtracting (5.8) from (5.6) we get the relation I
Kc(z-zO)[f(z)-f(zO)]
= (2~n
zET
C
(8'f+(x'),K c (z-x')K(x'-zO)),
,
zO E T
C
(5.9)
.
Suppose C is a regular cone so that Kc(z) f:. 0, z E T C . Divide (5.9) by Kc(z - zO) and, in (5.9), in accordance with formula (5.7), make the substitution (5.10) As a result we obtain the representation
or, using the definition (5.1) of the Schwartz kernel,
f(z) = i(8'f+(x'),Sc(z - x';zo - x')) + ~f(zo), z E TC ,
zO E T C .
(5.11)
Formula (5.11) is called the generalized Schwartz representation. This completes the proof of the following theorem. If C is an acute cone} then any fu.nction f( z) of the class H (C) that satisfies the condition (5.5) can be represented in terms of its boundary valu.e f+ by the Poisson integral (5.7) and can also be represented in terms of the imaginaTy part of its boundaTy value by the formula (5.9). And if, besides, the cone C is regular, then for any such function f(z) the generalized SchwaTtz representation (5.11) holds true. THEOREM.
13. EQUATIONS IN CONVOLUTION ALGEBRAS
171
12.6. A generalization of the Phragrn.en-Lindeloftheorem. The Phragmen-Lindelof theorem in the theory of holomorphic functions is defined as any generalization of the maximum principle to the case of unbounded domains or to more general (than continuous) boundary values. Here we give one such generalization of the maximum principle that will be used later on in Sec. 21.1.
If the boundary value f+(x) of a function f(z) of the class H(C), where C is an acute cone) is bounded: If+ (x) I ::; M! X E jRn, then we also have f( z) :S M, z E T C ,. what is more, for f( z) we have the generalized Poisson representation THEOREM.
I
I
f(z) = REMARK.
J
Pc(x - x', y)f+ (x') dx',
(6.1)
For n = 1 this theorem was proved by Nevanlinna [18].
+
PROOF. Since Kc(x iy) E £2 for all y E C (see Sec. 10.2), it follows that f+(x)Kc(x - zO) E £2 for all zO ETc and, hence, the condition (5.5) is fulfilled for
s = O. By the theorem of Sec. 12.5, for the function f(z) the Poisson representation (6.1) holds; from this and from the property (1.5) of Sec. 11.1, of the kernel Pc
follows the estimate
If(z)
I::; M
f
Pc(x - x', y) dx' = M,
o
which completes the proof of the theorem.
13. Equations in Convolution Algebras Let f be a closed convex acute solid cone in jRn (with vertex at 0). Then the sets of tempered generalized functions 8' (f +) and 8' (f) form convolution algebras [S'(r) is a subalgebra of s'(r+)] (see Sec. 5.6.2) that are isomorphic to the algebras of the holomorphic functions H +(C) and H (C), respectively, where C = int f* , and the isomorphism is accomplished by the operation of the Laplace transform (see Sec. 12.2).
13.1. Divisors of unity in the H+(C) and H(C) algebras. As was shown in Sec. 4.9.4 the solvability of the equation a *u
== f
a and
f E S' (r +),
in the convolution algebra 8' (r +) reduces to the existence of a fundamental solution £ (the kernel of the inverse operator a- 1 *) of the convolution operator a*,
a*f=d,
(1.1)
in the same algebra 8'(f+). The equation (1.1) is equivalent to the algebraic equation
L(a]f
=1
(1.2)
in the algebra H+(C) with respect to the unknown function f(z) = L[f]. Therefore the question of the existence of a fundamental solution of the operator a* in the algebra S' (r +) reduces to the question of the possibility of dividing unity by the function fo(z) = L[a] in the H+(C) algebra. In other words, the question reduces to studying the di visors of uni ty in the H + (C) algebra: if f E H + (c), then we want to know under what conditions l/f E H+(C).
172
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
The necessary condition for this, I(z) #- 0, z E T C , is not a sufficient condition, as will be seen by the following simple example: f(z) = e- i / z E H(O, (0) since jf( z) I e- y/!zI2 S 1. However, 1/ f $. H + (0,00) since
=
1
f(z) EXAMPLE. If H(z)
H(C).
#-
---lL-
=e
2
1.. 1
1(1 ,>;2)
> eY
- ~ .
°is holomorphic and homogeneous in T
C
,
then H- 1 (z) E
The proof is similar to the one given for the Cauchy kernel (see the lemma in Sec. 12.5). 0 We first note that the study of divisors of unity in the H + (C) algebra reduces to studying the divisors of unity in its subalgebra, the H (C) algebra. Indeed, any function f(z) in H+(C) [that is, can be represented in the form
f(z)
f E Hi a ,{3)(C)
for certain a >0,0'
= e-i(z,e) fe(z),
°
and 13 ~ 0]
fe E H(C),
where e is an arbitrary point in int r such that (y, e) Lemma 1 of Sec. 4.4, such points exist). Indeed,
Ife(z) 1= lei(z,e) f(z) I :S Ilfll~a,(3)
2::
> alyl
(1.3) for all y E C (by
(1 + Iz12) cr/2 [1 + ~ -(3 (y)J,
so that Ie E H(cr,(3)(C). From the result of Sec. 12.4 we have the following theorem. THEOREM. For I E H (C) to be a divisor of unity in the H (C) algebra, it is necessary and sufficient that, for any cone C' ~ C and any number e > there exist numbers 0" ~ 0, (3' 2:: and M' > 0 such that
°
°
J
II (z ) I ~ M 'e - elyI (1 +
2) - a
1Z 1
I/
2
1
y I /3! ,
z E TC
J •
(
1.4)
The condition (1.4) is hard to verify. We now point to several sufficient criteria for the divisibility of unity in the H (C) algebra that follow from the theorem that has just been proved. 13.2. On division by a polynomial in the H(C) algebra. THEOREM. Suppose P(z) #- 0 is a polynomial, and a function f(z) is holomorphic in T C and PIE H (C). Then f E H (C) and the operation f --t P f has a continuous inverse in H (C). COROLLARY. If the polynomial P(z) does not vanish in T e , then ~ E H(C). Indeed, in that case, p(z) is a holomorphic function in T C and P ~
= 1 E H(C).
PROOF OF THE THEOREM. To prove this theorem we take advantage of the following result obtained by Hormander [see inequality (2.3) of Sec. 15.2 for p = OJ: For a given polynomial P(z) there are numbers 'm ~ 0 (an integer) and I< > such that for any t.p E cm (l~ 2n) the following estimate holds true:
t.
°
Icp(x,Y)1
°
s I< sup (x,y) (1 + IzI2)m/210(x,y)[P(z)
-~f(z)
1 ~ I(z) =
't.
fl
z)
is
< O. > 0 vanishes
By the theorem, 1/1 E H(C),. But if C;Sf(z) at some point in the C domain T , then, by virtue of the maximum principle for harmonic functions, 0, z E Let us maxI<j
2. If a function j(z) is holomorphic and 8'f(z) ~ 0 in yn = [.::: Y1 0] I then 1+
[/(z) I ~ v12!f(i) I m~x
1$.1$n
where i
2 IZ'1 J,
z E Tn
>
(3.6)
Yj
= (i, £, ... , i).
PROOF,
The holomorphic mapping Wj
=
Zj -
+
Zj
'I
.,
'" --J
z·
Z
transforms the tubular domain Tn on
N, follows from the definition of Or and from its homogeneity (see b)).
r
Let f E 5 1 (f) (see Sec. 4.5). By the primitive of order with respect to the cone f we call the convolution
DEFINITION.
f( -ad (€L
fe-a)
= Or * f.
0',
(1.1)
= 1, f = [0,00), C = (0,00), Kc(z) = ~, Or (z) = *m *.... * = *om, times m=O,l, ...
EXAMPLE. For n
°° °
8cm(~) = ,sCm) (~),
Therefore, the operator O¥* for 0' > 0 is the (fractional) antiderivative of order 0:; for a = 0 it is the identity operator; for a < 0 it is the (fractional) derivative (see 4.9.5). 1. Iff E S'(f), then its primitive fe-a) for all sufficiently large a is continuous in ~n J the following representation is valid LEMMA
>N (1.2)
180
2 INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
and the inequality
(1.3) holds for some C > 0, T > 0, where m is the order of f. (In (1.2) TJ is an arbitrary COO-function, 18131](01 ~ C fJ , f, E lP?n, which is equal to 1 in 1" and equal to 0 outside f2', E > 0 is arbitrary.)
Let m denote the order of f E S'(f) (see Sec. 5.2). By virtue of the reasoning above, Or E cmCIl~n) for all sufficiently large Q' > Nand supp Or C f. Using the standard reasoning (cf. Sec. 5.6.2), one can deduce representation (1.2) from this fact and from representation (6.2) of Sec. 5.6. Representation (1.2) implies the continuity of /(-0:) (f,) in]Rn and inequality (1.3) if we note that the set of the functions PROOF.
eE ]Rn}
{( -+ 7J(()Or(f, - (), is continuous in 8 m with respect to and estimate d)
f, and make use of inequality (2.3) of Sec. 5.2
If(-a)(f,)1 ~ Ilfll-m 117](f,')e~(f, - f,')llm < Cllfll_mlf,l n (a-1)+m. The lemma is proved.
A function f(f,) has an asymptotic g(f,) of order a in the cone if, for any E int f, there exists the limit
DEFINITIONS. 1.
f
If, I -+
as
00
D
e
1(\~)WQf(~) = gC~I)
(14)
and there exist constants M and R such that 1f,I-Qlf(f,)I~M,
1f,I>R,
f,Eintr.
(1.5)
2. A generalized function /(0 taken from S' (f) has a quasiasymptotic g(O of
order
Q'
at
00
if
k -+ 2'. A generalized function order C\' at 0 if
j (f,) taken from
pO: f(px) -+ g(x),
in
00
5'.
S' has a qUQsiasymptotic
P -+ +0
in
(1.6)
9(f,)
of
5'.
3. A function j(z) holomorphic in TC has an asymptotic h(z) of order a at 0 in T C if
(i) lim pO: f{pz)
p-t+O
= h(z),
(1.7)
(ii) there exist numbers M, a and b such that
< M 1 + 1- !a 7
palf(pz)!
-
~~(y)'
0< p ~ 1,
(1.8)
Definition 2 implies that the quasiasymptotic 9 of order 0: at CX) (if exists) belongs to SI (f) and is a homogeneous generalized function of degree Q' (d. Sec. 5.7),
g(tf,) = tag(f,),
t > O.
Its primitive g{ -N) is a homogeneous generalized function from S' (f) of the homogeneity degree ex + nN.
14. TAUBERIAN THEOREMS FOR GENERALIZED FUNCTIONS
EXAMPLE.
J(O has the quasiasymptotic
J(~)
of order n = -n at
181 00.
In order that f E S' (f) has the quasiasymptotic 9 of order n at 00 it is necessary and sufficient that its Fourier transform has the qUQsiasymptotic g of order a + n at O. This assertion follows from Definitions 2 and 2', from equality (3.7) of Sec. 6.3.5:
1
F
[k-af(k~)] = po+n j(px),
p = ~ > 0,
and from the continuity of the operation of the Fourier transform in Sf. D In particular, if f E S'(f) has a quasiasymptotic 9 of order a at CXJ, then f(-N), -00 < N < 00, also has the quasiasymptotic g(-N) of order a + nN at 00. The assertion follows from Definition 2, from the equality
-----
I(-N)(x) = 1C~(x)j(x), and from the homogeneity of the kernel Kc(x) (see Sec. 10.2). D If a function has the ordinary asymptotic, then it also has the quasiasymptotic of the same order. More exactly, the following lemma is valid. LEMMA
r
>
-n in the cone same order n at 00.
Q'
taken from S'(f) has the asymptotic g(~) of order as I~I -t 00, then f also has the quasiasymptotic 9 of the
2. If a function
PROOF. It
f(~)
follows from (1.4) and (1.5) that
k~of(k~) --+ 1~lo9 C~I)
=
9(0,
k --+
00
almost everywhere in
jR"
(we assume that 9 is continued by zero onto the whole IR?n) and, moreover,
Ik- ce f(k~)1 ::; MI~la,
I~I
> R/k,
~ E IR?n.
Let t.p E S. Then (k- a I(k~), t.p)
J J f(kE,)t.p(~) d~ J f(k~ho(E,) d~ Jg(~ho(~) d~
= k- a I(k~)R/k
+ k- a
-t
= (g,l,O),
k
~ 00,
I~I 0 and has the asymptotic lim
f{ --q) (~)
{-++oo ~q+a+1
(Fig. 29 depicts the domain lxj
< yf3
=
C
r(o:
(3.2)
+ 1 + q)
in the half-plane y
> 0.)
The proof of the theorem follows from the General Tauberian theorem of Sec. 14.2. 0
14. TAUBERIAN THEOREMS FOR GENERALIZED FUNCTIONS
187
14.4. Tauberian and Abelian theorems for nonnegative measures. In this case the General Tauberian theorem of Sec. 14.2 is simplified, namely. condition b) can be omitted (and then Condition (2) should be automatically fulfilled). Let Jl(d~) be a nonnegative measure with the support in the cone r (see Sec. 1.7). Its primitive J.L( -I) (e) = p * Or can be almost everywhere in ~n represented by the integral It (-
1)
(~) =
J
P ( de)
(4.1 )
I
.!l(O
where ~(e) = f n (E - f) (see Fig. 30). Its Laplace transform jl(z) can be expressed by the integral
jl(Z)
=
f
ei(z,{) p(dE),
(4.2)
r
If J.l E 1)/(f) is a nonnegative homogeneous measure, then its primitive 1l(-l)(E) is continuous in intf. LEMMA.
PROOF. Let
en --+ e n ~ 00, EE int f. I
Then for any
€
> 0 there exists
a~
>0
such that
o < /-l(-I) ((1 + ~)E) -
o)E) < 6. :s 1l(-I)(E):S 1l(-I) ((1 +o)E).
p(-l)((l-~)E)
(4.3)
Jl(-l)((1-
(4.4)
by virtue of homogeneity and monotonicity of the function p( -1) (E) with respect to r. Then, starting from some number N, for n > N the inclusions En E int r are valid and the inequalities
f
hold, which imply the inequalities J.L(-l)((l-
6)E)
:s jl(-l)(En):S J.L(-I) ((1 +o)E). E- r
Comparing these inequalities with (4.3) and (4.4), we obtain
n>
N,
Figure 30
o
which is what we set out to prove.
The General Tauberian theorem of Sec. 14.2 immediately implies the following theorem. For a nonnegative measure It( de) from S' (f) to have a quasiasymptotic 9 of order Q at 00 it is necessary that the following conditions hold THEOREM.
(1) lim pQ+n [J,(pz) = h(z)}
Z
E TC ,
p-++O
Q+n IJJ~ (pz )I :s M 1~~(y) + Izla 0 < P < I} () 2 p I
z E
TC
,
188
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
and it is sufficient that there exists a solid subcone C' C C such that (a) lim pa+n[J(ipy)=h(iy), YEC'. p-l- +0
(4.5)
In this case g(z) = h(z) and g(~) is a nonnegative homogeneous measure of Q with support in r which satisfies relations (2.8) for N = 1,2, ....
degree
In order to prove the theorem it is sufficient to note that condition b) (see (2.6)) of the General Tauberian theorem always holds for the functions p(z)
pa+n Ijl(px
+ ip'xe) I ::; pa+n / e-PA(e'0J.l(d~) ['
= pa+nh(ip'xe)
=,X-a-n(p'xt+nh(ip'xe) < ,X-a-n
sup pa+nh(ipe) O..e) I ~ c>..-n, 0 < p ~ 1, 0 < >..:S 1, Ixl:S 1. p ~ (a+n)
1
I
190
2. INTEGRAL TRANSFORMATIONS OF GENERALIZED FUNCTIONS
This and (5.4) imply estimate b):
po+n
jj(PX + i>.pe)!
= [p~(o+n) If+(px
M
+ ip).e) I] -; - < C~ >.- ~n, D
which proves the theorem.
The Tau berian theory presented in this section remains valid if we replace the scale (au tomodelling) function kO of order Q' by a regularly varying function p( k). A continuous positive function p(k), k E (0, <Xl) is called regularly varying, if for any k > 0 there exists the limit lim p(tk) t-+oo
p(t)
= C(k),
and the convergence is uniform with respect to k on any compact of the semi-axis (0,00). One can easily see that C(k)C(kt) = C(kkd; hence, C(k) = kO for some real a. The number a is called the order of automodellity of the regularly varying function p( k). Let us give some examples of the regularly varying functions of order a:
kG,
kOhl(l+k),
kOlnln(k+e),
k Oo (2+sinV'k).
Concerning the regularly varying functions see Seneta [92]. At present, the Tauberian theory of generalized functions is developing in other directions. The quasiasymptotic on the orbits of one-parametric groups of transformations preserving a cone and relating theorems of the Keldysh type and the comparison theorems are being investigated. Essential progress is achieved concerning the extension of theorems of the Wiener type on the generalized functions. All these results can be found in the book by Vladimirov at at [122] and in recent papers by Drozhzhinov and Zavialov [26, 27, 28].
CHAPTER 3
SOME APPLICATIONS IN MATHEMATICAL PHYSICS 15. Differential Operators with Constant Coefficients The theory of generalized functions has exerted a strong influence on the development of the theory of linear differential equations. First to be mentioned here are the fundamental works of L. Girding , L. Hormander, B. Malgrange, I.M. Gel'fand, L. Ehrenpreis of the 19508 devoted to the general theory of linear partial differential equations irrespective of their type. The results of these studies are summarized in the Analysis of Linear Partial Differential Operators in four volumes by Hormander [51] (1985). Big advances have been made in the theory of the so-called pseudodifferential operators [a generalization of differential and integral (singular) operators] 1 . 15.1. Fundamental solutions in V'. One of the basic and most profound results is the proof of the existence of a fundamental solution £(x) in 'D' of any linear differential operator P( 8) $. 0 with constant coefficients (see Sec. 4.9.3), that IS,
P(o)£(x)
= J(x)
(1.1 )
where
P(8)
L
=
c
acr 8 t,
( 1.2)
Icrl~m
is a differential operator of the mth order. This result was first obtained independently by L. Ehrenpreis [31] (1954) and B. Malgrange [73] (1953). Before proceeding to the proof of the existence of a fundamental solution, we will first prove two lemmas on polynomials. LEMMA
1. If
P(~)
L
=
L
aQe~,
lal=m
IQI~m
is an arbitrary polynomial oJ degree m real transformation of coordinates ~=C(I
laO'I i- 0,
> 1,
detC
then there exists a nonsingular linear
i- 0,
that transforms the polynomial P to the form
?(() = a~~m +
L
Pk(€~, ... ,~~)~~k!
a
1= o.
O p such that the following inequality holds:
(2.3) The existing proofs of this lemma are extremely complicated. We confine ourselves here to the proof of only the case n = 1. First we will prove (2.3) for the case P(~) f Setting 'l/J ~ 1;
and so forth. Consequently,
1I I, p(o) (-i~)
P( -i~)
-t 0,
I~I-t
00.
(6.1)
This result was obtained by Hormander [46]. The proof of the following theorem is similar to that of Theorem I.
I'. For an operator P( 8) to be elliptic, it is necessary and sufficient that, for any 0, every solution u(x) in V'(O) of the equation P(8)u = 0 be (real) THEOREM
analytic in (J.
The algebraic condition for ellipticity: for an operator P(8) to be elliptic, it is necessary and sufficient that its principal part
Pm(O) =
L lal=m
satisfies the condition Pm(E) =j:. 0,
ef:. O.
aa cYJI
15. DIFFERENTIAL OPERATORS WITH CONSTANT COEFFICIENTS
211
This result was obtained by Petrovskii [80] (l939L for classical solutions, by Weyl [124] (1940) for generalized solutions for the Laplace operator, and by Hormander (1955) (see [46]) in the general case. We now prove a theorem on wiping out isolated singularities of harmonic functions. A generalized function u( x) in 1J' (G) is said to be harmonic in a domain G if it satisfies the Laplace equation ~u = 0 in G (and then u E Coo via Theorem I). THEOREM
II. Let 0 E G. If the function u is harmonic in the domain G \ {O}
and
u(x) u(x)
= o(lxl-n+2), = o(ln lxI),
n>- 3', n
Ix]-+ 0,
= 2,
(6.2)
then u is harmonic in G. Let Un @ G. We introduce the function u(x), which is equal to u(x) in U Rand 0 outside U R- This function is integrable on ]R.n and by (3.7') of Sec. 2.3 it is the generalized function in 1J'(IR n ) such that PROOF.
From this, by the theorem of Sec. 2.6, we have Llu = -
~: 85
R
-
{)~ (u8 SR ) +
L
a
ca o 8
III
]R.n.
(6.3)
lal~m
Since i1 is of compact support, by using (1.11), we obtain, from {6.3L it
= En *' ~it = -En * ;: 05R =
V(O) n
-
En
*
:n
(u8 SR
+
L
COlEn
*8
Ot
§
!OtIS;m
a + V(1) + """ c en [; n La
(6.4)
1001S;m
where
En(x) £2(X)
= kn lx]-n+2,
n -> 3',
1
= 211" in Ixl
is the fundamental solution of the Laplace operator, and V~O)(x) and VJ1)(x) are surface potentials of a simple and double layer on the sphere SR (see Sec. 4.9). From the representation (6.4) for Ixl < R and from the condition (6.2) it follows that COl. = 0, so that
Ixl < R, whence it follows that u(x) is a harmonic function in the ball UR. The proof of the theorem is complete. D
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
212
15.7. Hyperbolic operators. Let C be a convex open cone in ~n with vertex at O. The operator P(8) is said to be hyperbolic relative to the cone C if it satisfies the condition: there is a point Yo E JR." such that
P(Yo - iz)
i- 0
for all
z E Te .
(7.1 )
For the operator P( 8) to be hyperbolic relative to a cone C, it is necessary and sufficient that it have a (unique) fundamental solution E(x) in the algebra V' (C* ) which solution can be represented as THEOREM.
1
Eo E S'(C*),
(7.2)
where the point Yo E lR n is defined in (7.1).
If the operator P( 8) is hyperbolic relative to the cone C then the polynomial P(yo - iz) does not vanish in the tubular domain T C . Therefore 1/ P(yO - iz) E H(C) (see Sec. 13.2) so that PROOF. NECESSITY.
I
1
p[ -i(iyo + z)] Setting (
= z + iyo
I
= L[£o](z),
Eo E 8'(C*).
(7.3)
we obtain (see Sec. 9.2.3)
P( ~i() = L[EoJ(( - iyo) = L[Eo(x)e(Yo,x)]
1
whence follows the representation (7.2). SUFFICIENCY. If the operator P(B) has a fundamental solution of the form of (7.2), then the function L[Eo](z) is holomorphic in T C and, hence, by virtue of (7.3) the polynomia.l P(Yo - iz) does not vanish in T C since the operator P(8) is hyperbolic relative to the cone C. The uniqueness of a fundamental solution in the algebra V' (C*) was proved in Sec. 4.9.4. The theorem is proved. 0 1. The wave operator 0 is hyperbolic relative to the future light cone V+, and (see Sec. 13.5) EXAMPLE
D( -iz) =
-z5 + zi + ... + z~ i-
2. The differential operator ative to the cone (0,00). EXAMPLE
REMARK.
p(ft)
in
T V +.
(see Sec. 15.4.4) is hyperbolic rel-
The cone C is a connected component of the open set (see Hormander
[46])
[y: Pm(y) f. OJ. 15.8. The sweeping principle. Let P(8) be a differential operator with constant coefficients of order m that is defined by equality (1.2) in Sec. 15.1 and E(x) be its fundamental solut.ion. Suppose that 0 is an open set in JR. n , 80 is its boundary and eo(x) is its cha.racteristic function (see Sec. 0.2). Let u E V'(O) be the solution of the homogeneous equation P(8) = 0 in 0 which is representable in 0 in the form of the potential V = £ :+: f with the density f E V'.
16. THE CAUCHY PROBLEM DEFINITION.
213
If there exists a generalized function h E V', supp h C
ao, such
that
£
*h =
{u = £ * I, xE 0,
_
(8.1 )
XEJRn\O,
0,
then we say that the sweeping of the density
I on
00 occurs.
Let 0 be a bounded open set in }Rn. If the potential V = £ * I, I E V', eXlsts in V', satisfies the homogeneous equation P( 0) V = 0 in 0 and there exists the generalized function Oo(x)V(x) in 1>',2 then the sweeping of the density f on 80 occurs. THEOREM.
PROOF.
Write h = P(8)[80 V]'
h E V'.
(8.2)
By the hypothesis of the theorem, supp h C 80. Since the convolution £ and 00 V exists in V' (0 is bounded! see Sec. 4.3), by virtue of (8.2) 80 V can be represented as the potential (see Sec. 15.1)
(8.3)
=
The restriction of equality (8.3) onto 0 yields £ *h V = u, and its restriction onto IRn\O yields £*h = 0, and equalities (8.1) are proven. The theorem is proved. 0
=
Let 0 G be a bounded domain in = S and the potential
EXAMPLE.
boundary
aa
V =
1
!xln-2 * I ,
f
~n
supp f
E -n', v
with a piecewise smooth
c
TT1l n lJ\\.
\
G
exist in V' n C 1 (G). In this case formula (3.7 ' ) of 2.3.7 yields an explicit expression for h, h (x) = -
a (V 8s )( x). on 8s (x) - on
8V
(8.4)
16. The Cauchy Problem 16.1. The generalized Cauchy problem for a hyperbolic equation. Let
5 be a C-like surface of the class Coo and let 5+ be a domain lying above S (see Sec. 4.5); P(8) is a hyperbolic operator relative to the cone C of order m. We consider the classical Cauchy problem
P(8)u=f(x),
8kUI S = u.dx)' on k
x E 5,
xE5+, k
(1.1)
= 0,1, .. "m -1,
cm
(1.2)
cm-l
that is, the problem of finding a function u E (5+) n (5+) that satisfies the equation (1.1) in S+ and the condition (1.2) on S. For solvability of the Cauchy problem (1.1)-(1.2) it is necessary that f E C(S+) and Uk E Cm - k - 1 (S). Suppose that the classical solution u of the problem (1.1)-(1.2) exists and f E C(S+). We extend the functions f and u by zero onto S_ and denote the 2Concerning the multiplication of a generalized function by the characteristic function of an open set. see Sec. 1.10.
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
214
extended functions by f and U respectively. Then the function u(x) satisfies the following differential equations over the entire space IR n :
= j(x) +
P(8)u
(1.3)
where vktSs is the density of a simple layer on S with surface density Vk (see Sec. 1.7), uniquely defined by the functions {Uj}, by the surface S, and by the operator P(8). The generalized function :~k (VktSS) acts on the test functions
0 possesses the support Sxo (for odd n Xo
'2:
= 1),
3), U 'Eo (Jor even n or n
and satisfies the limiting relations, as
--+ +0, £n'EO (x)
PROOF.
[; ( n 0,
()
~
L.-
-I
(11 -a
v _ 1- a
I)
O 0) (see also Sec. 3.4),
('')~~f. >/) =(- 1)' (VJ~) .tiP») = (-I)k(V~O),CPvJ(k)) = (-I)k([Ul X oj * En, (xo + eo)) X
X
= (iik%=t). ~(x)>/>(xo)(u,(e), = ::::
I(
Ok£nxo(X)
ax~
I( * til
,1J(X)
Ok Enxo
ax~
,I.p
(
)
O. Similarly, by M o we denote the class of functions {f(x)} that are measurable in JR.n and that satisfy the following estimate for arbitrary € > 0:
(6.3) In (6.2) it may be assumed that the quantity CT,~ does not decrease with respect to T. If f EM, then the heat potential V exists in M, is expressed by the integral
(6.4) satisfies the estimate: for arbitrary
IV(x,t)l:s
€
> 0,
tCtJ~(f)
(1 _ 8fe)n/2
e 2 e:l x l
2
(6.5)
'
and satisfies the initial condition: for arbitrary R
V(x, t) IXI 0,
-t +0.
(6.6)
Indeed, since the functions E and f are locally integrable in lR"+l , it follows that their convolution V = f * E exists, is expressed by the formula (6.4), and it is
226
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
a locally integrable function in ~n+l if the function
IJ t
If(e, T)j f(x -
h(x, t) =
e, t -
r) de dT
o is locally integrable in IRn+l (see Sec. 4.1). We will prove that the function h satisfies the estimate (6.5). This estimate follows from the estimate (6.2), by virtue of the Fubini theorem, t
h(x, t)
< Ct,E: -
If
~12 11 2 e- ~+E: {
0
II
d€dr [4rr(t_r)]n/2
t
0,
x E ~n. Consequently,
V E C1(~+ X ~n) n C2(~+ X ~n). Finally, if Uo E M o n C, then\ by what has been proved, the potential y(O) E C (IH.~ x ~ n) n Ceo (IH. ~ x IH. n). Thus, the generalized solution u(x, t) defined by (7.1) belongs to the class C(~~ x ]Rn) n C2(~~ X ]Rn) and therefore is the classical solution of the heat
17. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T
C
229
equation (5.1) for t > O. Moreover, by (6.6) and (6.9), that solution satisfies the initial condition of (5.1) as well. Now this means that formula (7.1) will yield the solution to the classical Cauchy problem. The proof of the theorem is complete. 0 The uniqueness of the solution of the Cauchy problem for the heat equation may be established in a broader class, namely in the class of functions that satisfy in each strip 0 ~ t ~ T, x E lR n , the estimate REMARK.
I
lu(x, t) ~ CTearlxl2, (see, for example, Tikhonov [101]).
17. Holomorphic Functions with Nonnegative Imaginary Part in T C 17.1. Preliminary remarks. We denote by H + (G) the class of functions that are holomorphic and have nonnegative imaginary part in the region G. A function u(x , y) of 2n variables (x, y) is said to be plurisubharmonic in the region G c if it is semicontinuous above in G and its trace on every component of every open set [A: zO + Aa C G], zO E G, a E en, a '# 0, is a subharmonic function with respect to A. The function u(x, y) is said to be pluriharmonic in the region G if it is a real (or imaginary) part of some function that is holomorphic in G. Concerning plurisubharmonic and convex functions, see, for example, Vladimirov [105, Chapter II]. The following statements are equivalent: (1) A function u(x, y) is pluriharmonic in G. (2) A real generalized function u(x, y) in V'(G) satisfies in G the system of equations
en
82 u
--= 0, 8z OZk
1 < j, k ~ n,
Zj
j
= Xj + iYj.
(3) The functions u(x, y) and -u(x, y) are plurisubharmonic in G. Here,
8~; =H8~j -;8~J,
8~j =H8~j +i8~J·
From this it follows that every pluriharmonic function in G is harmonic with respect to every pair of variables (x j , Yj), j 1, ... In, separately and, hence, is a harmonic function in G,
=
~u = L (::~ + ::,) = 4 L 0:.2;z. = O. lSjSn
J
J
lSjSn
J
J
Therefore U E COO (G) (see Sec. 15.6). We denote by P + (G) the class of nonnegative pluriharmonic functions in the region G. Let the function j(z) belong to the class H+(T c ) so that ~f E P+(T c ), where the cone C is a domain. Without loss of generality, we may assume that the cone C is convex. Indeed, by the Bochner theorem the function j(z) is holomorphic (and single-valued) in the hull of holomorphicity Tch C of the domain T C and assumes the same values in T ch C as in T C (see, for example, Vladimirov, [105, Sec. 17 and Sec. 20].
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
230
Furthermore, the cone C may be assumed to be different from the entire space Otherwise, f(z) is an entire function and the condition ~f(z) > 0 in en leads via the Liouville theorem for harmonic functions to the equation c.sf(z) = const in and, hence, f(z) = const in Finally, we may assume that '2Jf(z) > a in T C . Indeed, if '2Jj(zO) = 0 in some point zO E T C , then, by the maximum principle for harmonic functions, r;Jf(z) 0 in T C , and then f(z) const in T C . The function f(z) of the class H + (T C ) satisfies the following estimate (see Sec. 13.3): for any cone C' @ C there is a number M(C') such that
]Rn,
en.
en
=
=
I I-
f(z) < M(C') 1 +
2
z I
Iyl
1
(1.1 )
,
Consequently, f E H(C) (see Sec. 12.1). Now let C be a (convex) acute cone (see Sec. 4.4) and let f E H+(T c ). By virtue of the estimate (1.1), f (z) possesses a spectral function 9 (~) taken from S'(C*) (see Sec. 12.2), f(z) = £(g]. From this, using the definition of the Laplace transform (see Sec. 9.1)' we have, for all z E T C ,
'2Jf(x
.
+ zy)
where g(e) --t g*(e)
;i
=
f(z) - j(z)
= F
2i
= g( -e).
[9({)e-(Y,{) - g*(e)e(Y,Oj 2i (x)
(1.2)
From (1.2) we derive the equation
[g(e)e-{Y'O - g* (e)e(YI~)]
= F x- 1[~f(x + iy)] (E),
y E C.
(1.3)
E S.
(1.4)
Let f+(x) be a boundary value of f(z) in S', that is,
!
f(x
+ iy)ff'(x) dx
--t (f+,
cp),
Y --t 0,
Y E C,
ip
Then 9 = F-1U+] and '2Jf+ is a tempered nonnegative measure (see Sec. 5.3). We denote it by 11 = '2J f +. Passing to the limit in (1.3) as y --t 0, y E C in S' (see Sec. 12.2), and using (1.4), we obtain (1.5 ) so that
-ig(E)
+ ig*(E) »
0
(see (1.4) of Sec. 8.1)
is a positive definite generalized function by virtue of the Bochner-Schwartz theorem (see Sec. 8.2). Let us now prove the following uniqueness theorem for functions of the class H+(T C ) [and the class P+(TC)]. THEOREM.
a E C" and ~b
If f E H+(TC) and J1. = O.
= 2sf+ = 0,
then f(z) = (a, z)
+ b,
where
If U E P+(T c ) and its boundary value J1 = 0, then u(x, y) ~ (a, V), where a E C*. COROLLARY.
Since J1. = 0, it follows that, by (1.5), the spectral function 9 [in S'(C"')] of the function f satisfies the condition 9 g* and, hence, since -C* nCO< {O} PROOF.
=
=
17. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T C
(cone C· is acute!), the supp 9
= {O}.
g(~) =
231
By the theorem of Sec. 2.6,
2:
caaa8(~),
lalsN
so that j(z) is a polynomial. But f E H+(T c ) and the estimate (1.1) shows that the degree of that polynomial cannot exceed one, so that f(z) = (a, z) + b, z E T C . But ~f(z)
=
(~a,
y)
+ (~a, x) + ~b 2: 0,
and therefore ~a E C" and 8'a = O. Furthermore, from ~b O. The proof of the theorem is complete.
=
~f+(x)
= 0 it follows that 0
This theorem is an elementary variant of Bogolyubov 's "edge-of-thewedge" theorem (see, for example) Vladimirov [105, Sec. 27]). REMARK.
EXAMPLES OF FUNCTIONS OF THE CLASS
H +(G).
-t
(1) If f E H+(G), then E H+(G) (see Sec. 13.3). (2) If C is an acute cone, J.l 't. 0 a nonnegative measure on the unit sphere, supp J.l C pr C .. , then
(3) -J;2 E H +(T v +) (see Example 2 of Sec. 10.2).
17.2. Properties of functions of the class P+(TC). Every function u(x, y) of the classP+(TC ) is an imaginary part of some function f(z) of the class H+(T C ). Therefore it satisfies the estimate (1.1), and its boundary value in 5' is a nonnegative tempered measure J.l = '2sf+ = u(x, +OL so that, by (1.4)
!
y --+ 0,
u(x, y)R
(2) If f E C
n .coo,
then the integral
!
is a continuous function in T C (3) For the Poisson integral
!
(2 ..5)
f(x - x')Pc(x - x', Y)J.l(dx') .
Pc(x - x', Y)J.l(dx') = J.l * Pc
the Fourier transform formula
(2.6) holds true. (4) The following limiting relations hold:
!
!
Pc(x - x ' ,Y)Il(dx') -t 11,
Pc(x - x', y')u(x ' , y) dx' -t u(x, V),
Y -t 0,
YE C
Y' -t 0,
Y'
E C,
zn
S',
(x, y) E T C .
(2.7)
(2.8)
(5) There is a function vc(Y) with the following properties: (a) vc(y) is nonnegative and continuous in C; (b) vc(Y) -t 0, Y -t 0, Y E C; (c) the following representation holds u(x, y)
=
!
Pc(x - x', Y)lt(dx ' ) + vc(y),
(x, y) ETc.
(2.9)
(6) If C is a regular cone, then
!
u(x',y')Sc(z-xl;ZO-xl)dx'-t! Sc(z-x';zo-x')J.l(dx'),
, ° '
y -t,
Y E C, '
C'
@ C,
z E
C
T ,
za E
(2.10)
C
T ,
where Sc is the Schwartz kernel of the tubular domain T C (see Sec. 12.5).
1. Since u E P+(TC ) implies that u E P+(TCI), C 1 c C, it follows that all the above-enumerated statements hold true also for an arbitrary (open) convex cone C 1 C C. REMARK
C
17. HOLOMORPHlC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T
REMARK
233
2. The limiting relation (2.7) also holds on functions of the form
It'(x)=1P(x)Pc 1 (x,y'L
1/JECn£oo,
y'EC11
CICC.
(2.11)
3. The estimates (2.2) and (2.2'), for n = 1, C = (0,00) (upper halfplane). follow from the Herglotz-Nevanlinna representation (see Sec. 18.2 below). In the general case, they have been proved by Vladimirov (in [111] for C = ~+; in [114, II] for C = V+, n = 4; in [116] for the general case). REMARK
4. The representation (2.9) was obtained in two special cases by Vladimirov in [111] (C = IR.f-) and in [114, II] (C = V+, n = 4). REMARK
To prove the theorem, fix
> 0 and set
€
IE: (z)
= 1_
I(z) iE/(z) ,
=
where I E H+(TC) is such that 0 (see Lemma 1 of Sec. 4.4).
IIKb(x + 2iy)f(x + 2iy)11 2 < M 2 (C') [lyl2 + Ll- 3n - 2 (y)]
,
y E
C'
.
(4 5) .
Therefore
G'.
(4.6)
The estimate (4.6) holds true if distance ~(y) is replaced by the lesser distance ~'(y) (from y to BC /). Applying the lemma of Sec. 10.5 (for a O, s 0, 'Y == ~n + 1
=
=
242
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
and C = C'), we conclude that the function K~(z)j(z) is the Laplace transform of the function 91 (€) =
ob· * 9
Bc •
* (}c· * 9
[see Sec. 9.2.7] taken from £;,(C*) for all Sf < -~n - 1 and C f lE C, where 9 is the spectral function of the function f: j(z) = £[g]. Hence 91 E £;(C*) for all s < - ~n - 1. The theorem is proved. 0 To prove the corollary, set
Then in the convolution algebra S' (C·) we have, in the case of a regular cone C [see Sec. 4.9.4 and Sec. 13.1],
Be: *91 = Be: * (O~. *g) = (Oe= * Bb.) *g = J * 9= 9· 0
The function 91 with the indicated properties is unique. The results ofSecs. 17.2-17.4 have been obtained by Vladimirov [116].
17.5. Indicator of growth of functions of the class P+(TC ). In Sec. 17.2 we st.udied t.he growth of functions of the class P + (T e ) as y -+ 0, Y E C, and as Ixl -+ 00. Here we will investigate the growth of such functions as Iyl -+ 00, y E C. First we will prove the following lemmas.
1. If u E P + (T C ), where C is a convex cone, then for every bounded region D C IR n and for every point y E C there is a number to > 0 such that JOT all (xO, yO) E r D the function u(x O, yO + ty)(t - t o )-1 does not inc;:ease with respect to t on (to, 00). LEMMA
=
Fix zO xO + iyo E T D and y E C. Since the cone C is open and convex, there is a number to = to(YO, y) such that yO + ty E C for all t > to. Therefore the function u(x O+ uy, yO + (T + to)Y) belongs to the class P +(Tl) [with respect to the variables (u, T)), and so it can be represented by the formula (see Sec. 18.1) PROOF.
I
00
0
0
)
T
( ux+Uy,Y+(T+to)Y=7r
°
J-l(xO,yoJy;du') a 0 ( )2 2+ (X,y,y)T , (J' -
+T
(J"
(5.1 )
-00
0 and the measure J.l > where a > - 0 satisfies the condition of growth [see (2.2')]
-00
Putting u
= 0 in (5.1), dividing by T
I
and setting
T
=t -
to
> 0,
we get
whence, by the B. Levi theorem, we conclude that Lemma 1 holds true.
0
17. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T C LEMMA 2. Suppose the function f(x} is convex on the set A. xo E A and z E ~n the function
1
i[J(xO
+ tx) -
243
Then for all
f(x O)]
does not decrease with respect to t on the interval [0, to] provided that all the points xO + tx, 0 :::; t :::; to, are contained in A.
By the definition of a convex function (see Sec. 0.2), the function f(x + tx) is convex with respect to t on [0, to] and, hence, for arbitrary 0 :S t < l t :::; to, PROOF.
O
f(xO
+ Ix)
(1 -:;) ::; :,f(XO + I' x) + (1 -:') J(x°),
=f
[:' (I' x
+ xo) +
xO]
that is
t1 [J(xo + tx) -
1
f(xO)] :::; t l [J(XO
+ tlX) - f(xO)]
o
which completes the proof of Lemma 2. LEMMA
T = ~n + the functio,,! D
3. If the function u(x, y) is plurisubharmonic in the tubular domain iD and is bounded from above on every subdomain T D ', D' @ D, then
= sup u(x, y) x
M(y)
(5.2)
is convex and, hence, continuous in D. PROOF.
for all 0
.. are homogeneous of degree of homogeneity 1; for example,
'(
/\ u;
ry )
I'1m m(try) = r I'1m m(try) = t-+oo t t-+oo tr
. m(t'y) = r hm = r.-\(u; y),
t'
t'-+oo
r
> O.
From this, a.nd also from (5.7) and (5.5) follows the inequality
>..( u; y)
:s:
h( u; y)
(5.8)
y E C.
We will now prove the following theorem. If u E P +(T c ), where C is a convex cone, then the growth indicator h(u; y) is nonnegative, concave, homogeneous of degree of homogeneity 1 in C, and THEOREM.
>..(u; y)
. u ( X°,yO + t y) = h(u; y) = t-+oo hm ,(xo, yO) E en, t
y E C.
(5.9)
For (XII, y") E r C the function tu(xO, yO + ty) does not increase with respect to t E (0,00) and the following inequality holds true: h(u; y) :::; u (x yO + y), ( x 0 , yO) E T C ,yE C. (5.10)
°,
PROOF.
We will prove that for every y E C the function . u(X O, yO + ty) u(xO, yO + ty) = I'1m --=--...;.....--~ 11m t-+=
t
t-+oo
t - to
(5.11)
17. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T C
245
does not depend on (xO, yO). For this it suffices to prove, by virtue of the Liouville theorem, that for every y E C the nonnegative function (5.11) is pluriharmonic with respect. to (xo, yO) in en. That is, it is pluriharmonic in every tubular domain TD = ffin + iD, where D @ ffin. By Lemma I, the function (5.11) in the domain T D is the limit of a nonincreasing sequence of functions u(xO, yO + ty)(t - to)-I, t ~ 00, t > to, of the class 'P+ (T D ) and therefore is itself pIuriharmon ic in T D (see Sec. 17.1). Thus, by (5.5), the second of the equalities (5.9) holds, and, by Lemma 1, the function tu(xO, yO + ty) does not increase with respect to t for t > 0 if yO E C. Therefore
Putting t = 1 here, we obtain the estimate (5.10). From this estimate we derive
h(u; y) :; m(y),
yE C,
so that, by (5.6), h(u; y) ::; A(U; y),
yE C.
This inequality together with the inverse inequality (5.8) is what yields the first of the equalities (5.9), from which fact it follows that the indicator h(u; y) is a convex function in C. This completes the proof of all assertions of the theorem. 0 A more general theory of growth of plurisubharmonic functions in tu bular domains over convex cones is developed in Vladimirov [117]. REMARK.
17.6. An integral representation of functions of the class H + (T C ). We established here that a function of the class H+(TC), where C is an acute regular cone, is representable in the form of a sum of the Schwartz integral and a linear term if and only if the corresponding Poisson integral is a pluriharmonic function in TC. We first prove a lemma that generalizes the Lebesgue theorem on the limiting passage under the sign of the Lebesgue integral (see Vladimirov [114, IV]). Suppose the sequences £1 have the following properties: LEMMA.
Uk
(x) and
Vk
(x), k = 1, 2, ... I of functions in
(1) ludx) I ::; vdx)' k = 1,2, ... , almost everywhere in ~n,(2) udx) ~ u(x), vk(x) ~ 1'(x) E £1, k ----t 00, almost everywhere in IP?Tl;
(3)
f Vk(X) dx f v(x) dx,
Then u E
---t
£1
k
- t 00.
and
f
udx) dx -t
f
u(x) dx,
k -+
00.
(6.1)
From (1) and (2) it follows that u E £1 and Vk(X) ± udx) ~ 0, k = 1,2, ... , almost everywhere in jRn. Applying the Fatou lemma to the sequences of functions Vk ± Uk, k ~ 00, and making use of (3), we derive the following chain PROOF.
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
246
of inequalities:
f
[v (x) ± u(x)] dx
~ limk-.+oo = limk-.+oo
= whence we derive
limk-+oo
f
f f
v(x) dx
[vk(x) ± Uk(X)] dx
Vk(X) dx + limk-.+oo
+ limk-+oo
f
f ±ud
x )] dx
±Uk(X)] dx,
f uk(x)dx~ f u(x)dx~limk-.+oo f
uk(x)dx,
o
which is equivalent to the limiting relation (6.1). THEOREM
(Vladimirov [118]). Let f E H+(T c ), where C is an acute (convex)
cone. Then the following statements are equivalent: (1) The Poisson integral
f
(6.2)
Pc(x - x', Y)J.l(dx'),
is a pluriharmonic function in TC. (2) The function c;,s f( z) is representable in the form
~f(z) =
f
(6.3)
Pc(x-x',Y)J.l(dx')+(a,y),
for a certain a E C" . (3) For all y' E C the following representation holds:
~f(z+iy')= fpc(x-X',y)8!(x'+iy')dxl+(a,y),
zET c .
(4) If C is a regular cone, then for an arbitrary zO E TC, the function be represented as
f(z) = i
f
Sc(z - x'; zO - x')p(dx') + (a, z)
+ b(zo),
z E Te,
(6.4)
f (z)
can
(6.5)
where b(zO) is a real number. Here, b(zO) = ~f(zO) - (a, xO) and (a, y) is the best linear minorant of the growth indicator h(~f; y) in the cone C.
Under the hypothesis of the theorem, the best linear minorant of the nonnegative convex function h(~fj y) of degree of homogeneity 1 exists in the cone C (see Sec. 17.5). For example, h(C;Sv0;y) = nand (a,y) = 0 in V+. REMARK.
PROOF.
Let
f
E
H+(TC). (1) -+ (2). The function v(x, y)
= ~f(x) -
f
Pc(x - Xl Y)J.l(dx') I
belongs to the class P+(T C ) and its boundary value, as y ~ 0, y E C, is equal to 0 (see Sec. 17.2). By a corollary to the theorem of Sec. 17.1, v(x, y) = (a, y) for some a E C·. The representation (6.3) is proved.
17. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN T
C
247
We now prove that (a, y) is the best linear minorant of the function h(~f; y) in the cone C. From (6.3) and (5.5) it follows that (a, y) is a linear minorant of h in C. Suppose (a', y) is another linear minorant of h in C, that is
(a', y) S; h('2sf; y),
yE C.
(6.6)
The function
h (z) = f(z) - (a', z),
C;Sfdz)
= ~f(z) -
(a', y)
belongs to the class H + (T C ) since
'J'f(z)
> h(CSf;y) > (a',y),
by the theorem of Sec. 17.5 and by virtue of (6.6). Furthermore, since ~Jl+ 0, we have F
[e- Y1 {1(1- 8 )J] = F [(1- T)(e-Y1€1f)] = F [(1- /}2 - y2 - 2y/}sgn~ + 2yo(€))(e-YI~lf(€))] 2
= (1 + x 2 -
y2)F[e- Y1 €IJ(e)]
+ 2ixyF[sgnee-YI€IJ(~)] + 2yf(0).
(1.5)
Taking into account the equalities
00
F[Sgn~e-YI{I](x)=2ife-Y€sinx~d~=x22xi + y2
l
o
we obtain
F
1
[f(~)e-YI(I] = 2 1r F[J] * F[e-Y1€1] y
= 1r F
[J(~) sgn ~e-YI(I]
=
f
u(dx') (x - X')2 + y2'
2~ F[J] * F[sgn~e-YI(IJ
= 2. 1r
f
(x - x')o-(dx') . (x - x'F + y2
Substituting the resulting expressions into (1.5) and taking into account (1.4) for k = 0, we obtain (1.3) for n = 1. The case of n > 1 is considered in similar fashion if one notices that every operator Tj operates only on its own variable ~j, if one applies. the Fourier transform technique with respect to some of the variables (see Sec. 6.2 and Sec. 6.3), and if one takes advantage of equations of the type (1.4). The proof of Lemma 1 is complete. D
18. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN Tn
251
2. Let the function v(~) be continuous and bounded in IRn and let supp v C JR+. Then the solution of the equation LEMMA
=-Tl.
8; ... a~ u(~) =
(1 -
ai) ... (1 -
8~)v(~)
exists and is unique in the class of continuous functions in which functions satisfy the estimate
(1.6)
jRn
with support in ~,
Iu (€) I < C (1 + €;) ... (1 + ~~).
(1.7)
The solution of equation (1.6) is unique even in the a.lgebra. V'(~) and is representable in the form (see Sec. 4.9.4) PROOF.
u
= En * (1 - ai) ... (1 -
8~)v.
(1.8)
Let us represent the right-hand side of (1.8) as u(~)
= (1 =
ai) ... (1 - 8~ )£n '" v {[O(€d6 - 6(6)] x ... X
x (ejl -
ejJ ... (€j"
[O(~n)~n - 6(~n)]}
*v
- {flJ d{it ... d€jk'
It remains to note that each summand in the last sum is a continuous function that satisfies the estimate (1.7). This completes the proof of Lemma 2. 0 3. Suppose u(e) is a continuous tempered function in IR n . Then the solution of the equation LEMMA
(1 - 8;) ... (1 - o~)v(€) =
0; ... a~u(e)
(1.9)
exists and is unique in the class of continuous tempered functions.
The solution of equation (1.9) is unique even in the class S' since the Fourier transform of the generalized function (1 (1- 8~)6(€), equal to n (1 + xi) . .. (1 + X~)l does not vanish anywhere in IR . We will prove its existence: PROOF.
an ...
£(~) = ~e-I(d- ···-I{n I 2
an ...
is the fundamental solution of the operator (1 (1 - 8~). Since ute) is tempered, the convolution £ * u exists (see Sec. 4.1). Therefore, the solution v of equation (1.9) can be expressed in the form of a convolution: v =
E * ar
... a~u
= of ... a~£ * u
= {[-0(6)+ ~e-I('I] x ... x [-o({nl + ~e-I(.I]} * = (_I)n u (€)+
L
(_1)n-k 2k
.l~k~~
.11 2£, 6 > 2E,
2
,
E, E ~n-2.
Put (1.12) The
{)O/x, E Coo
n.c oo for Va, and Xl': satisfies the equation
(1-8?)(1-a~)X€(e)=O,
{1>2e,
6>2£,
E,E~n-2.
(1.13)
Fix 6 > 0 and let 2£ < 6. From the equation (1.13) and from the bounded ness of the function (1 - 8i)Xt (~) we derive the relation
(1- 8i)Xf:(el,6,f.)
= (1 -
8i)x,(6,6,f.)e-(~1-(5),
that is,
(1- ai)[x,(6,6,E,) - X,(6,~2,€)e-(€I-o)]
= 0,
6
~
6.
(1.14)
Similarly, from the equation (1.14) we derive the relation
That is, by (1.12),
(1- 8~) ... (1- o~)[vf:(~1,6,E,) - v,(6,6,E,)e-({1-O) - v£ (~l, 6, f.)e-({2- 8) + 11, (6, 6, (,)e-({dE2- 20 )] = 0,
6
~ fJ,
62:
6,
From this, by uniqueness of the solution of the last equation (via Lemma 3), follows the equality
v,(e)
= ve(fJ,6,t)e-({1-6) +v€(6,o,~)e-({2-0)6 2: 6,
6
~ 0,
Ve(O,fJ,~)e-({1+{2-20),
~ E jRn-2.
Passing to the limit here as € --+ 0 and, furthermore, as J --+ 0, we obtain the representation (1.10). This completes the proof of Lemma 4. 0
18. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN Tn
LEMMA
263
5. The equation
L:
a? " .8~u(~) +
aa8a6(~) = 0,
(1.15)
lSlal~N
provided that u E C, supp u 1 S 10:1 N.
s
PROOF.
c iif;.,
is possible only when u(~) = 0 and aa = 0,
In the algebra "D'(~L the equation (1.15) is equivalent (see Sec. 4.9.4)
to
u(~) = - En
L
*
ao/ja o
l~lal$N
L
aa8at:n(~)
1~lal$N
n
-L: j=1
al=···=aj=l ai+l=" ·=an=O
n
-L L
aa{}aEn(~).
(1.16)
j=l ai::::O
Each term in the second sum of (1.16) contains at least one o-function or their derivatives with respect to anyone of the variables ~j, 1 :S j :S n , and the combinations of those o-functions and their derivatives in all terms are distinct. The other summands in (1.16) are locally integrable functions, whence we conclude that a a = 0 if there is a f such that Q;j ~ 2, and (1.16) takes the form n
u(E) = -
L: L j=:1
Ql::: ..
aaO(~O:l) ... O(~aj )(aj+l O((aj+l) ... Ea n O((a n
)·
·=aj::::l
Uj+l =:"'::::Qn::::O
From this, taking into account the properties of the function u, it is easy to derive, by induction on n, that all aa 0 and u(~) O. Lemma 5 is proved. 0
=
LEMMA
=
6. The general solution of the equation
a? ... 8~ u(~) = 0
(1.17)
in the class of continuous functions with support in -~ U ~ is expressed by the formula
(1.18) where C is an arbitrary constant.
=
Function (1.18) satisfies (1.17) since 8r., .8~£n(±O o(~). Let u(E) be an arbitrary solution to (1.17) taken from the class under consideration. Then the function u+(e) On(~)u(~) satisfies (1.17) in IR n \ {O} and hence (see Sec. 2.6) PROOF.
=
a; ... a~u+ (€) = L
co:a a e5(€)
o:SlalSN
=
coo; ... 8~[n(~) +
L ISlalSN
CaaUd(~)
(1.19)
254
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
y' R
o +1--
-R
R
x'
-R Figure 42
for certain N and ca. By Lemma 5, the equation (1.19) is possible only for C a = 0, lad ~ 1 and u+(~) = CO£n(~). Similarly, we derive that u-(~) = Bn(-~)u(O = C~£n(-~) so that u(~) u+(~) + u-(e) co£n(e) + c~£n(-e). But by virtue of (1.17)
=
=
ai ... a~u(e) = coar ... a~£n (e) + c~ar ... a~£n (-e) =(co + c~)6(e) = 0, so that c~ = -co and the representation (1.18) is proved. The proof of Lemma 6 is complete. 0 18.2. Functions of the classes H+(Tl) and P+(T 1 ). We first consider the case n 1. Suppose the function f E H + (T 1 ), that is, f( z) is holomorphic and ~f(z) = u(x. y) ~ 0 in the upper half-plane T 1 so that ~f E P + (T 1 ). Recall that fez) satisfies the estimate (see Sec. 13.3)
=
(2.1)
y> 0, and the measure J.L = SSf+
= u(x, +0) satisfies the condition (see Sec. /
J.l(dx) -1"":"'+-x-':-2
17.2) (2.2)
< 00.
Let € > 0 and R > 1 and denote by CR and -CR semicircles of radius R centred at 0, as depicted in Fig. 42. By the residue theorem we have
L/ + f) R
fez + i€) = _1 1+z 2 21ri
R
f(( + i€) d(
f(i + it) (1+(2)((-z)+2i(z-i)'
y> 0,
Izl < R.
CR
(2.3)
18. HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN Tn
255
Analogously, for the function f~~~i;> which is meromorphic in the lower half-plane y < with the sole simple pole -i, we have
°
I
j«( + ie) d( (1 + (2)« - z) Sending R to
00
f(i + ie) 2i(z + i) ,
y> 0, Izi < R.
(2.4)
in (2.3) and (2.4), and using the estimate (2.1), according to which
(1 + Rei", + ie
1
J
1r
:S
M
R d'f'
-=1
(Rsin 0.
-00
Adding together the resulting equalities, we derive an integral representation for the function f(z + ie):
f (z + ic ) =
1~z2 II
I
00
(1
u(x',c)dx'
+ x'2)(x' _ z) + zu
(
0,1
+ c ) + ':Jl! Z + ze 0.
(2.5)
Separating the imaginary part in (2.5L we obtain an integral representation for the function u(x, y + c):
I
00
u(x,y+e) == Y 1r
u(x',e) [(
x-x
';2 +y2- l+x1'2] dx'+yu(O,l+e),
-00
y
> 0.
(2.6)
Passing to the limit in (2.5) and (2.6) as c ---+ 0, and making use of the limiting relation (2.3) of Sec. 17.2, we obtain the necessity of the conditions in the HerglotzNevanlinna theorem (see Nevanlinna [78]).
256
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
For the function f( z) to belong to the class H +(T 1 ), it is necessary and sufficient that it be representable in the form THEOREM 1.
00
f(z)
= ~, /
+ az + b
(1 + x' z)J.L(dx')
1 + x,2)(x' - z)
11" -00 00
=i /
+ az + h,
Sl(Z - x'; i - x')J1(dx')
(2.7)
y> 0,
-00
where the measure J.l is nonnegative and satisfies the condition (2.2), a > 0, and b is a real number. The representation (2.7) is unique, and /-l r;s f +, b ?Rf(i),
=
=
00
a =
CJf(i) _ ~ /
jJ(dx)
1+x
1T'
2
=
lim CJf(iy) , Y-+OO
(2.8)
Y
-00 00
J.l(dx')
Y / c.}f(z) = -1T'
( x-x ')2
+ y 2 + ay,
(2.9)
y> 0.
-00
The sufficiency of the conditions of Theorem I is straightforward. For the function u(x,y) to belong to the class P+(T 1 ), it is necessary and sufficient that it be representable in the form COROLLARY.
J( 00
u(x, y)
= -1T'Y
J.l(dx') x-x 'F + y 2
+ ay,
(2.9)
y> 0,
-00
where a here,
>0
and the measure J.l is nonnegative and satisfies the condition (2.2);
p, = u(x,
+0)
and
a
=
lim u(O, y) . y
y-+oo
From the representation (2.9) it follows that the Poisson integral is a harmonic function in T 1 . By the theorem of Sec. 17.6, the representation with the Schwartz kernel holds with respect to any point zO E T 1 [formula (2.7) for zO = 1]. REMARK.
In terms of the spectral function g(O of the function f(z) (see Sec. 17.1), the class H + (T 1 ) is characterized by the following theorem (Konig and Zemanian [62]). THEOREM
II. For a function f(z) to belong to the class H + (T 1 ), it is necessary
and sufficient that its spectral function g(~) have the following properties:
(a)
-ig(~)
+ ig"' (~) » 0, iU"(~) + iwS'(~),
(b) g(O = where a 2: 0 and u(~) is a continuous function with support in [0,00) which function satisfies the growth condition J
(2.10) Here, the expansion (b) is unique, the number a is defined by (2.8), and ~f(z) is defined by (2.9).
18.
HOLOMORPHIC FUNCTIONS WITH NONNEGATIVE IMAGINARY PART IN
Tn
257
COROLLARY. For the measure J.l to be a boundary value of the function u(x, y) of the class P+(T 1 ), J.l == u(x,+O), it is necessary and sufficient that J.l == F[v"], where v" » 0, v is a continuous *-Hermitian function satisfying the growth condition (2.10), and v(O) == O. In this case the junction v with the indicated properties is unique to within the summand iC~J where c is an arbitrary real number. This follows from Theorem II for v == u + u· (necessity) and for u == 8v (sufficiency) if we take advantage of (1.5) of Sec. 17.1, Jl ~F[-ig + ig·].
=
PROOF OF THEOREM II. NECESSITY. Let f E H+(T 1 ). Condition (a) was proved in Sec. 17.1. To prove condition (b), rewrite representation (2.7) as [compare with (2.5)]
J 00
f(z)
=
1 + z2 7r
Jl(dx ' ) . (2)( )+~j('l)z+b 1 + x' x' - z
-00
== i(l + z2)(fT * KI(x' + iy)) + ~f(i)z + b,
U
=
7r(1
(2.11)
+Jl r,2) .
Since Kt{x + iy) E 1-l s (for all sand y > 0) [see (2.5) of Sec. 11.2] and J fT(dx') [see (2.2)], the Fourier transform formula of the convolution u *](1 holds:
p-l[U * KI] == F[u](-e)F-l[Kd(~) == e-Y€O(~)v(~),
< 00 (2.12)
where v(~) = F[u]( -~) a continuous positive definite (and, hence, bounded) function (see Sec. 8). Now, using (2.11) and (2.12), we compute the spectral function 9 (~) (see Sec. 9): g(~)
= i(1 -
(2)[O(~)v(~)]
+ iCJj(i)t5'(~) + M(~).
By Lemma 2 of Sec. 18.1 there exists a continuous f;.mction in [0 1 00) that sa.tisfies the estimate (2.10) and is such that
Ul (~)
(2.13) with support
(1 - 8 2 ) {O(~)[ v(O - v(O)]} == a2Ul (~). Therefore, (2.13) takes the form
g(~) = ill'
[u,( 0, 6 < 0 leE 1R n -2] .
By Lemma 4 of Sec. 18.1, the function X(€) can be represented as
X(e) = e-l{dx(O, 6, i)
+ e- I(2I X(€ll 0, €) -
e- l{d-16I x (O, 0,
E\
(3.14)
€ E G+_.
In accordance with the induction hypothesis, for the functions that follow
- = X2 ... n(6,€L -
x(0,6,~)
- = X13 ...n(6,e), x(O, 0, €) = X3 .. n (€),
X(~l, O,~)
the corresponding representations (3.13) hold true. Substituting them into (3.14), we obtain (3.13) in the domain G+_. The representation (3.13) occurs also in other domains of the type G+_ that do not contain -~u~. From the uniqueness of the representation (3.13) in the indicated domains of the type G+_ it follows that the appropriate representations (3.13) coincide in the intersections of those domains. Hence, the representation (3.13) holds true everywhere outside -~ U~. By introducing the function 4>1...n
(€)
= X(€) -
L
1:
2 0, .. . ,(en,y) > 0] is an n-hedral acute cone. Then
We denote by A the (nonsingular) linear transformation
z --+ (
= (1 = (e 1, z), ... , (n = (en
I
z»)
= Az.
(1.1 ) GI
The transformation ( = Az maps biholomorphically the domain T onto the domain Tn, and the transformation €' A-I T ~ maps the cone C'· onto the cone R;.. In the process, the derivatives a (a 1 , ... , an) pass into the derivatives (8i {)~), 8j &~J., via the formulae
=
{)' =
=
I ' .• ,
8;=
L
1 0 in T C , whence, by the arbitrariness of C' C C, it follows that ~f(z) > 0 in T C which is what we set out to prove. The theorem is proved. 0 I
19.2. Positive real matrix functions in T C • For an N x N matrix function A (z) to be positive real in T C , where C is an acute (convex) cone in ~n, it is necessary and sufficient that its spectral matrix function Z(€) have the following properties: THEO REM.
(a) (Z(~)a
+ Z*(~)a,a)
a E eN,
»0,
(2.1)
=
(b) for any n·hedral cone C' (y: (el,y) > O, ... ,(en,y) the cone C, it is (uniquely) representable in the form
L
Z(e) = (el,8f·· .(en,8)2ZCJ(~) +
> 0] contained in
Zg)8jcS(~),
(2.2)
l~j~n
where the matrix function ZC' (~) is a continuous tempered function in ~ n with support in ~, and the matrices Z~P, j = 1, ... n, are real symmetric and such that I
'L...J "'
(j) ~ 0, YjZc'
-,
(2.3)
Y E C.
l~f~n
Here the following equation holds:
f
m
(Z * 0,
cP = (CPl' ... , CPN) E SXN.
(2.4)
From (b) of the theorem it follows that the spectral function Z(~) is real and its elements Zkj E S'(C"'), so that the matrix function A(z) L[Z] is holomorphic in the domain TC, ~here C int C** (see Sec. 12.2), and satisfies the condition of reality A(z) A( -z). Let us now verify that the generalized function 9a(e) = (Z(e)a) a) satisfies, for all a E eN I the conditions (a) and (b) of the theorem of Sec. 18.1. Condition (a) is fulfilled by virtue of (2.1): PROOF. SUFFICIENCY.
=
=
=
ga(~)
+ g~ (0 = (Z(~)a + Z· (~)a, a) »
O.
Conditions (b) are fulfilled by virtue of (2.2) and (2.3): ga (~) = (el, 8)2 ... (en, 8)2(ZCI (e)a, a)
+
L::
(Z~j) a , a)8j6(~),
l:S-j~n
where (ZCI (e)a, a) is a continuous tempered function in n~n with support in C'*, and
L
(j)
Yj(Zcl a,
a) ~ 0,
-='
Y E C.
l~j~n
That is, the vector
(zg) a, a), ... , (Zb~) a, a)) E C'·. Noting that 9a(e) is the spectral function of the function (A(z)a, a), we derive from the theorem of Sec. 18.1 that ~(A(z)a) a) ~ 0, z E T C) a E eN. That is) ~A(z) ~ 0, z E T C . Thus, the matrix function A(z) is positive real in T C .
270
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
NECESSITY. Let A(z) be a positive real matrix function in T C . Then for every vector a E eN the function (A(z), a) is positive real in r C . By the theorem of Sec. 18.1 , its spectral function 9 a (~) taken from S' (C·) has the following properties: (a') ga(~) + g:( REMARK
0
1. For n == 1 the theorem has been proved by Konig and Zemanian 2 by Vladimirov [113].
2. In ~2, any convex open cone C is dihedral, that is, C=
[y: (el,Y) > 0, (e2,Y) > 0],
and for that reason we can take the cone C itself for the cone C' in the representation (2.2) .
20. Linear Passive Systems 20.1. Introduction. We consider a physical system obeying the following scheme. Suppose the original in~perturbation u( x) (Ul (x), ... , UN (x)) is acting on the system. as a result of which there arises an out-perturbation (response of the
=
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
272
system) f(x) = (fdx), ... I fN(X)), Here, by x = (Xl,"" Xn ) are to be understood the temporal, spatial and other variables. Suppose the following conditions have been fulfilled: (a) Linearity: if to the original perturbations UI and U2 there correspond perturbations II and h then their linear combination Q'Ui + f3u2 is associated with the perturbation aJI + f3f2. (b) Reality: if the original perturbation u is real, then the response perturbation f is real. (c) Continuity: if all components of the original perturbations u(x) tend to 0 in [', then so do all components of the response perturbation f(x) tend to o in V'. (d) Translational invariance: if a response perturbation f( x) is associated with the original perturbation u(x). then, for any translation h E lR n , to the original perturbation u(x + h) there corresponds a response perturbation f(x + h). The conditions (a)-(d) are equivalent to the existence of a unique N x N matrix Z(x) = (Zkj(X)), Zkj E v,(~n), which connects the original u(x) perturbation and the response perturbation f(x) via the formula (see Sec. 4.8) (1.1 )
Z*tt=f.
Let us impose on the system (1.1) yet another requirement, the so-called condition of passivity relative to the cone r. Suppose r is a closed, convex, sohd cone in lR n (with vertex at 0). (e) Passivity relative to the cone f: for any vector function cp(x) in V following inequality holds:
~
J
(Z
* cp, cp) dx > O.
X
N
the
(1.2)
-r Note that the function (Z * cp, cp) E V (see Sec. 4.6), so that the integral in (1.2) always exists. Furthermore, because of the reality of the matrix Z(x) the condition of passivity (1.2) is equivalent to the condition
J
(Z
* .e, e E int f, in (2.1) and passing to the limit as >. -+ +00
(so that - f + >'e -+ lR n , Fig. 44L we obtain from (2.1) the inequality (2.2). 20.2.3. Causality with respect to the cone r: supp Z(x)
c r.
0
(2.3)
V:
N and let A be a real number. Substituting
I, 1J(t) 0, t < 0 and e E int f*. Then the function T/((e, x)) E OM and for that reason 1J((e,x))/kj E 8' (see Sec. 5.3). Furthermore, the support of the generalized function
=
=
is compact, by virtue of Lemma 1 of Sec. 4.4 (see Fig. 22; the cone f is assumed to be acute!), so that Skj E S' (see Sec. 5.3). Conclusion: Zkj E S'. D
276
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
20.2.6.
The condition of passivity (1.2) holds in the strong form:
~/
(Z
* lp, rp) dx 2:
0,
lp
E
(2.6)
S'XN.
-r Indeed, fix 'P taken from Sec. 5.1). Then by (1.2)
sxN
and let lpv E 'D xN , lpv ---* 'P, v ---*
~ !(Z*lpv,VJv)dx 2 0,
v
= 1,2,...
00
in
sxN
(see
(2.7)
.
-r By 20.2.5, Zkj E 8 ' , 1 ~ k, j < N, and therefore Zkj is of finite order (see Sec. 5.2). Denoting by m the largest of the orders and using the estimate (6.4) of Sec. 5.6, we obtain, for k = 1, ... N, I
(a) I(Z
* JPv)k(x)1 < C (1 + IxI 2)m/2 1~~~ Ilrpvjllm,
(,8) (Z
* ipv)k(X) IXI~!l (Z * If'h(x),
v ---*
v = 1,2, ....
00
for arbitrary R> 0, from which we conclude that passage to the limit is possible as v -+ 00 under the integral sign in the inequality (2.7); we thus obtain the inequality (2.6). [] 20.2.7. The existence of impedance:
Z(() = L[Z] = F [Z(x)e-(q,X)] (p),
(= p+ iq
is a holomorphic matrix function in the domain T C = ~n + iC 1 where C = int r*. This follows from the properties 20.2.3 and 20.2.5: Zkj E S'(f) (see Sec. 9.1).
o
20.2.8. The condition of reality of impedance:
Z()
= Z( -(),
(2.8)
This follows from the reality of the matrix Z(x).
[]
20.2.9. The property of positivity of impedance: ~Z()
> 0,
(E T e .
(2.9)
Indeed, let the function 17e (x) be such that TIe E ceo; 17e (x) = 1, ry,(x) 0, X rt f'; laary,(x)1 :::; CCH:' Then, for all (E T C (see Sec. 9.1),
=
ip((;x)
X
E fe /2;
= 77e(-x)e- i ((,x) E 8,
and for all a E eN the vector function aip E SXN; therefore, using the formula (6.4) of Sec. 4.6, we have
(Z*aip,arp)
= [(Za,a) * 1. _ [(e, x)] 1Jc (X ) - 1J - - . f~
0 \
0]
l
We set
c
Then for all
O, ... ,(en,q) > 0]. Then C ' c C , e E 0 ' and C'· [x: (e,x) > O, ... ,(e~,x) ~ 0]. For the cone G' , the representation (3.1) holds true. Taking into account that representation, we transform the right-hand side of the inequality (3.4) to the following form:
=
-!
(Z
* (c,oTJe),epf]e)dx =
-
-r~
I
(el,0)2 ... (e n ,0)2(ZC'
* (c,o'f}e),'PT/e)dx
-r~
For the quantity h (c) we have the estimate
Ih(e)! < Cl
L l~k,j~N
!
(el,o)2 ... (e n ,0)2
Ixl 0 is such that suppc,o CUR. In the inner and outer integrals in (3.6) we make a change of the variables of integration via the following formulae, respectively:
x -----t B x = y = [YI = (e, x), ... , Yn = (e~, x)] ,
x' -+ B x' = y i •
Then the cone C'· goes into the cone ~ = [yl: y~ 2: 0, ... , y~ ~ 0] the ball UR goes in to a bounded domain contained in some ball UR 1 , the strip 0 < (x, e) < c goes into the strip 0 < YI < c, and the derivative (ek' &) into the derivative Ok (see Sec. 19.1). Setting I
ZC',kj(B-1y')
= Vkj(Y'),
ai ... a~?pj (y)
epdB-1y) = 1/Jl(Y), = Uj(Y),
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
280
we obtain from (3.6) the estimate
!li+
or
Vkj (Y/)Uj (y - Y/)17
( /) Yl - Y £
1
dy' dy
Iy-y'l 0] and so forth. By means of an m-fold repetition of that process we obtain that the matrix Z(x) defines a passive operator relative to the cone f m = [x: (e 1 , x) > 0, ... , (em, x) ~ 0],
/
(Z
* 'P, 'P) dx ~ 0,
'P E V:
N
.
(3.10)
-r m
But the convex cone C* = [x: (x, q) > 0, q E 0] may be approximated from above by arbitrarily close m-hedral cones r m as m ---t 00. Therefore, passing to the limit as r m ---t C* under the condition of passivity (3.10), we obtain the condition for passivity for the cone C* = (int f*)* = r 1 which is what we set out to prove. 0 Combining Theorem I, the theorem of Sec. 19.2, and the remark of Sec. 20.2, we obtain THEOREM II. The following conditions are equivalent:
(a) The matrix Z(x) defines a passive operator relative to an acute cone
r.
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
282
(b) The matrix Z(x) satisfies the weak condition of passivity (2.13) relative to the cone r. (c) The matrix Z(x) satisfies the condition (2.5) and the conditions (b) of the lemma. (d) The matrix Z(x) satisfies the condition of dissipation (2.2) and the conditions (b) of the lemma.
20.4. Multidimensional dispersion relations. The results obtained in Sec. 20.3 permit deriving (multidimensional) dispersion relations (see Sec. 10.6) that connect the real and imaginary parts of the matrix Z(p) - the boundary value of the impedance Z((). For the sake of simplicity of exposition, we confine ourselves to the case of the cone C = lR+. Let us first prove the following lemma. LEMMA.
The general solution of the matrix equation
8; ... 8;Z(x)
=0
(4.1)
in the class of real continuous *-Hermitian matrix functions in lR n with support in -~ U ~ is given by the formula
Z(x)
= ZO[£n(x) -
(4.2)
£n(-x)L
where Zo is an arbitrary constant real skew-symmetric matrix. PROOF.
By Lemma 6 of Sec. 18.1 we have
Zkj(X)
= ZO,kj[£n(X) -
£n(-X)] ,
1 < k,j
< N,
=
where ZO,kj are arbitrary real numbers. From this and from the conditions Zjk(X) Zkj( -x) it follows that ZO,kj = -ZO,jk, that is, Zo = The representation (4.2) is proved. The lemma is proved. 0
-z6'.
We denote by N( -lR~ U~) the class of +-Hermitian matrices that are the Fourier transforms of real continuous tempered matrix functions in IR" with support .
ii'ii"
~
-w;+ U IR+. For a matrix of the class N( -~ Unt;.), all matrix elements belong to the space of generalized functions V~2 (see Sec. 10.1). From the lemmajust proved it follows that the generalized solution of the matrix In
equation
pi·· .p~M(p) -n
=0
=-n
in the class N (-lR.+ U lR+) is given by the formula
M(p) = iZ(O)8 1
··
·8n~[i"Kn(P)],
(4.3)
where Z(O) is an arbitrary constant real skew-symmetric matrix. Indeed, passing to the Fourier transforms in (4.2) and using the definition of the kernel ICn(p) (see Sec. 10.2), we have
M (p)
= Zo {F[£n] -
F[En]}
= 2iZoSF[En ](p) = 2iZo8'F(r9 n (X)Xl ... x n ] = iZ(O)Ol" ·8"~[inICn(P)], where
Z(O)
= 2(-1)"Zo.
(4.4)
o
20. LINEAR PASSIVE SYSTEMS
283
THEOREM. In order that the matrix Z (x) should define a passive operator rel· ative to the cone ~, it is necessary and sufficient that its Fourier transform Z(p) satisfy the dispersion relation
~i(p) = (2~)nP~" .p~(M * ~Kn) + iZ(O) -
2:
Z(j)Pj,
(4.5)
lsJ$n
where the matrix M(p) is a solution in the class N(-~: U~) of the equation p~ ... p~M(p) = ~Z(p).
(4.6)
Here the matrix ~Z(p) is such that for all a E eN the generalized function (~Z(p)a, a) is a non-negative tempered measure in ~n,o the matrix ZeD) is real, 1, ... , n, are real, constant constant, skew-symmetric, and the matrices z(j), j and positive. In the dispersion relation (4.5), matrices [M(p), Z(O), Z(l), ... , zen)] are unique up to additive terms of the form
=
(4.7) where A is an arbitrary constant real skew-symmetric matrix. REMARK 1. For n = 1 the theorem was proved by Beltrami and Wohlers [4]; for n 2 2, it was proved by Vladimirov [113]. REMARK 2. The actual growth of the measure (~Z(p)a, a) is such that the measure (~Z(p)a, a)
(1 is finite on
~n
+ PI) ... (1 + p~)
(see the theorem of Sec. 18.4).
PROOF OF THE THEOREM. NECESSITY. Suppose the matrix Z(x) defines a passive operator relative to the cone By Theorem I of Sec. 20.3, the matrix Z (x) has the following properties:
IPl:.
(a)
(Z(x)a
+ Z· (x)a, a) »
(4.8)
0,
(b)
Z(x)
= 8r'"
8~Zo(x)
+
L
Z(J)8j o(x),
(4.9)
lsj~n
where the matrix-function Zo(x) is continuous, real, and tempered in ffi.n with support in the cone ~; the matrices Z(j), j 1, ... , n, are real constant and positive. Passing to the Fourier transform in (4.8) and (4.9), we conclude that for all a E en the generalized function
=
(mZ(p)a, a)
1 = -F[(Z(x)a + Z· (x)u, u) ] 2
(4.10)
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
284
is a non-negative tempered measure in Sec. 8.2) and
~n
(by the Bochner-Schwartz theorem; see
L
Z(p) = (-l)npi·· .p~F[Zo](p) - i
Z(j)Pj,
(4.11)
l~j~n
~Z(p) = (_~)n pi ... p~F[Zo(x) + Z~(x)](p).
(4.12)
We set
M(p)
=
(_l)n 2
F(Zo(x) + Zo(x)](p).
(4.13)
The matrix M(p) belongs to the class N(-~ Uw:.) and by (4.12) it satisfies the equation (4.6). Furthermore, taking into account the equalities (see Sec. 10.1 and Sec. 10.2)
F[Zo](p) = F ((Zo(x) + Z~(x))en(x)]
(2~)n F [Zo (~) + Z~ (x)] * F[OnJ = 2(-lt M * K n , we rewrite relations (4.11) as
Z(p)
= (2~)nPI" .p~(M * Kn) -
L
i
Z(J)Pj.
(4.14)
l~j~n
Separating the real and imaginary parts in (4.14), we obtain the dispersion relation (4.5) (for Z(O) = 0) and the relation ~Z(p)
2
2
= (21r)n Pl
..
2
'Pn(M
* ~Kn),
(4.15)
which is equivalent to the relation (4.6) by virtue of (4.6) of Sec. 10.4: 2
M = (21r)n M
* ~Kn.
(4.16)
Suppose the matrix Z(x) is such that its Fourier transform Z(p) satisfies the dispersion relation (4.5), where the matrix M(p) is a solution in the class N(-~: U~) of the equation (4.6), and that matrix is such that for any a E en the generalized function (~Z(p)a, a) is a non-negative tempered measure; the matrix Z(O) is real and skew-symmetric, and the matrices Z(j), j = 1, ... n, are real, constant and positive. By (4.16), the equation (4.6) is equivalent to equation (4.15), which, together with the dispersion relation (4.5), yields SUFFICIENCY.
I
Z(p)
= (2~)n pi· .. p; (M * Kn ) -
z(O) -
~
i
z(j)Pj,
(4.17)
l~J~n
whence, using the inverse Fourier transform, we obtain
Z(x)
= 8; .. ·8~Zl(X)
- Z(O)6(x)
+
L l~j::;n
Z(j)8j 6(x),
(4.18)
20. LINEAR PASSIVE SYSTEMS
285
=
where ZI(X) 2(-1)n F-l[M](x)O(x) is a real continuous tempered function in with support in the cone ~. Noticing that
8;·· ·8~Zt{x)
a; .. ·8~[ZI(X) -
- Z(O)o(x) =
jRn
Z(O)En(x)],
we obtain that the matrix Z(x) satisfies the condition (4.9). The condition (4.8) is also fulfilled, by virtue of (4.10) and the Bochner-Schwartz theorem (see Sec. 8.2). By Theorem II of Sec. 20.3, the matrix Z(x) defines a passive operator relative to =-1l the cone lR+. We now prove the uniqueness of the dispersion relation (4.5) up to additive terms of the form (4.7). Suppose the representation (4.5) occurs with other matrices h by what has b een proved, [M 1 , Z 1(o) , ZI(1) , ... I ZI(n)] . Ten,
M(p) - Mt{p) = iA8 1 ... 8n~[inKn(P»), where A is some constant real skew-symmetric matrix. From this, by subtracting the distinct representations (4.5) for SSZ(p). we obtain
(2~)n Api· .. P~ [81 ... an ~(in K n ) * ~Kn] + i[Z(O) - ziG)] -
L
[Z(j) - z~j)]Pj
= O.
(4.19)
l:Sj:Sn
Passing to the inverse Fourier transform in (4.19) and using the formulae (4.4) and (2.8) of Sec. 10.2, we obtain the equality
- ~ A8; ... a~ { [En (x) - En (- x)]
[On (x)
- On (- x) ) }
:L
+ i[Z(O) - zi O)]6(x) +
[Z(j) - Zi j )]8jo(x)
15j5 n
= -~A8i ... a~ [£n(X) + En ( -x)] + i[Z(O)
- Z~O)]6(x)
L
+
[Z(j) - ZP)]OjO(x)
l::;j::;n
= i[Z(O) - ziG) -
A]o(x)
L
+
[Z(j) - Z~j)]oj6(x)
= 0,
l:Sj:Sn
which is only possible for Z(O)
= Z~O) + A,
Z U) -- ZU) 1 ,
This completes the proof of the theorem.
j
= 1, . .. ,n.
o
20.5. The fundamental solution and the Cauchy problem. The fundamental solution of the passive operator Z* relative to the cone r is any matrix A(x), A kj E V', that satisfies the convolution matrix equation
Z*A=Io(x).
(5.1 )
The operator A* is also said to be the inverse of Z* (compare Sec. 4.9.4), and the matrix function A(() - the Laplace transform of the matrix A(x) - is called the admittance of the physical system.
3 SOME APPLICATIONS IN MATHEMATICAL PHYSICS
286
The passive operator Z * relative to the cone r is said to be non-singular (respectively, completely non-singular) if r is an acute solid cone and det Z(() :j:. 0, ( ETc, where C = int r'" (and, respectively, if for any a E eN, a :j:. 0, there exists a point (0 E T C such that
a) > 0.)
W(.2((0)0.,
(.5.2)
The equivalent definition of a non-singular passive operator Z * is: Z * is passive if there exists a point (0 ETc such that det Z((o) :j:. (Drozhzhinov [22]). If the operator Z * that is passive relative to the cone r is completely nonsingular, then
°
atZ(() > 0,
(5.3)
Indeed, by Theorem I of Sec. 20.3, the function (Z(()a, a) is holomorphic and W(Z(()a, a) 2:: 0 in T C . But then, by (5.2), the inequality W(Z(()a., a) > 0 holds if a i- (see the reasoning in Sec. 17.1), which is equivalent to (5.3). D From this it follows that any completely non-singular passive operator is also a non-singular passive operator relative to the same cone. Furthermore, for an operator Z* that is passive relative to an (acute solid) cone r to be completely non-singular, it is necessary and sufficient that the equality
°
(Z(x)o.,o.)
= igb(x)
be impossible for any a E eN, a i- 0, and for any real g. Indeed, if the operator Z * that is passive relative to the cone non-singular, then (5.4), which is equivalent to the equality
(Z{()a, a)
°
= ig,
(5.4)
r
is completely
( ETc,
is impossible by (5.2) for any a i- and for any real g. Conversely, suppose the operator Z * that is passive relative to the acute solid cone r is not completely non-singular. Then, for some a i- 0, we would have ~(Z(()a, a) ::; 0, ( E T C . On the other hand, by Theorem I of Sec. 20.3, the function (Z(()a, a) is holomorphic and 3r(Z(()a, a) 2:: 0 in T C and therefore W(Z(()a, a) = 0 in T C. Hence, (.2()a, a) = ig, where 9 is a real number so that (5.4) holds for certain a :j:. 0 and for certain real g. 0 I. Every non-singular passive operator relative to a cone r has a unique fundamental solu.tion that determines a non-singular passive operator relative to that same cone r. THEOREM
Let Z * be a non-singular passive operator relative to a cone r so that Z(() is a positive real matrix in T C (by Theorem I of Sec. 20.3) and det Z(() i- 0, ( E T C . We will prove the existence and uniqueness of the solution of equation (5.1) in the class of matrices A(x) that define non-singular passive operators relative to r. Applying the Laplace transform to equation (5.1), we obtain an equivalent matrix equation PROOF.
z(()A(()
= I,
(5.5)
Equation (5.5) is uniquely solvable for all ( E T C and its solution - the matrix function A(() = Z-l(() - is holomorphic and deL4.(() i- 0 in T C • Furthermore, from the equality Z(() = Z(-(), ( E T C , and from (5.5) it follows that
20_ LINEAR PASSIVE SYSTEMS
Z«)A( -() =
287
I, that is, - -1
Z
-
«) = A «)
~
= A (-C) , -
C
( ET .
mZ«) > 0, ( E T C , and ~Jt4.(e) = A:+ «)[~z«)] A(e) 2: 0,
Finally, from the condition
from (5.5) we derive
( E TC .
(5.6)
Consequently, the matrix A(O is positive real in T C . By Theorem I of Sec. 20.3 the matrix A(x) defines a non-singular passive operator relative to the cone r. The matrix A(x) is unique. The proof of Theorem I is complete. 0 If the passive operator Z * is completely non-singular, then its inverse operator A* is completely non-singular. COROLLARY.
Indeed, since mZ«) > 0 and det A(e) f:. 0, it follows, by (5.6), that ~A«) > 0, (E T e . 0 Let r be a closed convex acute cone, C int f*, let 5 be a C-like surface, and let S+ be a region lying above S (see Sec. 4.4). By analogy with Sec. 16.1 we introduce the following definition. By the generalized Cauchy problem for an operator Z * that is passive relative to the cone f with source f E V' (8+) xN we call the problem of finding, in JRn, a solution u(x) taken from V'(S+)xN of the system (1.1). As in Sec. 16.1, the following theorem is readily proved.
=
THEOREM
r
II. If a passive operator Z* is non-singular relative to a (solid) cone
then the solution of its generalized Cauchy problem exists for any f in V' (.c,+) x N is unique, and is given by the formula I
I
(5.7)
u==A*f.
If S is a strictly C -like surface and f E S' ($+) x N, then the solution of the generalized Cauchy problem for the operator Z * exists and is unique in the class 8 ' (S+) xN [and is given by the formula (5.7)]. COROLLARY.
This follows from Theorem II and from the results of Sec. 5.6.2. 0 Thus, passive systems behave in similar fashion to hyperbolic systems (see Sec. 16.1, Hormander [46, Chapter 5], Friedrichs [33], Dezin [15]). 20.6. What differential and difference operators are passive operators? A system of N linear differential equations of order at most m (with constant coefficients) is determined by the matrix (compare Sec. 15.1)
L:
Z(x).=
ZaaaJ(x),
(6.1)
O~lal~m
where Za are (constant) N x N matrices. THEOREM I. For a system of N linear differential equations with constant coefficients to be passive relative to an acute cone r, it is necessary and sufficient that
Z(x)
=
L
Zj 8j15(x)
+ ZoJ(x),
(6.2)
1 '5-j '5- n
where Zl, ... , Zn are real symmetric N x N matrices such that El<j0
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
288
PROOF. NECESSITY. Suppose the differential operator Z * defined by formula (6.1) is passive relative to the cone f. Then by Theorem I of Sec. 20.3 the matrix function
(6.3) °:51 a l:5 m is positive real in T C . Therefore, for every a E eN the function (Z(()a, a) is holomorphic and ~(Z(()a, a) ~ 0 in T C . Therefore, that function satisfies the estimate (1.1) of Sec. 17.1 and, hence, all matrix elements Zkj(() satisfy that estimate:
(_i()a Za,kj :S M(G') 1 ~ 1(1
L
2 ,
q1
O xlal 2 ,
are cones with vertex at 0, and f
r
C f
e·
(6.7)
20. LINEAR PASSIVE SYSTEMS
The cones f
LEMMA.
c
and f
289
are closed and acute: f
r
c
= f + r r; I E int r~. r
The mapping (6.6) is continuous from eN (from JR.N) into ~n and, by virtue of the inequality (6.7), is of a compact nature, that is, the pre-image of any compact set is a compact set. Therefore the cones r c and r r are closed. Furthermore, from the equalities PROOF.
(Zja, a) = (Zjb, b) + (ZjC, c),
a
= b + ic,
j = 1, .. -,n,
we conclude that f c = f r + f r . Finally, by the inequality (6.7) the plane (l,x) = 0 has only one point in common with the cone r c - its vertex. Therefore f c and f r are acute cones and 1 E int r~ (see Lemma 1 of Sec. 4.4). The lemma is proved. 0 Notice that the cones r c and shows: ZI > 0, Z2 0 and r c and
=
THEOREM
rr
rr
may not be solid, as the following example lie in the plane X2 O.
=
II. In order that the matrix (6.2) define a passive completely non-
singular operator, it is necessary and sufficient that the malT'ices Zl, . , . , Zn be real symmetric, that the matrix Zo be real and ~Zo 2: 0, and that there exists a t1eetor I E ~n such that
L
lj Zj
+ ~Zo > O.
(6.8)
l~j~n
Here, the passivity and the complete non-singularity of the operator Z", occur in the case of any acute cone r that contains the cone r c, and 1 E int r* .
Suppose the matrix (6.2) defines a passive and completely non-singular operator with respect to a certain (acute) cone r. Then the conditions of Theorem I are fulfilled and, by (5.3) and (6.5), PROOF. NECESSITY.
-
~Z(()
=
~
Lt qjZj
+ ~Zo > 0,
( E Tc ,
C
= int r*,
l~j~n
so that the condition (6.8) holds for all q E C. SUFFICIENCY. Let the matrices Zo, _.. ,Zn in (6.2) satisfy the conditions of Theorem II. Suppose r is an acute cone containing the cone r c and such that lEe = int r*. From this if follows that (q, x) ~ 0 for all q E C, x E r c C f, that IS,
(q, x)
=
L
qj(Zja, a) ~ 0,
q E C,
I:
q E C.
l~j~n
This means that
qjZj ~ OJ
l~j~n
By Theorem I, the matrix Z(x) defines a passive operator relative to the cone Furthermore, it is given that lEe and so, by (6.5) and (6.8),
mZ(il) =
L
IjZ}
r.
+ ~ZO > 0,
l~j~n
so that the operator Z * is completely non-singular relative to the cone f (see Sec. 20.5). The proof of Theorem II is complete. 0
3. SOME APPLICATIONS IN MATHEMATICAL PHYSICS
290
Theorem II states that the matrices Ll<j 0,
( ETc I
(6.11)
I O. REMARK.
20.7. Examples. Let us denote by Y+(a) = [(x, t): at > Ixl] the future light cone in 1R 4 , which corresponds to the speed of propagation a: V+ = y+ (1) (compare Sec. 4.4). . 20.7.1. Maxwell's equations. The principal part of the approximate differential operator is of the form 4
aD
--rotH
oxo
oB
'
~ uXo
+rotE,
(7.1)
4S pecification of div D and div B in the system of Maxwell's equa.tions is not essential for our purposes; actually, these are consistency conditions.
20. LINEAR PASSIVE SYSTEMS
where Xo
291
= ct, c is the speed of light in vacuum, x = (xo, x) D
= c * E,
and
= J.l * H,
B
(7.2)
where c and Ii are 3 x 3 matrices called tensors of dielectric and magnetic permeability respectively. If c and f.l are constant matrices that are multiples of the unit matrix, c fIJ(x), Ii f.lIJ(x), then the system (7.1)-(7.2) becomes
=
=
BE
8H
c--;:;- - rot H,
Ii--;:;-
uxo
uXQ
+ rot E.
(7.3)
-+ The system (7.3) is passive with respect to the cone V (l/VEii) by virtue of the inequality
[E ( :~ , E) - (E, rot H)
/ - V+(1/ ftIi)
+ It
(~:. ' H) + (H, rot E)] dx ~ 0,
which holds for all E E Dr (I~4) x3 and H E Dr (I~ 4) x3. Here, N To prove the inequality (7.4) we make use of the identity (H, rot E) - (E, rot H) = div(E
X
== 6,
n
(7.4)
= 4.
H)J
by virt.ue of which the left-hand member of (7.4) is equal to
-"fiPlxl
/
/
liP
-00
0
8~o
(c1E1 2 + IlI H I2 ) dxo dx + / -00
=
/
div(E x H) dx
Ix/