Subharmonic functions of finite order in a cone. I - Springer Link

I1(I -- K~)

2 (l -F Kn) ~" I1 =

]~ ~

de~. ~.

--V~"

~< m a x

l+X ~ ~< 2c,

from where, taking into account (29), there follows the stability of the operator U. LITERATURE CITED 1. 2. 3. 4.

5. 6. 7.

8. 9.

R. S. Phillips, "The extension of dual subspaces invariant under an algebra," in: Proc. Internat. Sympos. Linear Spaces (Jerusalem, 1960), Pergamon, Oxford (1961), pp. 366-398. Yu. L. Daletskii (Ju. L. Daleckii) and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence (1974). T. Ya. Azizov and I. S. Iokhvidov, Linear Operators in Spaces with an Indefinite Metric, Wiley-Interscience, Chichester, Great Britain (1989). S. A. Kuzhel', "J-nonexpansive operators," Teor. Funktsii Funktsional. Anal. i Prilozhen. (Khar'kov), No. 45, 63-68 (1986). J. Bognfir, Indefinite Inner Product Spaces, Springer, Berlin (1974). M. G. Krein and Yu. L. Shmul'yan, "On fractional linear transformations with operator coefficients," Mat. Issled., 2, No. 3, 64-96 (1967). B. Sz.-Nagy and C. Foias, Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam (1970). F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar, New York (1955). M. G. Krein and Yu. L. Shmul'yan, "On plus-operators in a space with an indefinite metric," Mat. Issled., 1, No. 1, 131-161 (1966).

SUBIIARMONIC FUNCTIONS OF FINITE ORDER IN A CONE. 1 (GENERAL THEORY)

A. Yu. Rashkovskii and L. I. R o n k i n

UDC 517.55

In this paper, consisting of three parts, we investigate subharmonic functions in a cone. Here we consider a series of problems, arising from investigations in the domain of entire functions. We shall dwell on this in some detail. In the theory of entire functions (of one and several variables) the questions regarding the growth of functions and the distribution of their zeros are of main interest. These same questions are fundamental also in the investigation of classes of functions, related in a certain sense to the entire ones, namely, functions that are holomorphic in a cone, a tubular domain, in domains of the form G x C k, etc. To a significant degree, the interest in the indicated problems is due also to the large role played by them in the applications of the theory of functions to partial differential equations, to convolution equations, to probability theory, and to the theory of generalized functions. One of the fundamental methods for the solving of the problems, related to the growth and the distribution of the zeros of holomorphic functions, is their reduction to corresponding problems for the wider class of subharmonic functions. Here one starts from the fact that ln]f(z) 1, where the function f(z) is holomorphic, is plurisubharmonic and, consequently, subharmonic, while the volume of the null set of the function f in some domain G is equal to/z(G), where u is the Riesz associated measure (the Riesz mass) of the function In ] f(z) [. The most complete study of the growth of subharmonic functions and of the distribution of their Riesz mass has been carried out for the class of functions of finite order in R m (i.e., functions u(x) such that u(x) _< a Ix [P + b) and for its subclass of functions of completely regular growth.

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i lkh Prilozheniya, No. 54, pp. 74-89, 1990. Original article submitted January 12, 1989. 0090-4104/92/5804-0347512.50,1992 Plenum Publishing Corporation

347

In its initial definition (see [1]), given for entire functions of one variable by B. Ya. Levin and A. Pfluger in 1937--1938, a function of completely regular growth has been characterized as function for which the logarithm of its modulus outside a sufficiently scarce set is asymptotically close to a positively homogeneous function (more precisely, to its indicator). The fundamental fact of the Levin--Pfluger theory is the equivalence of the completely regular growth of an entire function to the existence of a definite regularity of the distribution of its roots. The mentioned definition and the basic facts of the theory have been extended in 1961--1962 by V. S. Azarin to subharmonic functions in the space R m, m > 2. Subsequently one has found a new approach to the investigation of functions of completely regular growth. P. Z. Agranovich and L. I. Ronkin [2] and V. S. Azarin [3] have shown that the completely regular growth of a subharmonic function u(x) is equivalent to the convergence in D' of the family of functions t-Pu(tx), t --, ~.* Another definition, also equivalent to the initial one, has been given by L. Gruman

[41. The situation becomes significantly more complicated at the consideration of functions, not in the entire space R m, but in some of its cone. Already for functions, holomorphic in the half-plane, the connection between growth and root distribution has a character, different from that for entire functions, and requires the introduction of new concepts. The corresponding theory, including the theory of functions of completely regular growth, has been constructed by N. V. Govorov [5]. It has been extended to subharmonic functions in the half-plane by A. I. Kheifits. t We mention that N. V. Govorov's constructions are based on refined point estimates and their transfer to the multidimensional case seems to be unlikely. Moreover, remaining within the framework of N. V. Govorov's approach to these questions, it is not even clear how to formulate the corresponding results in the multidimensional case. However, it turned out that, just as for entire functions or for functions that are subharmonic in R m, the regularity of the growth of the function u(x) in the half-plane is equivalent to some special convergence of the functions t-Pu(tx). This fact, as well as some other new facts on functions of completely regular growth in a half-plane, have been established earlier by one of the authors [6]. The approach to the construction of the theory of functions in a half-plane, presented there, has been extended by us to functions that are subharmonic in a cone. The obtained results, as well as preliminary results on arbitrary subharmonic functions of finite order in a cone, have been communicated in [7]. This paper contains an extended presentation of the results that have been given in [7] without proof and in a weaker form. Here we consider function u(x), subharmonic in a cone of the space R m and satisfying in it the condition u(x) < a I x [P + b. To each of these functions we associate its Riesz mass and a real measure (charge) on the boundary of the cone, which is the limiting value of the function u(x) in the sense of weak convergence. The first part of the paper is devoted to integral estimates of functions and of their associated measures. For the determination of these estimates, first we justify some limit processes, we establish integral formulas of the type of Carleman's formula (for functions that are holomorphic in the half-plane) and of the type of Green's formula. The obtained results, representing, in our opinion, an interest in its own right, together with the representation of subharmonic functions in a cone, given in [8], constitute the foundation for the investigation of functions of completely regular growth in a cone, presented in the second part of this work. We give some criteria for regular growth, including the equivalence of the complete regularity of the growth of the function u(x) to a special regularity of the distribution of the generated measures: Riesz and boundary measures. We have obtained formulas connecting the characteristics of the growth of the function with the characteristics of the distribution of their associated measures. In parallel investigations, a series of facts regarding functions of regular growth in a cone have been established by L. Gruman [4]. We mention that in his definition of such functions, L. Gruman did not start from the above mentioned traditional definition of functions of completely regular growth in C or R m, as done by us, but from the definition of the ray of regular growth, given by him earlier for functions in R m. It can be shown that the two definitions are equivalent. However, the results are different since the problems considered by us and by L. Gruman are different.

* An investigation of the weak convergence of the family {t-Pu(tx)}, t --- oo (and of more general families), in the case when u(x) is not subharmonic but a generalized function of slow growth with support in a convex pointed cone, has been carried out by V. S. Vladimirov, Yu. N. Drozhzhinov, B. I. Zav'yalov. Moreover, the formulation of the problems and the nature of the obtained results are essentially different from those from the theory of functions of completely regular growth. tA. I. Kheifits, On subharmonic functions of completely regular growth in the half-plane. Dokl. Akad. Nauk SSSR, 239, No. 2, 282-285, 1978.

348

1. Fundamental Concepts. On the sphere S 1 = {x ~ Rm: [ X I = 1} we consider a domain F with a twice smooth boundary and in this domain F we consider the boundary value problem A*~o + )1~o = 0, ~o] or = 0, where A* is the spherical part of the Laplace operator A, having in spherical coordinates the following form: .0 ~

m-- I 0

A = b'~ q-

r

I A*

Orq-~

"

By 2j = 2j(F), 0 < 21 < 2 2 _< )l3 _< ... we denote the eigenvalues of this boundary value problem. We shall assume that the domain F is such that the eigenfunctions ~oj = ~o~',corresponding to the eigenvalues 2j, belong to the class C2(i-) and 0~Ol/0n > 0 on OF (here and everywhere in the sequel, 0/0n denotes differentiation along the inner normal). We denote by K = K r the cone {x ~ Rm: x/Ix I ~ F}. The elements of the Euclidean volume in R m and of the (m - 1)dimensional volume on OK, induced by the metric of R m, will be denoted by dw and dcr, respectively, while the element of the (m - 1)-dimensional volume of the sphere S"R will be denoted by d S R. All the functions ~oj are assumed to be extended into the I qDl12dSt = I.'~ We also set B R = {xCRm:! x I "< R},

cone K by equalities q0i (x) = % r-~--~ and be normalized by condition r

S R = g B R , KR----K n B~, K~, R=-KR"\ff~, I'R ~ K ~ SR, rr, n = OK tq I Q ~; Om= (m - - 2) f dS~ (m > 2), O2 = 2 n; $1

•

=

[-m

+ 2 +-

V

-

2) 2 + 4

.

We mention that the functions q)i (x) [ x ]~[are harmonic in the cone K, belong to the class C2(K \ {0}), and vanish on 0 \ {0}. In the sequel, for the sake of brevity, the function ~ol will be denoted by ~o, while the numbers x~ by ~c-+. The set of functions, subharmonic in the domain G C R m, will be denoted by SH(G). By/z =/~u we shall denote the def

1

Riesz measure of the subharmonic function u(x). In other words, p, = ~ An, where the Laplace operator A is considered as an operator in the space D' of generalized functions. It is known that the first partial derivatives of the function u ~ SH(G) exist almost everywhere and are locally summable functions, defining the corresponding derivatives in the space D'(G). We need additional information on the derivatives of subharmonic functions. LEMMA 1. Assume that the domain G C R m contains the sets Fr, a < r < b. Assume that the functions u @ SH(G), while the sequence of functions uj ~ SH(G) rq C~(G), decreasing monotonically, converges for j --- ~ to the function u. Then for any

r ~ (a, b), satisfying the condition ~t(Fr) = 0, and for any function r] ~ C~(Fr) = {p ~ C~(Sr): supp ~0 C Fr} there exists the limit

li m I Oui - d " 1,..| ~ n o,.,

independent of the selection of the sequence uj. In addition, for each function r / ~ D(Ka,b) for almost all r ~ (a, b) we have the equality

Xim f0.,

f0,,

,.

(1)

Proof. We consider some domain f2 C C G with a piecewise smooth boundary, satisfying the conditions: 1) i"r C 0f2, 2) #(0f~) = 0. We denote by Q(x) solution of the Dirichlet problem in domain f2 with boundary condition Q ] 0 ~ ,. P, = 0, Q[ rr ~ ~/. According to Green's formula, we have

f o,, ui ~OOds _ ~'r Or rl dS, = ; O A uid o~ , Since uj $ u, we have lira

;

oo

u i ~-~ ds =

u ~ ds and the sequence Auj converges weakly to the measure 0rn~ = Au. From

349

the latter, taking into account that/~(Of~) = 0, we conclude that

lt:m ! A u j Q d t ~

Ora f

(2)

Thus, there exists lira

au~

- - 0~

S

Qd p, q-

(3)

u Tn

We note that the right-hand side of the equality (3) does not depend on the selection of the sequence {uj}, while the left-hand side does not depend on the selection of f2. For the proof of the validity of the equality (1) it is sufficient to show that for any function v(t) E D((a, b)) we have the equality b

a

v (r) lim 1-.|

Pr

~lauldS,dr= "~

v

(Ixl) ~ ~ d~o.

lfa, b

Taking into account the definition of the generalized function 0u/Or, we obtain that ra

~(a, b

b

f ou, I v(r) lii~ ,(Ix[)n~dco------~-~.lim ~ u1~xt[ v(lxl)rl(xl-~jox,ld~o=lim t.~ 0 the function ~PR(X) = 7R(

I xl )~o(x), where 7R(t) = if- -- R r- - K+ff+. We note that the function ~PR is harmonic in Kr, R for

any r E (0, R) and vanishes on F R t3 Fr, R. LEMMA 2. Let u(x) be a function, subharmonic in some neighborhood G of the set Kr, R. In this case, if/~(Fr) = 0, then

l(r, R

J

Pr

f

T'r

(4)

rR

Pr, R

Proof. First we prove (4) in the case when u E C2(G). For this we make use of the Green formula, applied to the domain Kr,R and to the functions u(x) and WR(X). Then we obtain 0m

~R d p = - - , Kr , R

FR

~R an - - u

dSn

a~n On

~

- -

U

I" n to dSR - - ~R (r) {,( a ~u to dSr 4- V'~ r r~~ u tOdS, H- Pr, (R) 6 Thus, under the assumption that u E C2(G), equality (4) is proved. =

350

dSr - -

Pr x

ann 3-n - - u ,

u ~'R (I x I) -O~ ~ d o 9 R

do=

Assume now that u(x) is an arbitrary subharmonic function in the domain G D ~r,R' In each domain G', such that f(r,R C G' C C G, there exists a sequence of functions uj E C ~ ( G ') f3 SH(G'), decreasing monotonically to u(x). As we have just proved, equality (4) holds for the functions uj. Further, obviously, for j --, + we have

I u, tp dS~ + ~! u + dSd

Ft.

PR

r,.R

P,.R

From Lemma 1, by virtue of the assumption #(I'r) = 0, made in the lemma, we derive lirn rJ, ~ q~dS~ = r~ Finally, equality lira t~

~RAuido=

f Kr, R

I ~RAudo K~, R

holds by the same considerations as equality (2). Thus, in the equality (4), valid, as already mentioned, for the functions uj, for j --, o~ it is possible to take the limit, which yields the existence of the equality (4) for the function u(x). The lemma is proved. 3. Integral Estimates. Before formulating Lemma 3, in which we give estimates of subharmonic functions, important for our purpose, we agree to denote by Dj( .... ..., ...), j = 1, 2 .... , functions that depend for a given cone K only on the parameters mentioned in them, and locally bounded for the considered values of these parameters. LEMMA 3. Let u(x) be a subharmonic function in a neighborhood G of the set Kt,o~ and assume that/~(~) = 0. Assume further that for some p > x + and certain positive numbers a, b, c, and d we have the inequalities

(5)

u(x)~* R"- (1 - - 2x--w~), we obtain

j" lullxP.~ndc= ! lullxt~-~.-(Ixl*---lxl*+R*--•

rx,R

~f lul[xl=+•176

r' 0 and we note that in this case the functions V h in D'(K4,t,=) converge to the function u, then we obtain the estimates (13) and (14). In order to obtain (15), we make use of the equality (4) for the domain KL2R and the function Vh(X). We have

Ova,d KX, 2R

r~

"q- ?2R (~') ,I Vhcs dSx - - Y2R (2R) r'[2RVatp dS2R + q-

J

O(p

Vh?2R ([x])0-h da

F~, 24

*We note that in [9] one has obtained an estimate of the quantity/x(K~,R) when K' = K s F ' c C F . 353

(here ~th =

Pv. =

,)

~ AVh . F r o m here, making use of (8) and (12), we conclude that %'2~([ x I) q~d t ~ ~< c [~,~- - -

X x+

(2R) x--~+]

--

d [x%. TM

--

Kh, 2R or ~+-I

- -

(2R) x--x+]

-5 D4 (2R) ~

+

+ D, •R x--1 -- • Ro+m_,2o+m_~+ ~_ ~ D~Ro+~-~+ ~-. On the other hand, [

"f2R(lxllq~dlxt~>/

min ?2R(t)

>~Rx-(l--2~'--'~+)

S

.[ ~dlxn q)dlxh"

K43-,R Consequently,

S q~dlxh ~< 1 - -R--x2x--x+ S ~;2.~ (I x l) q) d ~ 0}, we have the equality

lira I Qq~dish = "J Qq~ dB, VQ E C' (K~., R)(u~ = P.h); h-O K~, R

KL, R

c) the functions Uh, considered as functionals on the space of functions from C(0K) with bounded supports in OK \ {0}, converge weakly for h --- 0 to some measure v = v u (on OK); d) the functions Uh7,, considered as functionals on the space C(I'r) for all r, excluding a countable set A2u = {r: Iv [ (0Fr) > 0}, converge weakly for h --> 0 to some measure v(r) = v(ur); e) if r ~ Au = Alu U Au2 and h --- 0, not assuming values where ffh(i'r) :~ 0, then for any function r/ E C2(i'r) there exists the limit r

lim

h~+0 Fr

def

ePrl~F dSr = Q (rl, r, u);

f) if in addition the function u satisfies in the cone K the inequality u(x) _< a Ixl p + b, then for any R > r we have

~ dl v [ ~< CR ~

Pr, R 354

C -= C (u, r).

(19)

h G h3 = ~ \ G, G~ = G \ {K + hx~ G~ = G \ {G~ U G~}, Proof. We denote G = Kt, R, G~ = G + hx~ G~ = G f~ G I,

~Oh(X) = ~o(x + hx0), Oh(X) = Q(x + hx~ Then

,,,:,

,,,:,.,

,,

The function u(x) is bounded from above in the domain K2/2,R. Therefore (see the estimates (13) and (14) of Lemma 4*) I u (x) I do) < oo;

(20)

G

(21)

A! ~ I u (x) I &0 = 0 (8).

From here it follows that for h --- 0 we have

SluE & o - + o, i = 3, 4, 5,

and, therefore, lim

h~O

~uQdo~=O,i=4,5; fuQ_h&o-~O. h

Taking into account the continuity of the function Q, we also have lira ~ u [Q-h - - Q] do) = 0. h~0

oh

Thus, statement a) is proved. For the proof of statement b), instead of the relations (20) and (21) we make use of the inequality (22)

~ tp d u / i n f qo(x Further we note that ,,= ~ ,~.\, \ A6[h x. R, x(G h

q- hx ~ >t ch, e > O. Consequently,

x~6

(23)

S

e,