MATHEMATICS

V. S. AZARIN

ON THE INDICATOR OF A FUNCTION SUBHARMONIC IN A MULTIDIMENSIONAL SPACE

(Presented by Academician S. N. Bernstein on 25 III 1961)

A function subharmonic in the whole space is called a function of order \(\rho\), if*

\[ \varlimsup_{R \to \infty} \ln^{+} M_u(R)\cdot(\ln R)^{-1}=\rho<\infty, \tag{1} \]

and of normal type \(\sigma\) at order \(\rho\), if

\[ \varlimsup_{R \to \infty} M_u(R)\cdot R^{-\rho}=\sigma<\infty, \tag{2} \]

where \(M_u(R)\) denotes the maximum of \(u^{+}(Q)\) in the ball \(K_R\) of radius \(R\) with center at the origin \(O\).

To characterize the growth of a function \(u(Q)\) of finite order and normal type in different directions, the concept of an indicator of growth is introduced, analogously to how this is done for entire functions.

Let \(\mathbf{x}\) be a unit vector, and let \(u(R\mathbf{x})\) be the value of \(u(Q)\) at the end of the vector \(R\mathbf{x}\), whose origin is at the point \(O\).

Definition. The indicator of a function \(u(Q)\), subharmonic in the whole space, of order \(\rho\), is the function

\[ h(\mathbf{x})=\varlimsup_{R \to \infty} u(R\mathbf{x})\cdot R^{-\rho}. \tag{3} \]

In the present article the properties of such an indicator are investigated and, in particular, a property analogous to the trigonometric convexity of the indicator of an entire function is proved.

Along with the notation already introduced, we shall use the following: \(E_m\) is \(m\)-dimensional space; \(\mathbf{p}, \mathbf{q}, \mathbf{r}\) are vectors of this space, and \(p, q, r\) are their absolute values; \(S_R\) is the sphere of radius \(R\) with center at the point \(O\); \(\sigma_1\) is the area of the sphere \(S_1\).

By F. Riesz’s theorem, to every subharmonic function there corresponds a certain distribution of masses in space. We shall assume (this does not restrict generality) that some neighborhood of the origin is free of masses. Unless the contrary is specified, by subharmonic functions we shall mean functions subharmonic in the whole space, of finite order, satisfying this condition.

From the definition of the indicator it is clear that the indicator of a subharmonic function of normal type is bounded above. However, in contrast to the indicator of an entire function, the indicator of a subharmonic function is, generally speaking, discontinuous and may even take the value \(-\infty\) on a set of vectors everywhere dense in \(S_1\). One can verify this by taking, as an example, a subharmonic function with masses distributed along the rays \(\{\lambda \mathbf{x}_j\}\) \((0<\lambda<\infty)\), where \(\{\mathbf{x}_j\}\) is a countable set of vectors everywhere dense in \(S_1\). The set of points,

* \(f^{+}(Q)=\max [f(Q),0]\).

where the indicator of a subharmonic function of normal type is infinite, nevertheless not too large, as the following theorem shows.

Theorem 1. The indicator of a subharmonic function of normal type is summable over the unit sphere.

We omit the proof. Let us note that the order of growth, the type, and the indicator can also be defined for a function subharmonic in the cone \(K_\Omega\) determined by a certain domain \(\Omega\) on the unit sphere. To this end, in equalities (1), (2), (3) one should regard \(u(Q)\) as a subharmonic function in \(K_\Omega\), take the maximum \(u^+(Q)\) in (1) and (2) over \(Q \in K_\Omega \cap S_R\), and in (3) regard \(x \in \Omega\).

The indicator \(h(\theta)\) of an entire function of finite order (as well as of a function holomorphic inside an angle and, in general, of a subharmonic function of order \(\rho\) inside an angle in the plane) has the property of trigonometric convexity, consisting in the following: if \(H(\theta)\) is a linear combination of the trigonometric functions \(\cos \rho\theta\) and \(\sin \rho\theta\), satisfying at the ends of a sufficiently small interval \((\theta_1,\theta_2)\) the conditions \(H(\theta_1) \geq h(\theta_1)\), \(H(\theta_2) \geq h(\theta_2)\), then \(h(\theta) \leq H(\theta)\) throughout the interval \((\theta_1,\theta_2)\) (see (1), p. 73). We note that \(H(\theta)\) is a solution of the equation \(u''+\rho^2 u=0\).

The main result of this note is the theorem that the indicator of a subharmonic function of order \(\rho\) in a certain cone with vertex at the origin has an analogous property with respect to solutions of the equation

\[ L[u]+\rho(\rho+m-2)u=0, \tag{4} \]

where \(m\) is the dimension of the space, and \(L[u]\) is the spherical operator:

\[ L[u]=(\sin^2\theta_2\cdots \sin^2\theta_{m-1})^{-1}\frac{\partial^2}{\partial\theta_1^2}u+ \]

\[ +(\sin^2\theta_3\cdots \sin^2\theta_{m-1})^{-1}\sin^{-1}\theta_2\cdot \frac{\partial}{\partial\theta_2}\sin\theta_2\frac{\partial}{\partial\theta_2}u+ \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ +\sin^{2-m}\theta_{m-1}\frac{\partial}{\partial\theta_{m-1}} \sin^{m-2}\theta_{m-1}\frac{\partial}{\partial\theta_{m-1}}u \tag{5} \]

(\(\theta_i\) are spherical coordinates; \(0\leq \theta_1\leq 2\pi\), \(0\leq \theta_k\leq \pi\), \(k=2,\ldots,m-1\)).

Theorem 2. Let \(h(x)\) be the indicator of a function \(u(Q)\) of finite order \(\rho\) and normal type, subharmonic in the cone \(K_{\Omega_1}\); let \(\Omega\) be a domain on the unit sphere \((\Omega\subset \Omega_1)\), bounded by a curve \(\omega\) having bounded curvature, and suppose that the first boundary-value problem for equation (4) has a unique solution in \(\Omega\). Let \(\varphi(x)\) be a continuous function defined on \(\omega\). If \(H^\varphi_{\Omega}(x)\) is the solution of equation (4) taking the value \(\varphi(x)\) on \(\omega\), and if for \(x\in\omega\) \(\varphi(x)\geq h(x)\), then

\[ H^\varphi_{\Omega}(x)\geq h(x) \]

for every \(x\in\Omega\).

We shall precede the proof of the theorem with a number of lemmas.

Lemma 1. Let \(G(p,t)\) be the Green function of the Laplace equation for the cone \(K_\Omega\) determined by the domain \(\Omega\), and let \(Y_1(x)\) be the eigenfunction corresponding to the first eigenvalue \(\lambda_1\) of the boundary-value problem

\[ L[Y]+\lambda Y=0,\qquad Y(x)\big|_{x\in\omega}=0. \]

Let \(\rho_1\) be the positive root of the equation

\[ z(z+m-2)-\lambda_1=0. \]

Then, for some \(M>0\), depending only on \(K_\Omega\), for \(p,t\in K_\Omega\) and \(2p<t\), the inequalities hold:
\[ G(p,t)\leq M Y_1(t^{-1}t)\,p^{\rho_1}\cdot t^{-\rho_1-m+2}, \]
\[ \frac{\partial}{\partial n_t}G(p,t)\leq M\frac{\partial}{\partial n_1}Y_1(t^{-1}t)\,p^{\rho_1}\cdot t^{-\rho_1-m+1}, \]
where \(\partial/\partial n_t\) is the normal derivative with respect to \(t\) at the point \(t\), belonging to the conical surface \(K_\omega\); \(\partial/\partial n_1\) is the derivative along the normal to \(K_\omega\) at the point \(x=t^{-1}\cdot t\;(\in S_1)\).

With the aid of Lemma 1, Lemmas 2 and 3 are proved.

Lemma 2. Let \(\varphi(t)\) be a continuous function on \(K_\omega\) satisfying the inequality
\[ |\varphi(t)|<At^\rho, \]
where \(A\) is some constant. Then the integral
\[ I(r)=\int_{K_\omega}\frac{\partial}{\partial n}G(r,t)\,\varphi(t)\,d\sigma_t \]
converges and represents a harmonic function with limiting values \(\varphi(t)\) on \(K_\omega\). Moreover,
\[ I(r)\leq Br^\rho, \]
where \(B\) is a constant.

Lemma 3. For a subharmonic function in \(K_\Omega\) \((\Omega\subset\Omega_1)\) of order \(\rho<\rho_1\), the integral
\[ H(r)=\int_{K_\omega}\frac{\partial}{\partial n}G(r,t)\,u(t)\,d\sigma_t \]
exists and the inequality holds
\[ u(r)\leq \int_{K_\omega}\frac{\partial}{\partial n}G(r,t)\,u(t)\,d\sigma_t. \tag{6} \]

The proof of this lemma is based on a generalization of the Phragmén–Lindelöf principle given in the work \((^2)\).

We now proceed to the proof of Theorem 2. Write inequality (6) for the subharmonic function \(u(Rr)\):
\[ u(Rr)\leq \int_{K_\omega}\frac{\partial}{\partial n}G(r,t)\,u(Rt)\,d\sigma_t. \]
Dividing by \(R^\rho\) and taking \(\overline{\lim}\), we obtain
\[ r^\rho \overline{\lim}_{R\to\infty}\frac{u(Rr)}{(Rr)^\rho} \leq \overline{\lim}_{R\to\infty} \int_{K_\omega}\frac{\partial}{\partial n}G(r,t)\,t^\rho \frac{u(Rt)}{(Rt)^\rho}\,d\sigma_t, \]
and, applying Fatou’s lemma as \(R\to\infty\) to the function
\[ \bigl[C-u(Rt)(Rt)^{-\rho}\bigr]\frac{\partial}{\partial n}G(r,t)\,t^\rho, \]
where \(C>\sigma\) is the type of \(u(Rt)\), we obtain
\[ r^\rho h(r^{-1}r)\leq \int_{K_\omega}\frac{\partial}{\partial n}G(r,t)\,t^\rho h(t^{-1}t)\,d\sigma_t. \]

Let \(\varphi(x)\) be a continuous function satisfying the condition of the theorem. Obviously,

\[ \int_{K_\omega} \frac{\partial}{\partial n} G(r,t)\, t^\rho \cdot h(t^{-1}t)\, d\sigma_t \le \int_{K_\omega} \frac{\partial}{\partial n} G(r,t)\, t^\rho \varphi(t^{-1}t)\, d\sigma_t = H(r). \]

Since \(H(r)\) and \(H_\Omega^\varphi(r^{-1}r)\, r^\rho\) have the same boundary values and are both of order \(\rho<\rho_1\), they coincide (see \((^{2})\)) and, consequently,

\[ r^\rho h(x) \le r^\rho H_\Omega^\varphi(x) \]

for \(x\in\Omega\), whence the assertion of the theorem follows.

The property of the indicator expressed by Theorem 2 (which may naturally be called its “subsphericity”) makes it possible to characterize more precisely the analytic properties of the indicator, namely:

Theorem 3. The indicator of a subharmonic function \(u(Q)\) of normal type in \(E_m\) is summable over any closed smooth contour \(\omega\) with bounded curvature that lies in a domain \(\Omega\) satisfying the requirements of Theorem 2.

In conclusion I express my deep gratitude to B. Ya. Levin, who posed the questions considered in this paper.

CITED LITERATURE

1. B. Ya. Levin, Distribution of zeros of entire functions, Moscow, 1956.
2. J. Deny, P. Lelong, 224, 1024 (1947).