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Reports of the Academy of Sciences of the USSR
- Volume 146, No. 4
MATHEMATICS
V. S. AZARIN
ON SUBHARMONIC FUNCTIONS OF COMPLETELY REGULAR GROWTH IN MULTIDIMENSIONAL SPACE
(Presented by Academician S. N. Bernstein on 20 IV 1962)
In this paper, the known theorems on the connection between the growth indicator and the distribution of zeros of entire functions of finite order \(\rho\) and completely regular growth \(\left({}^{3}\right.\), Chs. II, III) are carried over to functions subharmonic in the whole space of \(m\) dimensions. Along the way, the analytic properties of the indicator of a function subharmonic in the whole space or in a certain cone are investigated. This part of the paper adjoins the author’s preceding note \(\left({}^{5}\right)\).
Introduce in the space \(E_m\) of \(m\) dimensions the spherical system of coordinates by the formulas:
\[ \begin{aligned} x_1 &= r\sin\theta_0\sin\theta_1\ldots\sin\theta_{m-2},\\ x_2 &= r\cos\theta_0\sin\theta_1\ldots\sin\theta_{m-2},\\ &\cdots\\ x_m &= r\cos\theta_{m-2} \end{aligned} \tag{1} \]
\[ (0\leq r<\infty;\quad 0\leq \theta_0<2\pi;\quad 0\leq \theta_i\leq \pi;\quad i=1,2,\ldots,m-2). \]
We shall use the following notation: \(S_R\) is the sphere of radius \(R\); \(K_R\) is the ball of radius \(R\); \(dV\) is an element of the surface of the unit sphere \(S_1\); \(dl\) is an element of the surface of an \((m-2)\)-dimensional contour on \(S_1\); \(\mathbf{x}\) is a vector from \(S_1\); any vector \(\mathbf{r}\in E_m\) with origin at zero will be written in the form \(\mathbf{r}=r\mathbf{x}\), where \(r=|\mathbf{r}|\), \(\mathbf{x}=\mathbf{r}\cdot r^{-1}\;(\in S_1)\), and the value of a function \(u\) at the end of this vector as \(u(r\mathbf{x})\).
Denote by \(L\) the following differential operator of second order:
\[ L=\sum_{i=0}^{m-2}\frac{1}{\Pi}\frac{\partial}{\partial\theta_i}\cdot\frac{\Pi}{\Pi_i}\frac{\partial}{\partial\theta_i}+\rho(\rho+m-2), \]
where
\[ \Pi=\prod_{j=0}^{m-2}\sin^{j}\theta_j,\qquad \Pi_i=\prod_{j=i+1}^{m-2}\sin^2\theta_j,\qquad \rho\geq 0. \]
Let an \((m-2)\)-dimensional contour \(l\) be given on the sphere \(S_1\) by the equations
\[ \theta_i=\theta_i(\lambda_1,\ldots,\lambda_{m-2}),\qquad i=0,1,\ldots,m-2. \]
Denote by \(\partial/\partial n\) the derivative along the normal to this contour:
\[ \frac{\partial}{\partial n} = \sum_{k=0}^{m-2} \frac{\sqrt{\Pi}}{\Pi_k}\cdot D_k \left[ \sum_{j=0}^{m-2}\Pi_j^{-2}D_j^2 \right]^{-1/2} \frac{\partial}{\partial\theta_k}, \]
where \(D_k\) are the Jacobians:
\[ D_k=(-1)^k D(\theta_0,\ldots,\theta_{k-1},\theta_{k+1},\ldots,\theta_{m-2}) / D(\lambda_1,\ldots,\lambda_{m-2}), \]
\[ k=0,1,\ldots,m-2. \]
In this notation, for any sufficiently smooth functions, i.e. having, for example, continuous second derivatives, \(u\) and \(v\), defined on the sphere \(S_1\), and for an arbitrary domain \(V\) on this sphere bounded by a smooth contour \(l\), the generalized Green formula holds \((({}^{1}),\) p. 235):
\[ \int_V (uLv-vLu)\,dV=\int_l\left(u\frac{\partial}{\partial n}v-v\frac{\partial}{\partial n}u\right)\,dl. \tag{2} \]
We shall denote by \(W\) an arbitrary domain in \(S_1\), bounded by a smooth contour \(l\), in which the Dirichlet problem corresponding to the operator \(L\) is uniquely solvable. Denote by \(Y_W^\varphi(x)\) the solution of the equation
\[ LY=0 \tag{3} \]
with boundary conditions
\[ Y_W^\varphi(x)\big|_{x\in l}=\varphi(x). \]
Definition. A function \(h(x)\) is called subspherical if it satisfies the following conditions: 1) it is upper semicontinuous in \(S_1\); 2) it is finite on an everywhere dense set of points \(x\in S_1\); 3) if \(\varphi(x)\) is an arbitrary continuous function defined on \(l\)—the boundary of the domain \(W\)—and satisfying the inequality
\[ \varphi(x)\geq h(x)\big|_{x\in l}, \]
then the inequality
\[ Y_W^\varphi(x)\geq h(x) \]
holds throughout the domain \(W\).
As was shown in \(({}^{5})\), the concept of subsphericity is a generalization of the concept of trigonometric convexity and is analogous to the concepts of subharmonicity and convexity.
Equation (3) has a fundamental solution \(\Phi(\sin\theta)\), where \(\theta\) is the angle between the vector \(\vec{\xi}\), determining the point on \(S_1\) at which \(\Phi\) has a singularity, and the vector \(x\in S_1\). One can choose \(\Phi\) so that the condition
\[ \lim_{\varepsilon\to 0}\int_{\theta_{m-2}=\varepsilon}\frac{\partial}{\partial n}\Phi\,dl=1 \]
is fulfilled.
For a subspherical function there is a representation analogous to the F. Riesz representation for a subharmonic function.
Theorem 1. If \(h(x)\) is a subspherical function, then for any closed domain \(V\subset S_1\) there is a representation
\[ h(x)=Y(x)+\int_{\vec{\xi}\in V}\Phi\,d\Delta_{\vec{\xi}}, \]
where \(Y(x)\) satisfies equation (3) in the domain \(V\); \(\Delta_{\vec{\xi}}\) is an additive, nonnegative, upper semicontinuous set function (Radon measure), and the integral is understood in the Stieltjes sense.
We shall denote the measure of a set \(V\subset S_1\) by \(\Delta(V)\). For a twice continuously differentiable subspherical function \(h(x)\), the inequality \(Lh\geq0\) holds. This follows from the relation
\[ h(x)-Y_W^h(x)=-\int_W Lh\cdot G(x,\vec{\xi})\,dV, \]
obtained from formula (2) by substituting
\[ u=h-Y_W^h,\qquad v=G(x,\vec{\xi}), \]
where \(G(x,\vec{\xi})\) is the Green function of the operator \(L\) for the domain \(W\).
Applying the operator \(L\) to the function
\[ Y(x)=h(x)-\int_{\vec{\xi}\in V}\Phi Lh\,dV, \]
we see that \(LY=0\). Denoting \(d\Delta_{\vec{\xi}}=Lh\,dV\), we obtain the assertion of the theorem for a twice continuously differentiable function \(h(x)\).
The passage to the general case is based on the following lemma, which we state without proof.
Lemma. For every subspherical function \(h(x)\) there exists a sequence of twice continuously differentiable functions \(\{h_n\}_0^\infty\) such that \(h_n \geq h\) and \(\{h_n\}_0^\infty\) converges to \(h\) at every point, and moreover, for every domain \(V\subset S_1\) the condition
\[ \int_V Lh_n\,dV<\infty \]
is satisfied.
The completion of the proof of the theorem is carried out by the standard method (see, for example, \((^4)\), pp. 156–159).
Let now \(h(x)\) be the indicator of a function \(u(rx)\), subharmonic in the whole space, of finite order \(\rho\) and normal type (see \((^5)\)), i.e.
\[ h(x)=\overline{\lim_{r\to\infty}}\,u(rx)\,r^{-\rho}. \]
In \((^5)\) it was shown that \(h(x)\) has properties 2) and 3) from the definition of a subspherical function. Its upper semicontinuous regularization \(\widetilde h(x)\) is therefore a subspherical function \(\bigl(\widetilde h(x)=\overline{\lim}_{x'\to x}h(x')\bigr)\).
Theorem 2. The indicator \(h(x)\) can differ from its regularization only on such a set \(G\subset S_1\) that its intersection with any piecewise smooth contour \(l\) has zero measure on this contour.
The proof is omitted. We note only that it is based on A. Cartan’s theorem \((^2)\) on the upper envelope of a family of subharmonic functions.
Denote by \(\gamma\) an arbitrary subspace of \(m-2\) dimensions of the space \(E_m\), and by \(\psi\) the subspace orthogonal to it—a plane. Choose the coordinate system (1) so that the coordinate axes \(x_1\) and \(x_2\) lie in the plane \(\psi\). Then \(\gamma\) is defined by the equation \(\theta_1=0\). The equation \(\theta_0=0\) defines the “half-hyperplane” spanned by \(\gamma\) and the positive ray of the coordinate axis \(x_2\).
Definition. The \(\gamma\)-projection of a point \(Q\) with coordinates \((r,\theta_0,\theta_1,\ldots,\theta_{m-2})\) onto the “half-hyperplane” \(\theta_0=0\) is the point \(Q_\gamma^0\) with coordinates \((r,0,\theta_1,\ldots,\theta_{m-2})\). The set of \(\gamma\)-projections of points of a set \(C\subset E_m\) is called the \(\gamma\)-projection of the set \(C\) and is denoted \(C_\gamma^0\). The measure of the set \(C_\gamma^0\) on the hyperplane \(x_1=0\) is called the \(\gamma\)-measure of the set \(C\). It does not change under rotation of the coordinate axes \(x_1\) and \(x_2\) in the plane \(\psi\) and is denoted \(m_\gamma(C)\).
For a set \(C\subset E_m\) define the quantities \(\rho_R(C)\) and \(\omega_R(C)\) by the equalities
\[ \rho_R(C)=\sup_\gamma m_\gamma(C\cap K_R);\qquad \omega_R(C)=\operatorname{mes}(C\cap S_R), \]
where \(\operatorname{mes}\) denotes measure on the sphere \(S_R\).
Definition. A set \(C\) is called a \(C_0\)-set if it satisfies the condition
\[ \lim_{R\to\infty}\{\rho_R(C)+\omega_R(C)\}R^{-m+1}=0. \]
Definition. A subharmonic function in the whole space \(u(rx)\) of order \(\rho\) and of normal type is called a function of completely regular growth if the function
\[ h_r(x)=u(rx)r^{-\rho} \]
converges uniformly to \(h(x)\), the indicator of the function \(u(rx)\), under the condition that \(rx\to\infty\) in such a way that the endpoint of the vector \(rx\) does not belong to some \(C_0\)-set.
Let now \(\mu(K_R^V)\) denote the mass corresponding to the subharmonic function \(u(rx)\), contained in the sector \(K_R^V\) \((0\le r<R,\ x\subset V\subset S_1)\); \(h(x)\) is the indicator of \(u(rx)\); \(\Delta_x\) is the measure constructed by Theorem 1 for the regularized indicator \(\tilde h(x)\). We shall call the set \(V\) regular with respect to \(\Delta_x\) if the condition
\[ \Delta(V)=\sup_{\widetilde V\subset V}\Delta(\widetilde V) \]
is satisfied.
Theorem 3. In order that a subharmonic function in the whole space \(u(rx)\) of nonintegral order \(\rho\) and of normal type be a function of completely regular growth, it is necessary and sufficient that the limit
\[ \lim_{R\to\infty}\mu(K_R^V)R^{-\rho} = \frac{1}{\rho+m-2}\Delta(V) \]
exist for every set \(V\) regular with respect to \(\Delta_x\).
An analogous assertion holds for subharmonic functions of integral order.
These theorems solve one of the problems posed by B. Ya. Levin in his report at the Third All-Union Mathematical Congress in 1956.
Received
18 IV 1962
REFERENCES
¹ W. Feller, Math. Ann., 102, 633 (1930); UMN, vol. 8 (1941). ² H. Cartan, Bull. Soc. Math. France, 73, 74 (1945). ³ B. Ya. Levin, Distribution of the roots of entire functions, Moscow, 1956. ⁴ I. I. Privalov, Subharmonic functions, Moscow, 1937. ⁵ V. S. Azarin, DAN, 139, No. 5, 1033 (1961).