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UDC 517.947.42
MATHEMATICS
K. V. TEMKO, S. V. TEMKO
ON THE EQUILIBRIUM POTENTIAL
(Presented by Academician N. N. Bogolyubov on 4 V 1965)
§ 1. Let \(R_n\) be \(n\)-dimensional Euclidean space, \(n \geqslant 1\). Denote by \(\Pi\) an \(n\)-dimensional closed parallelepiped. Let \(\mathfrak{B}\) be a system of Borel sets from \(\Pi\), and let \(\mu\) be a measure defined on \(\mathfrak{B}\), i.e., a completely additive nonnegative set function defined on \(\mathfrak{B}\). Normalization means \(\mu(\Pi)=1\). If \(B \in \mathfrak{B}\) and \(\mu(B)=1\), then one says that \(\mu\) is concentrated on \(B\). This circumstance is denoted by \(\mu \prec B\).
We shall say that a function \(\varphi(t)\) is a kernel of the class \(\Phi\) if \(\varphi(t)>0\) is a continuous decreasing function defined for \(t>0\) and such that
\[ \lim_{t\to 0}\varphi(t)=+\infty \quad\text{and}\quad \int_0^1 t^{\,n-1}\varphi(t)\,dt<+\infty . \]
Denote by \(r_{PQ}\) the ordinary Euclidean distance between the points \(P=(x_1,\ldots,x_n)\) and \(Q=(y_1,\ldots,y_n)\) of \(R_n\). The Lebesgue–Stieltjes integral
\[ u(P)=\int_{\Pi}\varphi(r_{PQ})\,d\mu(Q) \tag{1} \]
defines at each point of \(R_n\) a function \(u(P)\), called the \(\Phi\)-potential generated by the measure \(\mu\). We note that all integrals encountered below will be understood in the Lebesgue–Stieltjes sense.
§ 2. Let \(F\) be a closed set from \(\Pi\), and let \(\mu\) be a measure concentrated on the set \(F\). Put
\[ I(\mu)=\iint_F \varphi(r_{PQ})\,d\mu(P)\,d\mu(Q), \tag{2} \]
\[ W(F)=\inf_{\mu\prec F} I(\mu). \tag{3} \]
Following Frostman \((^1)\), the \(\varphi\)-capacity of a bounded closed set \(F\) is the number \(C_\varphi(F)\) determined by the equation \(W(F)=\varphi[C_\varphi(F)]\). For an arbitrary set \(E\) from \(\Pi\), the \(\varphi\)-capacity is defined by putting \(C_\varphi(E)=\sup_{F\subset E} C_\varphi(F)\), where \(F\) is a closed subset of \(E\).
§ 3. Let \(F\) be a closed set from \(\Pi\), and suppose that for some class of kernels there exists a measure \(\mu^*\prec F\) such that \(I(\mu^*)=W(F)\); then the measure \(\mu^*\) is called an equilibrium measure, and the \(\varphi\)-potential \(u^*(P)\) generated by it is called the equilibrium potential.
In the present paper we consider the problem of existence and uniqueness of the equilibrium measure for a broader class of kernels \(\varphi(t)\) from \(\Phi\) than in \((^1,^5)\).
§ 4. On the existence of the equilibrium potential. Denote by \(F_\mu\) the support of the measure \(\mu\) (see \((^3)\)); it is known that the set \(F_\mu\) is closed. Using definition (3) and the selection theorem \((^1,^3)\), it is not difficult to show that if \(\varphi(t)\in\Phi\) and \(F\) is a closed set from \(\Pi\) of positive-
…of positive \(\varphi\)-capacity, then there exists an equilibrium measure \(\mu^* \prec F\). We shall say that the kernel \(\varphi(t)\in\Phi_1\), if \(\varphi(t)\in\Phi\), and if for the given \(n\ge 1\) there exists a number \(h>0\) such that \(0<h<n\) and \(t^h\varphi(t)\) does not decrease as \(t\) increases.
Lemma 1. Let \(\varphi(t)\in\Phi_1\), and let the \(\varphi\)-potential \(u(P)\), generated by some measure \(\mu\), be bounded on the support of this measure \(F_\mu\); then \(u(P)\) is bounded everywhere in \(R_n\).
Proof. Let \(P\notin F_\mu\); denote by \(P_0\) the point of \(F_\mu\) nearest to \(P\), or one of them if there are several. Such a point \(P_0\), at least one, exists, since \(F_\mu\) is closed. Since \(r_{P_0Q}\le 2r_{PQ}\) and \(\varphi(t)\in\Phi_1\), we find \(u(P)\le 2^h u(P_0)\), which proves the lemma.
Lemma 2. Let \(\varphi(t)\in\Phi_1\), and let the \(\varphi\)-potential \(u(P)\), generated by a measure \(\mu\), be continuous on the support of this measure; then it is continuous everywhere in \(R_n\).
Proof. Denote by \(S_\delta(P)\) the \(n\)-dimensional ball of radius \(\delta\) with center at the point \(P\). Put
\[ u_\delta(P)=\int_{S_\delta(P)\cap F_\mu}\varphi(r_{PQ})\,d\mu(Q). \tag{4} \]
From the continuity of \(u(P)\) on the closed set \(F_\mu\) it follows that for any \(\varepsilon>0\) there exists such a \(\delta(\varepsilon)>0\) that \(u_\delta(P)<\varepsilon\) for every point \(P\in F_\mu\). If, however, \(P\notin F_\mu\) and \(P\) is at distance less than \(\delta\) from \(F_\mu\), then, carrying out a proof analogous to the proof of Lemma 1, we find \(u_\delta(P)\le 2^h u_{2\delta}(P_0)\), which proves the lemma.
Lemma 3. Let \(\varphi(t)\in\Phi_1\), and let \(u(P)\) be a \(\varphi\)-potential generated by some measure \(\mu\); then
\[ m_r(P)\le Au(P), \tag{5} \]
where \(m_r(P)\) is the \(n\)-dimensional mean value of the \(\varphi\)-potential \(u(P)\); \(A\) is a constant independent of \(P\).
Proof. Substituting in \(m_r(P)\) the expression for \(u(P)\) from (1) and changing the order of integration, we find
\[ m_r(P)=\int_{F_\mu}d\mu(M)\cdot \frac{1}{V_r}\int_{\bar S_r(P)}\varphi(r_{PQ})\,dv_Q =\int_{F_\mu} f(r,P,M)\,d\mu(M), \tag{6} \]
where \(\bar S_r(P)\) is the closed \(n\)-dimensional ball; \(V_r\) is the volume of this ball; \(dv_Q\) is the volume element at the point \(Q\). Suppose that the point \(M\in \bar S_r(P)\); then, putting \(t=r_{MQ}\) and taking into account that \(r_{MQ}\le 2r\) and \(\varphi(t)\in\Phi_1\), we find
\[ f(r,P,M)\le \frac{n}{r^n}\int_0^{2r} t^{\,n-1}\varphi(t)\,dt \le \frac{2^n n}{n-h}\varphi(r_{PM}). \tag{7} \]
If \(r<r_{MP}\le 2r\), then, denoting by \(P_0\) the point of \(\bar S_r(P)\) nearest to \(M\) and taking into account that \((\varphi(r_{MQ})/\varphi(r_{P_0Q}))\le 2^h\) and \(V_{2r}/V_r=2^n\), we obtain \(\varphi(r_{MQ})\le 2^h\varphi(r_{P_0Q})\) and
\[ f(r,P,M)\le 2^{h+n}\frac{n}{n-h}\varphi(r_{PM}). \tag{8} \]
If \(r_{PM}>2r\), then \(r_{PM}\le r_{PQ}+r_{QM}\le 2r_{QM}\), and therefore \(f(r,P,M)\le 2^h\varphi(r_{PM})\). Hence, from (7) and (8), we obtain (5), where \(A=2^{h+n}n/(n-h)\). The lemma is proved.
Theorem 1. Let \(\varphi(t)\in\Phi_1\), and let \(F\) be a closed set from \(\Pi\) of positive \(\varphi\)-capacity; then: 1) \(W(F)\le u^*(P)\) nearly everywhere in the sense of \(\varphi\)-capacity on \(F\), and 2) \(u^*(P)=W(F)\) nearly everywhere in the sense of \(\varphi\)-capacity on \(F_{\mu^*}\).
Proof. Since \(\mu^*\) is an equilibrium measure, for arbitrary \(\varepsilon>0\) the inequality \(u^*(P)\le W(F)-\varepsilon\) cannot hold everywhere on \(F\). Consequently, there exists a point \(P_0\in F\), for which
\(u^*(P) > W(F)-\varepsilon\), and since the \(\varphi\)-potential is lower semicontinuous everywhere in \(R_n\), there exists a neighborhood \(O(P_0)\) such that \(u^*(P) > W(F)-\varepsilon\) for all \(P \in O(P_0)\). Let \(D\) be a subset of \(F\) where \(u^*(P) \leq W(F)-2\varepsilon\); the set \(D\) is closed. Suppose that \(C_\varphi(D)>0\). Define the set function \(\nu\), putting \(\nu=-\mu^*\) in \(O(P_0)\); \(\nu>0\) on \(D\) and \(\nu(D)=\mu^*[O(P_0)]=m\); \(\nu=0\) on \(D\cup O(P_0)\), and \(I(D,\nu)<+\infty\). Put \(\eta=\mu^*+h_1\nu\), where \(0<h_1<1\). We have \(\delta I=I(\mu^*+h_1\nu)-I(\mu^*)\geq 0\), since \(\eta \prec F\). On the other hand, we have \(\delta I<-h_1[2m\varepsilon-h_1 I(F,\nu)]\). Consequently, for sufficiently small \(h_1>0\) we obtain \(\delta I<0\). The latter is impossible. Thus, \(C_\varphi(D)=0\). The proof of the second condition is obvious.
Theorem 2. If, under the assumptions of Theorem 1, we have: 1) for \(n\geq 2\) the set \(F\) is the sum of a finite number of closed domains whose boundaries satisfy the Poincaré condition \((^1)\); 2) for \(n=1\) the set \(F\) is the sum of a finite number of closed and bounded domains, then the equilibrium measure \(\mu^*\) for \(\varphi(t)\in \Phi_1\) is such that: a) \(u^*(P)\geq W(F)\) everywhere on \(F\); b) \(u^*(P)=W(F)\) everywhere on \(F_{\mu^*}\); c) \(u^*(P)\) is continuous everywhere in \(R_n\); d) \(u^*(P)\leq 2^h W(F)\) everywhere in \(R_n\).
The proof of conditions a), b) follows from the fact that for \(\varphi\)-potentials generated by kernels \(\varphi(t)\in \Phi_1\), Lemma 3 holds, and therefore one can apply the method of proof proposed by Frostman in \((^1)\). Conditions c), d) follow respectively from Lemma 2 and Lemma 1.
We shall say that a kernel \(\varphi(t)\) satisfies the weak Frostman maximum principle if, from the fact that the continuous \(\varphi\)-potential \(u(P)\), generated by a measure \(\mu\), does not exceed some constant \(K\) on the support of this measure, it follows that it does not exceed this constant everywhere in \(R_n\).
Theorem 3. If, under the assumptions of Theorem 2, the kernel \(\varphi(t)\in\Phi_1\) satisfies the weak Frostman maximum principle, then there exists an equilibrium measure \(\mu^*\) for the given set \(F\) such that the \(\varphi\)-potential \(u^*(P)\) generated by it has the following properties: 1) \(u^*(P)=W(F)\) everywhere on the set \(F\); 2) \(u^*(P)\leq W(F)\) everywhere in \(R_n\).
The proof of this theorem follows immediately from the validity of the weak Frostman maximum principle and Theorem 2.
Theorem 4. If, under the assumptions of Theorem 1, the kernel \(\varphi(t)\in\Phi_1\) satisfies the weak Frostman maximum principle, then there exists an equilibrium measure \(\mu^*\) such that: 1) \(u^*(P)=W(F)\) almost everywhere, in the sense of \(\varphi\)-capacity, on \(F\); 2) \(u^*(P)\leq W(F)\) everywhere in \(R_n\); 3) if \(u^*(P_0)=W(F)\), then \(u^*(P)\) is continuous at the point \(P_0\).
The proof of conditions 1 and 2 of Theorem 4 is not difficult to obtain by using Frostman’s method from \((^1)\). Condition 3 follows from condition 2 and from the lower semicontinuity of the potential \(u^*(P)\).
We note that the set of points of \(F\) at which \(u^*(P)<W(F)\) is a set of type \(F_\sigma\).
Theorem 5. For the existence of an equilibrium \(\varphi\)-potential \(u^*(P)\) on a closed set \(F\) from \(\Pi\), it is necessary and sufficient that the kernel \(\varphi(t)\in\Phi_1\) satisfy the weak Frostman maximum principle.
The sufficiency of the condition follows from Theorem 4. The proof of necessity is not difficult to obtain by using the method of work \((^1)\). From Theorem 5 follows the result of work \((^5)\).
§ 5. On the uniqueness of the equilibrium measure. Without loss of generality one may assume that \(\Pi\) is an \(n\)-dimensional parallelepiped whose edges are such that \(-\pi \leq x_i \leq \pi,\ i=1,2,\ldots,n\).
Theorem 6. If the kernel \(\varphi(t)\) has positive Fourier coefficients, then, under the assumptions of Theorem 4, there exists a unique equilibrium measure \(\mu^*\) for the given set \(F\).
Proof. Let \(\mu_1^*\) and \(\mu_2^*\) be two equilibrium measures. Put \(\chi=\mu_1^*-\mu_2^*\), then \(I(\chi)=0\). Let \(u_1^*(P)\) and \(u_2^*(P)\) be \(\varphi\)-
equilibrium potentials generated respectively by the measures \(\mu_1^*\) and \(\mu_2^*\). Denote by \(c_{k_1,\ldots,k_n}^{(i)}\) the Fourier coefficients of \(u_i^*(P)\), \(i=1,2\). We have
\(c_{k_1,\ldots,k_n}^{(i)}=d_{k_1,\ldots,k_n}\gamma_{k_1,\ldots,k_n}^{(i)}\), where \(d_{k_1,\ldots,k_n}\) are the Fourier coefficients of the kernel \(\varphi(t)\), and \(\gamma_{k_1,\ldots,k_n}^{(i)}\) are the Fourier–Stieltjes coefficients of \(d\mu_i^*\). For \(n\ge 2\), putting
\[ \tau(t)= \begin{cases} (1-t^2)^\delta & \text{for } 0\le t\le 1,\\ 0 & \text{for } t\ge 1, \end{cases} \]
we obtain: if \(\delta>(n-1)/2\), then the spherical means \(S_{R,i}^{\tau}(P)\) of the Fourier series of \(u_i^*(P)\) (see (6)) are such that
\[ \lim_{R\to\infty} S_{R,i}^{\tau}(P)=2^l\Gamma(l+1)u_i^*(P) \tag{9} \]
at every point of discontinuity of the function \(u_i^*(P)\), i.e. almost everywhere in the sense of \(\varphi\)-capacity on \(F\) (since the sum of two Borel sets of \(\varphi\)-capacity zero is a set of \(\varphi\)-capacity zero). From the boundedness of \(u_i^*(P)\) and from (9)
\[ I(\varkappa)=\lim_{R\to\infty}\sum_{\rho^2\le R^2}\tau(\rho/R)d_{k_1,\ldots,k_n}\left(\gamma_{k_1,\ldots,k_n}^{(1)}-\gamma_{k_1,\ldots,k_n}^{(2)}\right)^2. \]
Hence we find that \(\gamma_{k_1,\ldots,k_n}^{(1)}=\gamma_{k_1,\ldots,k_n}^{(2)}\) for all possible combinations of integral values \(k_j\), \(j=1,2,\ldots,n\). From the uniqueness of the solution of the trigonometric moment problem (see (7)) it follows that \(\mu_1^*=\mu_2^*\). For \(n=1\) the proof of Theorem 6 evidently follows from consideration of Fejér means.
p. 6. On the Fourier coefficients of functions depending on the radius.
Lemma 4. Let \(n\ge 2\) and let \(f(P)\) be a function integrable in the Lebesgue sense on \(\Pi\) and depending only on \(r=\sqrt{x_1^2+\cdots+x_n^2}\) for \(P=(x_1,\ldots,x_n)\). Then its Fourier coefficients \(d_{k_1,\ldots,k_n}\) are such that
\[ d_{k_1,\ldots,k_n}=d(\rho), \]
where \(\rho=\sqrt{k_1^2+\cdots+k_n^2}\) and
\[ d(\rho)=\frac{\chi(n)}{\rho}\int_0^{\pi\sqrt n} r^{l+1}f(r)J_l(r\rho)\,dr; \tag{10} \]
\(\chi(n)\) is a positive function depending only on \(n\), \(l=(n-2)/2\), and \(J_l(z)\) is the Bessel function.
Theorem 7. Let \(n\ge 3\) and let the function \(t^{\,n-2}\varphi(t)\) be strictly decreasing; then, for \(\varphi(t)\in\Phi\), the kernel \(\varphi(t)\) has positive Fourier coefficients.
The proof of this theorem follows from Lemma 4 for \(n\ge 4\), and for \(n=3\) from expression (10).
Theorem 8. If \(n\ge 2\) and \(\varphi(t)\in\Phi\) is such that \(\{-\Phi_t'(t)\}\) is strictly decreasing, then \(\varphi(t)\) has positive Fourier coefficients.
This theorem is proved analogously to Theorem 7.
If the kernel \(\varphi(t)\in\Phi_1\), then, when the conditions of Theorems 4 and 7 (or 8) are fulfilled, there exists a unique equilibrium measure for the given set \(F\).
The authors consider it their pleasant duty to express their gratitude to Academician N. N. Bogolyubov for valuable remarks and useful discussion of the present work.
Received
30 IV 1965
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