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This article is cited in 4 scientific papers (total in 4 papers) Explicit form of fundamental solutions to certain elliptic equations and associated P. V. Paramonovab, K. Yu. Fedorovskiyab a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Saint Petersburg State University, St. Petersburg, Russia
Russian version:
Matematicheskii Sbornik, 2023, Volume 214, Number 4, DOI: https://doi.org/10.4213/sm9807 
Bibliographic databases:
    § 1. IntroductionRecently Mazalov[1] obtained criteria of Vitushkin type for uniform approximation of functions by solutions of general homogeneous second-order elliptic equations with constant complex coefficients in $\mathbb R^N$, $N\geqslant3$. These results give answers to both standard settings of the approximation problem under consideration: for individual functions and for classes of functions. Recall that these criteria are stated in terms of $B$- and $C$-capacities related to the partial differential equations under consideration. Our main aim is to study the geometric and metric properties of these capacities. In the harmonic case this problem is well known, and it was studied in details in the first half of the previous century in classical books and papers on potential theory (see, for example, [2] and [3], Ch. 2, § 3, and the literature cited there). The first (standard) results for the general case are presented in the final section of [4]. An essential step in the investigations of the capacities mentioned above is to find explicit expressions for fundamental solutions of equations under consideration. Despite the fact that for a large class of equations the corresponding formulae were obtained previously (see, for example, [5], § 6.2), the general case, as far as the authors are aware of, has almost not been considered in the literature. We present our (relatively simple and short) proofs of necessary statements in §§ 2 and 3. The main results of this paper are Theorem 1 (which describes the explicit form of fundamental solutions), Corollary 2 on two-sided estimates of fundamental solutions for a wide class of equations (for instance, all equations under consideration in the case of dimensions $N=3$ � $N=4$), and Corollary 3 (two-sided estimates for the corresponding $B_+$- and $C_+$-capacities, defined in terms of potentials of positive measures, via the harmonic capacity in the same dimension). Thus, the main metric properties of these capacities are identical (and similar to the harmonic capacity). Note that the problem of an upper estimate for $B$- and $C$-capacities themselves in terms of $B_+$- and $C_+$-capacities remains open. In § 4 some integral properties of fundamental solutions under consideration are obtained; they seem to be useful for further studies. AcknowledgementThe authors are deeply grateful to referees for their work on reviewing this work and for several useful comments and suggestions. § 2. Elliptic quadratic forms in $ {\mathbb R}^N$ with complex coefficientsLet $N\geqslant2$ be a fixed integer, and let $C$ be a symmetric $ N\times N $ matrix with complex entries $c_{mn}=c_{nm}$, $1\leqslant m,n\leqslant N$. Let $Q$ be the quadratic form in $\mathbb R^N$ determined by $C$, that is, for $x=(x_1,\dots,x_N)^t \in\mathbb R^N$, where the symbol $(\,\cdot\,)^t$ means the operation of matrix transposition. In what follows we use the notation $Q_N$ instead of $Q$ in order to highlight the dimension $N$ of the space on which $Q$ acts.Definition. The quadratic form $Q_N$ is called elliptic if $Q_N(x)\neq0$ for all $x\in\mathbb R^N_*$. Here and in what follows $\mathbb R^N_*=\mathbb R^N\setminus\{0\}$ and $\mathbb C^n_*=\mathbb C^n\setminus\{0\}$, $n\in\mathbb N$. Let us consider the concept of an elliptic quadratic form in detail since it will be essential for further constructions. Unfortunately, these authors were not able to find the necessary references in the literature despite a quite serious efforts applied. For example, in a classical book by Hörmander (see [5], § 6.2) the properties of relevant quadratic forms, which we establish in our Lemmas 4 and 5 below, are postulated (that is, assumed a priori) without any discussion. Therefore, below we present the proofs of all necessary statements, especially since they are based on some very simple ideas. Let us start with the case $N=2$. In this situation the definition of ellipticity can be reformulated as follows. A quadratic form
$$
\begin{equation*}
Q_2((x_1,x_2)^t)=c_{11}x_1^2+2c_{12}x_1x_2+c_{22}x_2^2
\end{equation*}
\notag
$$
in $\mathbb R^2$ is elliptic in and only if the roots $\lambda_1$ and $\lambda_2$ of the corresponding characteristic equation $c_{11}\lambda^2+2c_{12}\lambda+c_{22}=0$ are not real. Moreover, the form $Q_2$ is called strongly elliptic if $\lambda_1$ and $\lambda_2$ belong to the different half-panes ${{\mathbb C}_+ =\{z\in\mathbb C\colon \operatorname{Im}{z}>0\}}$ and ${{\mathbb C}_- =\{z\in\mathbb C\colon \operatorname{Im}{z}<0\}}$ of the complex plane $\mathbb C$. In the two-dimensional case it is convenient to treat $Q_2$ as a function of the complex variable $z$, that is, $Q_2(z)=Q_2((\operatorname{Re}{z},\operatorname{Im}{z})^t)$. Let $\mathbb T = \{z\in\mathbb C\colon |z|=1\}$ be the unit circle in $\mathbb C$ with the standard parametrization $\mathbb T=\{\varGamma_1(t)=e^{2\pi it}\colon t\in[0,1]\}$. Lemma 1. Let $Q_2$ be an elliptic quadratic form in $\mathbb R^2$. Then $Q_2$ is strongly elliptic if and only if $\Delta_{\mathbb T}\operatorname{Arg}{Q_2}=0$1. The latter condition is equivalent to the set $Q_2(\mathbb T)$ lying in an (open) half-plane in $\mathbb C$ whose boundary contains the origin. Proof. Assume that $\lambda_1\in\mathbb C_+$ and $\lambda_2\in\mathbb C_-$. It can readily be verified that $\Delta_{\mathbb T}\operatorname{Arg}(x_1-\lambda_1x_2)=-2\pi$ and $\Delta_{\mathbb T}\operatorname{Arg}(x_1-\lambda_2x_2)=2\pi$. Since $Q_2(z)=c_{11}(x_1-\lambda_1x_2)(x_1-\lambda_2x_2)$ for $z=x_1+ix_2$, we have $\Delta_{\mathbb T}\operatorname{Arg}{Q_2}=0$. Conversely, if both the characteristic roots $\lambda_1$ and $\lambda_2$ belong to $\mathbb C_+$ or to $\mathbb C_-$, then $\Delta_{\mathbb T}\operatorname{Arg}{Q_2}=\pm4\pi$. The first assertion of the lemma is proved.
In order to prove the second, observe that
$$
\begin{equation*}
\begin{aligned} \, Q_2(\mathbb T) &=\bigl\{Q_2(\cos(2\pi t),\sin(2\pi t))\colon t\in[0,1]\bigr\} \\ &=\bigl\{c_{11}\cos^2(2\pi t)+2c_{12}\sin(2\pi t)\cos(2\pi t)+c_{22}\sin^2(2\pi t)\colon t\in[0,1]\bigr\} \\ &=\biggl\{\frac{c_{11}+c_{22}}{2}+\frac{c_{11}-c_{22}}{2}\cos(4\pi t)+c_{12}\sin(4\pi t) \colon t\in[0,1] \biggr\}. \end{aligned}
\end{equation*}
\notag
$$
If $c_{11}=c_{22}$ or $c_{12}=\tau(c_{11}-c_{22})$ for some $\tau\in\mathbb R$ (both situations can occur in the strongly elliptic case), then the parametric representation in the last formula defines a straight line segment (containing the origin and traversed four times). In the other cases this parametric expression defines a doubly traversed ellipse that surrounds or does not surround the origin in the cases of the strongly elliptic and nonstrongly elliptic form $Q_2$, respectively. This follows immediately from the reasoning in the proof of the first part of the lemma. The proof is complete. Corollary 1. In the case of a nonstrongly elliptic form (and only in this case) $\Delta_{\mathbb T}\operatorname{Arg}{Q_2}=\pm4\pi$, and there exist two points $x\in\mathbb R^2_*$ and $y\in\mathbb R^2_*$ such that ${Q_2(x)=-Q_2(y)}$. As another corollary of Lemma 1, we note that the properties of ellipticity and strong ellipticity of quadratic forms are preserved under non-degenerate linear transformations of $\mathbb R^2$. Let $\mathbb S^{N-1}=\{x\in\mathbb R^N\colon x_1^1+\dots+x_N^2=1\}$ be the unit sphere in $\mathbb R^N$. Thus, $\mathbb T$ coincides with $\mathbb S^1\subset\mathbb R^2$ as a set, but in the definition of $\mathbb T$ a fixed orientation (determined by the parametrization $\varGamma_1$) is additionally assumed. Lemma 2. Let $Q_2$ be a quadratic form in $\mathbb R_2$, and let $N\in\{3,4,\dots\}$. An elliptic quadratic form $Q_N$ in $\mathbb R^N$ such that $Q_N((x_1,x_2,0,\dots,0)^t)=Q_2((x_1,x_2)^t)$ for $(x_1,x_2)^t \in\mathbb R^2$ exists if and only if $Q_2$ is strongly elliptic. Proof. Let $Q_2$ be a strongly elliptic quadratic form in $\mathbb R^2$, and let $V=\{tz\colon t\in\mathbb R_+:=(0, +\infty),\ z\in Q_2({\mathbb S^1})\}$. Take arbitrary coefficients $c_{nn}\in V$, $n\in\{3,\dots,N\}$. Then it follows immediately from Lemma 1 that the form $Q_N$ in $\mathbb R^N$ defined by is an elliptic form in $\mathbb R^N$.
Conversely, take a nonstrongly elliptic form $Q_2$ in $\mathbb R^2$ and assume that it is a restriction to $\mathbb R^2$ of some elliptic quadratic form $Q_N$ in $\mathbb R^N$ for some $N>2$. Let $\varGamma$ be a homotopy on the sphere $\mathbb S^{N-1}\subset\mathbb R^N$ that contracts the circle $\varGamma_1\subset\mathbb R^2_{(x_1,x_2)^t}$ (see above) to a point $a\in\mathbb S^{N-1}$ (we identify $\varGamma_1$ with the set $\{x\in \mathbb S^{N-1}\colon x_3=\dots =x_N=0\}$). Then the composition $Q_N\circ\varGamma$ is a homotopy in $\mathbb R^2_*$ contracting the cycle $\varGamma_2=Q_N\circ\varGamma_1=Q_2\circ\varGamma_1$ to the point $Q_N(a)$. But such a homotopy does not exist since $\Delta_{\varGamma_2}\operatorname{Arg}(z)=\Delta_{\varGamma_1}\operatorname{Arg}(Q_2)=\pm4\pi$ in view of Corollary 1. The lemma is proved. Lemma 3. Let $Q_N$ be a quadratic form in $\mathbb R^N$, $N\geqslant3$. The form $Q_N$ is elliptic if and only if the set $Q_N(\mathbb S^{N-1})\subset\mathbb C_*$ lies in an open half-plane in $\mathbb C$ whose boundary contains the origin. This is equivalent to the fact that the set $V_N=Q_N(\mathbb R^N_*)$ is a closed angle of size $\vartheta_{Q_N}<\pi$ in $\mathbb C_*$ with (punctured) vertex at the origin. Proof. Assume that $Q_N$ is an elliptic form. It suffices to prove that there is no pair of points $x\in\mathbb R^N_*$, $y\in\mathbb R^N_*$ such that $Q_N(x)= -Q_N(y)$. Indeed (arguing by contradiction), if such points $x$ and $y$ exist, then the restriction of $Q_N$ to the plane passing through $x$, $y$ and the origin is not strongly elliptic in view of Corollary 1. This contradicts the statement of Lemma 2. The converse is clear. The lemma is proved. It follows from Lemma 3 that for every elliptic quadratic form $Q_N$ in $\mathbb R^N$, $N\geqslant3$, there exists a number $\tau\in(0,1)$ and an angle $\vartheta\in(-\pi,\pi]$ such that the form $Q(x)=e^{i\vartheta}Q_N(x)$ satisfies the following conditions:
$$
\begin{equation}
|{\arg(Q(x))}|\leqslant \frac{\vartheta_{Q_N}}2 < \frac{\pi}2\quad\text{and} \quad \operatorname{Re}(Q(x))\geqslant \tau|Q(x)|\geqslant \tau^2|x|^2, \quad x\in \mathbb R^N_*.
\end{equation}
\tag{2.1}
$$
In what follows we assume that $Q_N$ is an elliptic quadratic form in $\mathbb R^N$ determined by the matrix $C=A+iB$, where $A=\operatorname{Re}{C}$ and $B=\operatorname{Im}{C}$. Note that the conditions stated in Lemma 3 can easily be verified for ‘diagonal’ forms $Q_N(x)=\sum_{n=1}^N c_{nn}x_n^2$ (that is, in the case when $C$ is a diagonal matrix). Several conditions for the diagonalization of quadratic forms can be found in [6], § 4.5. Proof. Arguing by contradiction assume that $\det{C}\,{=}\,0$. Then there exists $z\in\mathbb C^N$, $z\neq 0$, such that $Cz=0$. Let $z=x+iy$. The condition $Cz=(A+iB)(x+iy)=0$ holds if and only is the conditions $Ax=By$ and $Ay=-Bx$ hold simultaneously. Therefore, and Since both the matrices $A$ and $B$ are symmetric, the last condition implies that $Q_N(y)=-Q_N(x)$ (in particular, $x\neq 0$ and $y\neq 0$). This contradicts Lemma 3. The lemma is proved. Note that the condition $N\geqslant3$ is essential in Lemma 4, because the elliptic quadratic form $Q_2(x_1,x_2)=\frac14(x_1+ix_2)^2$ in $\mathbb R^2$ has the matrix such that $\det{C_2}=0$.Lemma 5. Let $N\geqslant 3$, let $Q_N$ and $C$ be as mentioned above, and let $Q'_N$ be the quadratic form determined by the matrix $C^{-1}$. Then $Q'_N$ is also elliptic. Proof. It follows from Lemma 4 that $\det{C}\neq 0$. Arguing by contradiction we assume that there exists $a\in\mathbb R^N_*$ such that $a^tC^{-1}a=0$. Let $z=C^{-1}a$, so that $z\in\mathbb C^N_*$. Then $a=Cz$ and $0=a^tC^{-1}a=(Cz)^tz=z^tC^tz=z^tCz$, since the matrix $C$ is symmetric.
As before, we put $x=\operatorname{Re}{z}$, $y=\operatorname{Im}{z}$, $A=\operatorname{Re}{C}$ and $B=\operatorname{Im}{C}$. Since $Cz=a$, we have $\operatorname{Im}((A+iB)(x+iy))=0$, which gives $Bx=-Ay$. Furthermore, since the following two equalities hold: It follows from the three equalities obtained that and Therefore, which contradicts Lemma 3. The lemma is proved.§ 3. Second-order elliptic operators with constant complex coefficients in $ {\mathbb R}^N$Let $N\geqslant2$ be a fixed integer, and let $Q$ be an elliptic quadratic form in $\mathbb R^N$ determined by the (symmetric) matrix $C$ with complex entries $c_{mn}$, $1\leqslant m,n\leqslant N$. This form defines a second-order elliptic partial differential operator with constant complex coefficients
$$
\begin{equation*}
\mathcal L=\sum_{m,n=1}^{N}c_{mn}\frac{\partial^2}{\partial x_m\,\partial x_n},
\end{equation*}
\notag
$$
which we call the operator associated with $Q$. The form $Q$ is in its turn called the symbol of $\mathcal L$. Consider the main examples. The first is the Laplace operator
$$
\begin{equation*}
\Delta_N=\sum_{n=1}^{N}\frac{\partial^2}{\partial x_n^2} \quad\text{in } \mathbb R^N.
\end{equation*}
\notag
$$
This operator is associated with the quadratic form $x_1^2+\dots+x_N^2$. The second example is the Bitsadze operator (the square of the Cauchy-Riemann operator) in $\mathbb R^2$. It is defined by the formula
$$
\begin{equation*}
\frac14\biggl(\frac{\partial}{\partial x_1}+i\,\frac{\partial}{\partial x_2}\biggr)^2=\frac{\partial^2}{\partial\overline{z}^2}.
\end{equation*}
\notag
$$
Its associated quadratic form $Q_2(x)=\frac14(x_1+ix_2)^2$ was mentioned before in our paper. As we mentioned above, all elliptic quadratic forms in $\mathbb R^2$ are divided into two classes: the class of strongly elliptic forms and the class of forms that are not strongly elliptic. In accordance with this classification, one says that a second-order elliptic operator $\mathcal L$ in $\mathbb R^2$ is strongly elliptic if its symbol is a strongly elliptic form (and nonstrongly elliptic otherwise). Thus, the Laplace operator $\Delta_2$ in $\mathbb R^2$ is strongly elliptic, while the Bitsadze operator is not. So this latter cannot be ‘lifted’ to an elliptic operator in $\mathbb R^N$ for any $N>2$. Note that the quadratic form, which is the symbol of the Bitsadze operator, has the matrix which was considered in § 2 as an example of a degenerate matrix of an elliptic quadratic form.Notice that the division of second-order elliptic operators in $\mathbb R^2$ into classes of strongly and nonstrongly elliptic operators is not formal: it is due to significantly different properties of these operators. The deep difference between the properties of strongly elliptic operators and nonstrongly elliptic ones, comes out in problems of the description of sets of removable singularities for solutions of the corresponding equations $\mathcal Lf=0$, in problems of approximating functions by solutions of these equations, as well as in conditions for solvability and uniqueness of solution of classical boundary value problems. Some examples of statements underlining this difference can be found, for example, in [7] or the recent paper [8]. Lemma 2 stated above is also an interesting example of the essential difference between two classes of operators under consideration. In this section we obtain explicit formulae for the fundamental solution of an arbitrary second-order elliptic operator $\mathcal L$ � $\mathbb R^N$, $N\,{\geqslant}\, 3$. Let $\mathcal L$ be associated with the quadratic form $Q$ defined by a matrix $C$. As shown in Lemma 4, this matrix is nonsingular, that is, $\det{C}\neq0$. Consider the matrix $D=C^{-1}$ and the corresponding quadratic form In view of Lemma 5 the form $\varLambda$ is elliptic. Let $\vartheta_{\varLambda}$ be the size of the angle $\varLambda(\mathbb R^N_*)$, so that $\vartheta_{\varLambda}<\pi$.Now let $S(z)$ be the ‘principal’ branch of the multi-valued function $\sqrt{z}$ in $\{{z\in\mathbb C}$: $-\pi<\arg{z}<\pi\}$ such that According to Lemma 3 there exists $\vartheta\in(-\pi,\pi]$ such that $|{\arg(e^{i\vartheta}\Lambda(x))}|\leqslant\vartheta_{\varLambda}/2<\pi/2$, which gives $\operatorname{Re}(e^{i\vartheta}\Lambda(x))>0$ for every $x\in\mathbb R^N_*$. Therefore, the function is real analytic in $\mathbb R^N_*$, homogeneous of order $1$ and, moreover, we have
$$
\begin{equation}
|{\arg(\Psi(x))}|<\frac{\vartheta_{\varLambda}}4<\frac\pi4 \quad \forall\, x\in\mathbb R^N_*.
\end{equation}
\tag{3.2}
$$
Proposition 1. Let $\mathcal L$, $Q$, $C$, $D$, $\varLambda$ and $\Psi$ be as mentioned above, and let for $x\in\mathbb R^N_*$. Then $\mathcal L\varPhi(x)=0$ for all $x\in\mathbb R^N_*$. Proof. Without loss of generality we can assume that $\theta\,{=}\,0$. Let $p=(2-N)/2$, so that $\varPhi(x)=\varLambda(x)^p_*$, where $\varLambda(\,\cdot\,)$ was defined in (3.1) and the symbol $*$ means that we are dealing with the corresponding principal branch $w^p_*= \exp(p\log_*w)$ of the multi-valued function $w^p$ in ${\mathbb C}\setminus (-\infty,0]$ (here $\log_*(w)=\log|w|+i\arg{w}$ is the principal branch of the multi-valued logarithm in ${\mathbb C}\setminus (-\infty,0]$).
Since $(w^p_*)'_w=(\exp(p\log_*w))'_w=pw^p_*/w$, we have
$$
\begin{equation*}
\frac{\partial\varPhi(x)}{\partial x_m}=p \frac{\varPhi(x)}{\varLambda(x)}\, \frac{\partial\varLambda(x)}{\partial x_m},
\end{equation*}
\notag
$$
and therefore
$$
\begin{equation*}
\frac{\partial^2\varPhi(x)}{\partial x_m\,\partial x_n}= p \frac{\varPhi(x)}{\varLambda(x)^2}\biggl( (p-1)\, \frac{\partial\varLambda(x)}{\partial x_m}\,\frac{\partial\varLambda(x)}{\partial x_n}+ \varLambda(x) \frac{\partial^2\varLambda(x)}{\partial x_m\,\partial x_n}\biggr)
\end{equation*}
\notag
$$
for $1\leqslant m,n\leqslant N$.
Since $p-1=-N/2$, to prove the equality $\mathcal L\varPhi(x)=0$ for $x\in\mathbb R^N_*$ it suffices to show that
$$
\begin{equation}
R(x)=\sum_{m,n=1}^{N}c_{mn}\biggl(-\frac{N}2\,\frac{\partial\varLambda(x)}{\partial x_m}\,\frac{\partial\varLambda(x)}{\partial x_n}+ \varLambda(x)\, \frac{\partial^2\varLambda(x)}{\partial x_m\,\partial x_n}\biggr)=0
\end{equation}
\tag{3.4}
$$
for every $x\in\mathbb R^N_*$, where the $c_{mn}$ are (as before) the elements of the matric $C$. Denoting by $d_{mn}$ ($1\leqslant m,n\leqslant N$) the elements of the matrix $D$ and taking into account the fact that this matrix is symmetric, we obtain
$$
\begin{equation*}
\frac{\partial\varLambda(x)}{\partial x_m}=2\sum_{j=1}^{N}d_{jm}x_j \quad\text{and} \quad \frac{\partial^2\varLambda(x)}{\partial x_m\,\partial x_n}=2d_{mn}.
\end{equation*}
\notag
$$
Furthermore, it can readily be seen that the following equation holds:
$$
\begin{equation*}
\frac{\partial\varLambda(x)}{\partial x_m}\,\frac{\partial\varLambda(x)}{\partial x_n}= 4\sum_{j,k=1}^{N}d_{mj}d_{nk}x_jx_k.
\end{equation*}
\notag
$$
The quantity $R(x)$ in (3.4) can be represented in the following form: Denoting the coefficients of the quadratic form $R(x)$ by $r_{jk}$, $1\leqslant j,k\leqslant N$, we obtain (in view of the symmetry of the matrix $D$) It remains to note that $(d_{j1},\dots,d_{jN})C(d_{1k},\dots,d_{Nk})^t$ is the element $d_{jk}$ of the matrix $D=DCD$, and $\sum_{m,n=1}^{N}c_{mn}d_{mn}=N$ since $DC=I$ (the identity matrix). Therefore, $r_{jk}=-2Nd_{jk}+2Nd_{jk}=0$, and the proof is complete. Now we can obtain an explicit formula for a fundamental solution for $\mathcal L$. Recall that a distribution $\varPhi_{\mathcal L}$ is called a fundamental solution for $\mathcal L$, if $\mathcal L\varPhi_{\mathcal L}=\delta_0$, where $\delta_0$ is the Dirac delta-function with support at the origin. Theorem 1. Using the notation of Proposition 1, the function where $\alpha_{\mathcal L}\in\mathbb C_*$ is a suitable constant, is a fundamental solution for $\mathcal L$.The proof of this theorem follows immediately from Proposition 1 and Theorem 3.2.3 in [5]. The explicit value of the coefficient $\alpha_{\mathcal L}$ can be obtained using Theorem 6.2.1 in [5]. For instance, it is well known that for $\mathcal L=\Delta_N$ (the Laplace operator in $\mathbb R^N$) we have $\varPhi_{\Delta}=\alpha_N |x|^{-N+2}$ for a certain negative constant $\alpha_N$. The following statement is a direct corollary of (3.2) and (3.3). Corollary 2. Let $N=3$ or $N=4$. Then for every operator $\mathcal L$ under consideration there exist $\lambda=\lambda_{\mathcal L}\in(-\pi,\pi]$ and $A=A_{\mathcal L}\geqslant 1$ such that
$$
\begin{equation}
\frac1{A |x|^{N-2}}\leqslant \operatorname{Re}(e^{i\lambda}\varPhi_{\mathcal L}(x))\leqslant |\varPhi_{\mathcal L}(x)|\leqslant\frac{A}{|x|^{N-2}}
\end{equation}
\tag{3.6}
$$
for all $x\in\mathbb R^N_*$. The analogous result also holds for operators $\mathcal L$ in $\mathbb R^N$, $N\geqslant 5$, which satisfy the following additional condition Moreover, it follows directly from (3.3) that for $N\geqslant 3$ the estimate
$$
\begin{equation*}
\frac1{A |x|^{N-2}} \leqslant |\varPhi_{\mathcal L}(x)|\leqslant\frac{A}{|x|^{N-2}}
\end{equation*}
\notag
$$
holds for all operators ${\mathcal L}$ under consideration.
§ 4. $\mathcal L$-analytic capacitiesLet $\mathcal L$ be a homogeneous second-order elliptic differential operator in $\mathbb R^N$, $N\geqslant3$, with constant complex coefficients. Let $\varPhi_{\mathcal L}(x)$ be its fundamental solution defined in (3.5). We are going to introduce capacities of two kinds that are related to the operator $\mathcal L$ and defined in terms of the classes of bounded functions (so-called $\mathcal L$-$B$-capacities) and continuous functions ($\mathcal L$-$C$-capacities, respectively). Capacities of the first kind are defined as follows. For a bounded set $E \neq \varnothing$ in $\mathbb R^N$ we put where $\langle T,\psi\rangle$ stands for the action of the distribution $T$ on the function $\psi\in C^{\infty}(\mathbb R^N)$ (in the case when this action is well-defined), and the supremum is taken over all complex-valued distributions $T$ with compact support $\operatorname{Supp}(T)\subset E$ satisfying the conditions $\varPhi_{\mathcal L}*T\in L^\infty(\mathbb R^N)$ and $\|\varPhi_{\mathcal L}*T\|_{\infty}\leqslant 1$. The symbol $*$ means here the convolution operation, while $\|\cdot\|_{\infty}$ means the norm in the space $L^{\infty}(\mathbb R^N)$. The symbol $\|f\|_E$ denotes the $\sup$-norm in the space $BC(E)$ of all bounded and continuous functions on a nonempty set $E \subset \mathbb R^N$ (in the case when $E=\mathbb R^N$ the subscript $E$ is dropped).It is convenient to introduce the so-called ‘metric’ (or ‘positive’) analogue of the capacity $\gamma_{\mathcal L}$: where the supremum is taken over all positive Borel measures $\mu$ with compact support $\operatorname{Supp}(\mu)\subset E$ satisfying the conditions $\varPhi_{\mathcal L}*\mu\in L^\infty(\mathbb R^N)$ and $\|\varPhi_{\mathcal L}*\mu\|_{\infty}\leqslant1$.Capacities of the second kind (the continuous versions of $\gamma_{\mathcal L}$ and $\gamma_{\mathcal L}^+$) are the capacities $\alpha_{\mathcal L}$ and $\alpha_{\mathcal L}^+$, which are defined in the same way, but the distributions $T$ in the definition of $\alpha_{\mathcal L}(E)$ and the measures $\mu$ in the definition of $\alpha_{\mathcal L}^+(E)$ are taken so as to satisfy the additional conditions $\varPhi_{\mathcal L}*T \in C(\mathbb R^N)$ and $\varPhi_{\mathcal L}*\mu\in C(\mathbb R^N)$, respectively. More detailed constructions of capacities related to elliptic differential operators and defined in terms of various spaces of functions in $\mathbb R^N$ can be found, for example, in [9]. In the classical (harmonic) case, when $\mathcal L=\Delta=\Delta_N$, $N\geqslant 3$, we have
$$
\begin{equation*}
\alpha_{\Delta}^+(E)=\alpha_{\Delta}(E)=\gamma_{\Delta}^+(E)=\gamma_{\Delta}(E).
\end{equation*}
\notag
$$
The key equalities in this chain are $\gamma_{\Delta}^+(E)=\gamma_{\Delta}(E)$ and $\alpha_{\Delta}^+(E)=\gamma_{\Delta}^+(E)$. They were obtained in [9], Theorem 3.1, and [2], Lemma XII, respectively. Notice that the harmonic capacities coincide with the corresponding generalized transfinite diameters in $\mathbb R^N$. In particular, these capacities are countably subadditive with constant $1$ (see [3], Ch. 2). It follows from Corollary 2, the definitions of capacities given above, and from [4], Remarks 1 and 2 and Lemma 3.2, that the following estimates hold (these statements in [4] and their proofs can be extended in a natural way to all dimensions $N\geqslant 3$). Corollary 3. Let $N=3$ or $N=4$, and let the operator $\mathcal L$ be as mentioned above. Then there exists $A=A_{\mathcal L}\geqslant 1$ such that
$$
\begin{equation}
A^{-1}\gamma_{\Delta}(E)\leqslant \alpha_{\mathcal L}^+(E)\leqslant \gamma_{\mathcal L}^+(E)\leqslant A\gamma_{\Delta}(E)\leqslant A^2\alpha_{\mathcal L}(E)\leqslant A^2\gamma_{\mathcal L}(E)
\end{equation}
\tag{4.2}
$$
for any bounded set $E\subset\mathbb R^N$. The analogous result holds for operators $\mathcal L$ in $\mathbb R^N$, $N\geqslant5$, satisfying the additional condition (3.7). The following questions appear quite natural.
All these questions can be clarified by reformulating them as follows: for which $N\,{\geqslant}\, 3$ and $\mathcal L$ are some or other specified properties fulfilled? Let us present some statements and conjectures related to the questions stated above. If what follows let $N \geqslant 3$, and let $\mathcal L$ be an operator with fundamental solution $\varPhi=\varPhi_{\mathcal L}$ that satisfies condition (2.1) for $\vartheta=0$. We start with the following statement of independent interest. where $B_R=B(0,R)$, and $\sigma$ denotes the surface Lebesgue measure on the corresponding sphere of integration. Proof. Since the function $\varPhi$ is homogeneous of order $2-N$, the following equalities can readily be verified:
$$
\begin{equation}
\int_{\partial B_r}\varPhi(x)\,d\sigma_x = \frac{r}{R}\int_{\partial B_R}\varPhi(x)\,d\sigma_x, \qquad \int_{B_R}\varPhi(x)\,dx = \frac{R}{2}\int_{\partial B_R}\varPhi(x)\,d\sigma_x.
\end{equation}
\tag{4.4}
$$
Therefore, we only need to verify the second inequality in (4.3) for $R=1$. We need one well-known property of fundamental solutions, which can be found, for example, in [10], Ch. 3, § 3.5 (namely, formula (4.6) below and Theorem 6 from this reference are used for the operator $T(f)=(\Delta\varPhi)*f$):
$$
\begin{equation}
\Delta\varPhi=\Delta\varPhi_{\mathcal L}=\lambda_0\delta_0+\varPsi,
\end{equation}
\tag{4.5}
$$
where $\delta_0$ is the Dirac delta function with support at the origin, $\lambda_0=\lambda_0(\mathcal L)\in\mathbb C$ and $\varPsi$ is some Calderon-Zygmund kernel in $\mathbb R^N$. Equality (4.5) is understood in the sense of distributions, and the function $\varPsi$ satisfies the following conditions: $\varPsi\in C^\infty(\mathbb R^N_*)$, $\varPsi$ is homogeneous of order $-N$, and
To both sides of (4.5) we apply the Fourier transform $T\mapsto\widetilde{T}$ acting in the space ${\mathcal S}'$ of tempered distributions. First, using the condition $\mathcal L\varPhi=\delta_0$ we obtain ${\widetilde{\mathcal L\varPhi}=(2\pi)^{-N/2}}$, that is This yields so that
$$
\begin{equation}
\widetilde{\Delta\varPhi}(y)=-|y|^2\widetilde{\varPhi}(y)=(2\pi)^{-N/2}\frac{|y|^2}{Q(y)}.
\end{equation}
\tag{4.6}
$$
On the other hand it follows from (4.5) that
$$
\begin{equation}
\widetilde{\Delta\varPhi}(y)=\lambda_0(2\pi)^{-N/2}+\widetilde{\varPsi}(y),
\end{equation}
\tag{4.7}
$$
where (taking (4.6) into account again) the function $\widetilde{\varPsi}\in C^{\infty}(\mathbb R^N_*)$ is homogeneous of order $0$. It follows from the definition of the Fourier transform of distributions that $\langle\widetilde{\Psi},\varphi\rangle= \langle\Psi,\widetilde{\varphi}\rangle$. Substituting in $\varphi(x)=\exp(-|x|^2/2)$ we have
$$
\begin{equation*}
\int_{\mathbb R^N}\widetilde{\varPsi}(y)\exp\biggl(-\frac{|y|^2}2\biggr)\,dy= \mathrm{p.v.}\int_{\mathbb R^N}\varPsi (x)\exp\biggl(-\frac{|x|^2}2\biggr)\,dx=0,
\end{equation*}
\notag
$$
which gives us
$$
\begin{equation*}
\int_{\partial B_1}\widetilde{\varPsi}(y)\,d\sigma_y=0
\end{equation*}
\notag
$$
because $\widetilde{\varPsi}$ is homogeneous of order $0$ as mentioned above.
In particular, it follows from (4.6) and (4.7) that
$$
\begin{equation}
\lambda_0=\frac1{\sigma(\partial B_1)}\int_{\partial B_1}\frac{|y|^2}{Q(y)}\,d\sigma_y\neq0,
\end{equation}
\tag{4.8}
$$
since $\operatorname{Re}(Q(y))>0$ for $y\in\mathbb R^N_*$ (this follows directly from (2.1)).
Now fix a radial function $\varphi(x)=\varphi(|x|)\in C^{\infty}_0(B_1)$ such that $\varphi(0)\neq0$. Then
$$
\begin{equation*}
\begin{aligned} \, \langle\Delta\varPhi,\varphi\rangle&=\int_{B_1}\varPhi(x)\Delta\varphi(x)\,dx =\langle\lambda_0\delta_0+\varPsi,\varphi\rangle \\ &=\lambda_0\varphi(0)+\int_{B_1}\varPsi(y)(\varphi(y)-\varphi(0))\,dy =\lambda_0\varphi(0)\neq0. \end{aligned}
\end{equation*}
\notag
$$
Therefore, $\displaystyle\int_{B_1}\varPhi(x)\Delta\varphi(x)\,dx\neq0$. Now, using the radial symmetry of $\varphi(x)=\varphi(r)$, $r=|x|$, the equality $\Delta\varphi(x)=\varphi''(r)+ ((N-1)/r)\varphi'(r)$, the homogeneity of $\varPhi$, and (4.4), we obtain
$$
\begin{equation*}
\int_{B_1}\varPhi(x)\Delta\varphi(x)\,dx= \int_{\partial B_1}\varPhi(x')\, d\sigma_{x'}\int_0^1r \biggl(\varphi''(r)+\frac{N-1}{r}\varphi'(r)\biggr)\,dr\neq0,
\end{equation*}
\notag
$$
which gives (4.3) directly. Theorem 2 is proved. Remark. The result stated in Theorem 2 remains true for every second-order strongly elliptic operator $\mathcal L$ in $\mathbb R^2$, while it fails for any second-order nonstrongly elliptic operator in $\mathbb R^2$. In fact, if the second-order operator $\mathcal L$ in $\mathbb R^2$ is strongly elliptic, then its characteristic roots belong to different half-planes with respect to the real axes, while if $\mathcal L$ is nonstrongly elliptic, then both of its characteristic roots belong to one of these half-planes. It follows from the proof of Lemma 3.1 in [11] that in the first case $\lambda_0\neq0$, where $\lambda_0$ was defined in (4.8), but $\lambda_0=0$ in the second case. Using this observation, the proof can be finished analogously to the proof of Theorem 2. Theorem 2 allow us to state the following conjecture, which seems quite plausible, at least for a sufficiently wide class of operators $\mathcal L$. Conjecture 1. Let $N\geqslant5$. Then there exists a constant $A=A(N,\mathcal L)$ satisfying $1\leqslant A<+\infty$ such that for every bounded set $E$ in $\mathbb R^N$, $E\neq\varnothing$, we have To examine this conjecture the following result in the form of an alternative seems to be useful. However, not a single case is known when option (A1) of the alternative below actually holds. Proposition 2. Let $N\geqslant5$, and let $\mathcal L$ be an elliptic operator in $\mathbb R^N$ with fundamental solution $\varPhi=\varPhi_{\mathcal L}$. (A1) Assume that there exists $(N-2)$-dimensional subspace $\mathcal X$ in $\mathbb R^N$ such that
$$
\begin{equation}
\int_{\partial B_1\cap {\mathcal X}}\varPhi(x)\,d\sigma_x=0.
\end{equation}
\tag{4.9}
$$
Then Conjecture 1 fails for the operator $\mathcal L$. More precisely, then $\gamma_{\Delta}(E)=0$ for every bounded set $E\subset {\mathcal X}$, but for every ball $B''\subset {\mathcal X}$ the inequality $\gamma_{\mathcal L}^+(B'')>0$ holds. (A2) Let for an $(N-2)$-dimensional subspace $\mathcal X$ of $\mathbb R^N$. Then $\gamma_{\mathcal L}^+(E)=0$ for every bounded set $E\subset\mathcal X$, $E\neq\varnothing$. Proof. Let us prove (A1). Without loss of generality we may assume that Let $y=(x_3,\dots,x_N)\in\mathbb R^{N-2}$. Then $\varPhi_{\Delta}(0,0,y)=C_N/|y|^{N-2}$, where $C_N<0$ is the corresponding constant. Let $E$ be an arbitrary nonempty compact set in $\mathcal X$, and let $\mu$ be a nonnegative measure on $E$ satisfying $\displaystyle\int_{E}d\mu_y>0$. To prove the property $\gamma_{\Delta}(E)=0$ it suffices to establish that the convolution $|y|^{2-N}\ast\mu$ is unbounded on $\mathbb R^{N-2}$. Indeed, the support $\mathop{\mathrm{Supp}}{\mu}$ of the measure $\mu$ lies in some closed cube $K_0$. We divide $K_0$ into $2^{N-2}$ equal (closed) cubes whose side faces are parallel to the corresponding side faces of $K_0$. Among these cubes we can find a cube $K_1$ satisfying the condition $\mu(K_1)\geqslant2^{2-N}\mu(K_0)$. Continuing this procedure recursively we construct a nested sequence of closed cubes $\{K_m\}$ with a common point $y_0$ such that $\mathop{\mathrm{diam}}(K_{m+1})=\mathop{\mathrm{diam}}(K_m)/2$ and $\mu(K_{m+1})\geqslant2^{2-N}\mu(K_m)$ for $m\in\mathbb N$. It follows directly from the fact that such a sequence of cubes exists that $(|y|^{2-N}\ast\mu)(y_0)=+\infty$, as required.
We continue the proof of (A1). Now let $B''$ be an arbitrary ball in $\mathcal X=\mathbb R^{N-2}_y$. In what follows we denote the ball in $\mathbb R^{N-2}$ with centre $y_0\in\mathbb R^{N-2}$ and radius $R>0$ by $B''(y_0,R)$ and the ball $B''(0,R)$ by $B''_R$. Consider a function $\varphi\in C^\infty_0(B'')$ satisfying the conditions $\varphi\geqslant0$ and $\displaystyle\int_{B''}\varphi(y)\,dy>0$, and the measure $\mu$ such that $d\mu_y=\varphi(y)\,dy$. Let $\varPhi$ be the fundamental solutions for the differential operator under consideration. Put $\varPhi_0(y)=\varPhi((0,0,y))$, so that Fix an arbitrary $y^0\in\mathbb R^{N-2}$ and take $R>0$ such that $B''\subset B''(y^0,R)$. Then
$$
\begin{equation*}
\begin{aligned} \, \bigl|(\varPhi_0\ast\mu)(y^0)\bigr| &=\biggl|\mathrm{p.v.}\int_{B''(y^0,R)}\varPhi_0(y^0-y)\varphi(y)\,dy\biggr| \\ &=\biggl|\int_{B''(y^0,R)}\varPhi_0(y^0-y)\bigl(\varphi(y)-\varphi(y^0)\bigr)\,dy\biggr| \\ &\leqslant A_1\int_0^R\frac1{\rho^{N-2}} \rho^{N-3} \rho\,d\rho\leqslant A_1R, \end{aligned}
\end{equation*}
\notag
$$
where the estimate is obtained by integration in spherical coordinates, and where $A_1=A_1(\varPhi,\varphi)\in(0,+\infty)$. Then we have $\|\varPhi_0\ast\mu\|_{\mathcal X}<+\infty$.
Now fix an arbitrary point $x^0\in\mathbb R^N\setminus\mathcal X$, for which we put $x^0=(x^0_1,x^0_2,y^0)$. Also let $\delta=\sqrt{(x^0_1)^2+(x^0_2)^2}$. We estimate $(\varPhi*\mu)(x^0)$. Assume, as above, that $B''\subset B''(y^0,R)$, and put $V=B''(y^0,R)\setminus B''(y^0,\delta)$. Then
$$
\begin{equation*}
\begin{aligned} \, &\bigl|(\varPhi\ast\mu)(x^0)-(\varPhi_0\ast\mu)(y^0)\bigr| \\ &\qquad\leqslant \int_{B''(y^0,\delta)}|\varPhi(x^0-y)|\,|\varphi(y)|\,dy +\biggl|\int_{B''(y^0,\delta)}\varPhi_0(y^0-y)(\varphi(y^0)-\varphi(y))\,dy\biggr| \\ &\qquad\qquad+\biggl|\int_{V}(\varPhi(x^0-y)-\varPhi(y^0-y))\varphi(y)\,dy\biggr| \\ &\qquad=I_1+I_2+I_3. \end{aligned}
\end{equation*}
\notag
$$
It is clear that $I_1\leqslant A_1 (1/\delta^{N-2}) \delta^{N-2}\leqslant A_1$, while $I_2\leqslant A_1\delta$ (this estimate was obtained above). Let us estimate $I_3$ using the inequality
$$
\begin{equation*}
|\varPhi(x^0-y)-\varPhi_0(y^0-y)|\leqslant A_2\delta\sup_{x\in[x^0,(0,0,y^0)]}|\nabla\varPhi(x-y)|\leqslant \frac{A_3\delta}{|y-y^0|^{N-1}},
\end{equation*}
\notag
$$
which gives immediately
$$
\begin{equation*}
I_3\leqslant\int_{\delta}^{R}\frac{A_3\delta}{\rho^{N-1}}\rho^{n-3}\,d\rho\leqslant A_4,
\end{equation*}
\notag
$$
where $A_2$, $A_3$ and $A_4$ are positive constants. Thus, $\|\varPhi_0\ast\mu\|_{\mathbb R^N}<+\infty$, which shows by definition that $\gamma_{\mathcal L}^+(B'')>0$, and part (A1) is proved.
Now let us establish (A2). As before, we assume that $\mathcal X=\{x\in\mathbb R^N\colon x_1\,{=}\, x_2\,{=}\, 0\}$ and $y=(x_3,\dots,x_N)\in\mathcal X$. Arguing by contradiction we assume that there exists a bounded set $E\subset\mathcal X$ such that $\gamma_{\mathcal L}^+(E)>0$. Then there exists a nonnegative measure $\mu$ with support $\mathop{\mathrm{Supp}}(\mu)\subset E$ such that $\displaystyle\int_{E}d\mu_y>0$ � $\|\varPhi_{\mathcal L}\ast\mu\|\leqslant1$. Set $B''_1=B_1\cap\mathcal X$ and fix a function $\varphi_1\in C^\infty_0(B''_1)$ satisfying the conditions $\varphi_1(y)\geqslant0$ and $\displaystyle\int_{B''_1}\varphi_1(y)\,dy=1$. Finally, for $\varepsilon>0$ we put $\varphi^\varepsilon(y)=\varepsilon^{2-N}\varphi_1(y/\varepsilon)$. Set $\mu^\varepsilon=\varphi^\varepsilon\ast\mu$ (the convolution is considered in $\mathcal X=\mathbb R^{N-2}$, and $\mu^\varepsilon$ is treated as a measure). Then
$$
\begin{equation*}
\mu^\varepsilon\geqslant0\quad\text{and} \quad \int_{\mathcal X}d\mu^\varepsilon > 0 , \qquad \|\varPhi_{\mathcal L}\ast\mu^\varepsilon\|\leqslant1.
\end{equation*}
\notag
$$
Furthermore, let $\psi^\varepsilon$ be the density of the measure $\mu^\varepsilon$ with respect to the Lebesgue measure in $\mathcal X=\mathbb R^{N-2}$, that is, $d\mu^\varepsilon_y=\psi^\varepsilon(y)\,dy$. Then $\psi^\varepsilon\in C^\infty_0(\mathcal X)$.
Since $\displaystyle\int_{\partial B''_1}\varPhi_{\mathcal L}(y)\,d\sigma_y\neq0$, there exists a number $\mu_0 \neq 0$ such that
$$
\begin{equation*}
\int_{\partial B''_1}\varPhi_{\mathcal L}\,d\sigma_y= \mu_0\int_{\partial B''_1}\varPhi_{\Delta}\,d\sigma_y.
\end{equation*}
\notag
$$
Let the point $y^0\in\mathcal X$ be such that $\psi^\varepsilon(y^0)>0$, and let $\mathop{\mathrm{Supp}}(\psi^\varepsilon)\subset B''(y^0,R)=B''$ for some $R>0$. Then on the one hand we have (recall that $\varPhi_{\Delta}=\alpha_N |x|^{-N+2}$ for some negative constant $\alpha_N$)
$$
\begin{equation*}
\int_{B''}\varPhi_{\Delta}(y-y^0) \psi^\varepsilon(y)\,dy=-\infty,
\end{equation*}
\notag
$$
but on the other hand
$$
\begin{equation*}
\biggl|\mathrm{p.v.}\int_{B''}\varPhi_{\mathcal L}(y-y^0) \psi^\varepsilon(y)\,dy\biggr|\leqslant1.
\end{equation*}
\notag
$$
It remains to observe that
$$
\begin{equation*}
\begin{aligned} \, & \biggl|\mathrm{p.v.}\int_{B''}\bigl(\varPhi_{\mathcal L}(y-y^0)-\mu_0\varPhi_{\Delta}(y-y^0)\bigr) \psi^\varepsilon(y)\,dy\biggr| \\ &\qquad = \biggl|\int_{B''}\bigl(\varPhi_{\mathcal L}(y-y^0)-\mu_0\varPhi_{\Delta}(y-y^0) (\psi^\varepsilon(y) - \psi^\varepsilon(y^0))\,dy\biggr| \\ &\qquad \leqslant A_5\int_0^R\frac1{r^{N-2}} r r^{N-3}\,dr\leqslant A_5R<+\infty, \end{aligned}
\end{equation*}
\notag
$$
where $A_5$ is some positive constant. The contradiction obtained completes the proof of (A2).
Proposition 2 is proved. Let us present some examples of the situation when the second case in Proposition 2 takes place. Proposition 3. Let $N\geqslant5$, and let the symbol $Q$ of the elliptic operator $\mathcal L$ in $\mathbb R^N$ have the form where $c_{p+1}\not\in(-\infty,0]$ and $p\in\{1,\dots,N-1\}$, and where $q=N-p\leqslant p$. As before, let $\mathcal X=\{x\in\mathbb R^N\colon x_1=x_2=0\}$. Then Proof. Without lost of generality we may assume that
$$
\begin{equation*}
\varLambda(x)=x_1^2+\dots+x_p^2+\lambda(x_{p+1}^2+\dots+x_N^2), \quad\text{where } \lambda=c_{p+1}^{-1}\not\in(-\infty,0]
\end{equation*}
\notag
$$
(here we use the notation introduced in § 3). Then
$$
\begin{equation*}
\varPhi(x)=\varPhi_{\mathcal L}(x)=c(\varLambda(x))^{-m}_*=c\exp\bigl(-m\log_*(\varLambda(x))\bigr),
\end{equation*}
\notag
$$
where $c\neq0$, $m=(N-2)/2$ and $\log_*$ denotes the principal branch of the multi-valued logarithm. Assume, furthermore, that $q\geqslant 2$ (we omit the easier case $q=1$). Let $y'=(x_3,\dots,x_p)$ and $y''=(x_{p+1},\dots,x_{N-1})$. Using this notation we have
$$
\begin{equation*}
\begin{aligned} \, I(\lambda) &:=\int_{\partial B''_1}\varPhi_{\mathcal L}(y)\,d\sigma_y \\ &= 2c\int_{|y'|\leqslant1}dy'\int_{|y''|\leqslant\sqrt{1-|y'|^2}} \frac{dy''}{\sqrt{1-|y'|^2-|y''|^2} \bigl(|y'|^2+\lambda(1-|y'|^2)\bigr)^m_*}. \end{aligned}
\end{equation*}
\notag
$$
Going over to the spherical coordinates in the last integral we obtain
$$
\begin{equation*}
I(\lambda)=c_1\int_{0}^{1}\rho^{p-3}\,d\rho\int_{0}^{\sqrt{1-\rho^2}} \frac{r^{q-2}\,dr}{\sqrt{1-\rho^2-r^2} (\rho^2+\lambda(1-\rho^2))^m_*},
\end{equation*}
\notag
$$
where the constant $c_1$ is not equal to zero. Put $a=\sqrt{1-\rho^2}$. Then (using the change of variables $r=a\sin{t}$) we arrive at
$$
\begin{equation*}
\int_{0}^{a}\frac{r^{q-2}\,dr}{\sqrt{a^2-r^2}}=c_2a^{q-2},
\end{equation*}
\notag
$$
where $c_2>0$ is an absolute constant. Therefore (in what follows $c_3\neq0$ is a suitable constant)
$$
\begin{equation*}
I(\lambda)=c_3\int_{0}^{1}\frac{\rho^{p-3}(1-\rho^2)^{(q-2)/2}}{(\rho^2+\lambda(1-\rho^2))^m_*}\,d\rho= c_3\int_{0}^{\pi/2}\frac{\sin^{p-3}t\cos^{q-1}t}{(\sin^2t+\lambda \cos^2t)^m_*}\,dt,
\end{equation*}
\notag
$$
where the change $\rho=\sin{t}$ has been made. The last integral equals $I(\lambda)$ also for $q=1$. Notice that
$$
\begin{equation*}
m=\frac{N-2}{2}=\frac{p+q}{2}=\frac{p-3}{2}+\frac{q-1}{2}+1.
\end{equation*}
\notag
$$
The following ‘handbook’ integral is known (see, for example, [12], formula 3.642, (1)):
$$
\begin{equation}
\int_{0}^{\pi/2}\frac{\sin^{2\mu-1}t\cos^{2\nu-1}t\,dt}{(a^2\sin^2t+b^2\cos^2t)^{\mu+\nu}}\,dt =\frac{1}{2a^{2\mu}b^{2\nu}}B(\mu,\nu),
\end{equation}
\tag{4.10}
$$
where $\operatorname{Re}\mu>0$, $\operatorname{Re}\nu>0$, $a>0$, $b>0$, and $B(\,\cdot\,{,}\,\cdot\,)$ is the Euler beta function. The integral $I(\lambda)$ corresponds to $\mu=(p-2)/2$, $\nu=q/2$, $a=1$ and $b=\lambda^{1/2}_*$. For ${\lambda > 0}$ it follows from (4.10) that
$$
\begin{equation*}
I(\lambda)=\frac{c_3}{2(\sqrt{\lambda})^q} B\biggl(\frac{p-2}{2},\frac{q}{2}\biggr)= \frac{c_4}{\lambda^{q/2}_*},
\end{equation*}
\notag
$$
where $c_4\neq0$ is a constant. Since $I(\lambda)$ is holomorphic for $\lambda\in\mathbb C\setminus(-\infty,0]$, in view of the uniqueness theorem we have $I(\lambda)=c_4/\lambda^{q/2}_*\neq0$ for all $\lambda$ under consideration. Proposition 3 is proved. 
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