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This article is cited in 3 scientific papers (total in 3 papers) On metric properties of P. V. Paramonovabc a Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia b Saint Petersburg State University, St. Petersburg, Russia c Moscow Center of Fundamental and Applied Mathematics, Moscow, Russia
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     § 1. IntroductionThe backgrounds of the issue can be found in [1]–[3]. Let
$$
\begin{equation*}
L_N(\mathbf x)=\sum_{i, j=1}^N c_{i j} x_{i} x_{j}, \qquad \mathbf x=(x_1, \dots, x_N) \in \mathbb R^N,
\end{equation*}
\notag
$$
be a homogeneous polynomial of degree two in $\mathbb R^N$ ($N \in \{2, 3, \dots\}$ is fixed) with constant complex coefficients $c_{i j}=c_{ji}$ that satisfies the condition of ellipticity: $L_N(\mathbf x)\neq 0$ for all $\mathbf x \neq 0$. Corresponding to $L_N(\mathbf x)$ is the elliptic differential operator
$$
\begin{equation*}
\mathcal L_N=\sum_{i, j=1}^N c_{i j}\,\frac{\partial^2}{\partial x_{i}\, \partial x_{j}}.
\end{equation*}
\notag
$$
An example: the operator corresponding to $L_N(\mathbf x)=\sum_{n=1}^N x_n^2$ is the Laplacian $\Delta_N$ in $\mathbb R^N$. Let $E$ be a nonempty subset of $\mathbb R^N$. Recall some notation: $\|f\|_E$ is the uniform ($\sup$-) norm of the bounded (complex-valued) function $f$ on $E$; $\omega_E(f, r)$ is the modulus of continuity of the bounded function $f$ on $E$ (for $E=\mathbb R^N$ we write $\|f\|$ and $\omega (f, r)$, respectively); $\mathrm{BC}(E)$ ($C(E)$) is the space of continuous bounded functions on $E$ (continuous functions on $E$, respectively) with the norm $\|\cdot\|_E$ (we do not use the topology on $C(E)$ for noncompact $E$). For an open set $U \neq \varnothing$ we let $C_0(U)$ ($C^{\infty}_0(U)$) denote the subspace of $\mathrm{BC}(U)$ (of $C^{\infty}(U)$, respectively) consisting of functions with compact support in $U$. For an open set $U \neq \varnothing$ in $\mathbb R^N$, let
$$
\begin{equation*}
\mathcal A_{\mathcal L_N}(U)=\bigl\{ f \in C^2(U)\mid \mathcal L_N f (\mathbf x)=0\ \forall\, \mathbf x \in U\bigr\}.
\end{equation*}
\notag
$$
We say that functions in this class are $\mathcal L_N$-analytic in $U$. It is well known that $\mathcal A_{\mathcal L_N}(U)\subset C^{\infty}(U)$ (for instance, see [4], Theorem 4.4.1). Let $\Phi_{\mathcal L_N} (\mathbf x)$ denote the standard (homogeneous of order $2- N$ for $N\geqslant 3$) fundamental solution of the equation $\mathcal L_N u=0$ in $\mathbb R^N$ (it is well defined for $N\geqslant 3$; for instance, see [4], Theorem 7.1.20, or [5], p. 161). For $N \geqslant 3$, by the $C\mathcal L_N$-capacity of a bounded nonempty set $E \subset \mathbb R^N$ (in the class of continuous functions) we mean the quantity
$$
\begin{equation}
\kappa_{\mathcal L_N} (E)=\sup_{T} \bigl\{|\langle T,1\rangle| \colon \operatorname{Spt} (T) \subset E,\, \Phi_N * T \in \mathrm{BC}(\mathbb R^3),\, \|\Phi_N * T\| \leqslant 1\bigr\},
\end{equation}
\tag{1.1}
$$
where the supremum is taken over all the indicated distributions $T$, $*$ denotes convolution, $\langle T,\varphi \rangle$ is the action of the distribution $T$ on the function $\varphi$ in the class $C^{\infty}(\mathbb R^N)$, and $\operatorname{Spt} (T)$ is the support of the distribution (function or measure) $T$. For $E=\varnothing$ we set $\kappa_{\mathcal L_N}(E)=0$. In terms of these capacities Mazalov [3] obtained criteria for uniform approximation of functions by solutions of the equation $\mathcal L_N u=0$ on compact subsets of $\mathbb R^N$ ($N \geqslant 3$). Section 3 of [6] is devoted to the properties of these capacities. Dimension $N=2$ has special features. Let
$$
\begin{equation*}
\mathcal L_2=c_{11}\, \frac{\partial^2}{\partial x_1^2} +2c_{12}\, \frac{\partial^2}{\partial x_1 \,\partial x_2}+c_{22}\, \frac{\partial^2}{\partial x_2^2}
\end{equation*}
\notag
$$
be an elliptic operator in $\mathbb R^2$, and let $\lambda_1$ and $\lambda_2$ be the roots of the corresponding characteristic equation $c_{11}\lambda^2+2c_{12}\lambda+c_{22}=0$. The operator $\mathcal L_2$ being elliptic is equivalent to the condition $\lambda_1, \lambda_2 \notin \mathbb R$. We say that $\mathcal L_2$ is strongly elliptic if the imaginary parts of $\lambda_1$ and $\lambda_2$ have distinct signs. The following result was established in [7]. Let $\mathcal L_2$ in $\mathbb R^2$ be an elliptic operator that is not strongly elliptic, let $X$ be an arbitrary (nonempty) compact set in $\mathbb R^2$, and let $f \in C(X)$. Then $f$ can be uniformly approximated on $X$ to any accuracy by $\mathcal L_2$-analytic functions defined in (own) neighbourhoods of $X$ if and only if $f$ is $\mathcal L_2$-analytic in the interior $X^{\circ}$ of $X$. For every strongly elliptic operator $\mathcal L_2$ this theorem fails. Relevant approximation criteria were given in [8] in terms of $C\mathcal L_2$-capacities, but these capacities have not been investigated, except in the harmonic case when $\mathcal L_2=\Delta_2$ (see [2]). Our aim here is to extend the main results from [2] to all strongly elliptic operators $\mathcal L_2$. Throughout what follows $\mathbf x=(x_1, x_2, x_3) \in \mathbb R^3$ and $\mathbf x'=(x_1, x_2) \in \mathbb R^2$. For an open ball $B=B(\mathbf x_0, r)$ in $\mathbb R^3_\mathbf x$ (with centre $\mathbf x_0$ and radius $r$) we let $B'$ denote the open disc $B'(\mathbf x'_0, r)$ in $\mathbb R^2_{\mathbf x'}$ (with centre $\mathbf x'_0$ and radius $r$). Fix an arbitrary strongly elliptic operator
$$
\begin{equation*}
\mathcal L'=\mathcal L_2=c_{11}\, \frac{\partial^2}{\partial x_1^2}+2c_{12}\, \frac{\partial^2}{\partial x_1 \,\partial x_2}+c_{22}\, \frac{\partial^2}{\partial x_2^2}
\end{equation*}
\notag
$$
in $\mathbb R^2$. By Lemma 2.1 in [6],
$$
\begin{equation*}
\mathcal L=\mathcal L_3=c_{11}\, \frac{\partial^2}{\partial x_1^2}+2c_{12}\, \frac{\partial^2}{\partial x_1 \,\partial x_2}+c_{22}\, \frac{\partial^2}{\partial x_2^2}+c_{11}\, \frac{\partial^2}{\partial x_3^2}=:\sum_{i, j=1}^3 c_{i j} x_{i} x_{j}
\end{equation*}
\notag
$$
is an elliptic operator in $\mathbb R^3$. Recall that any fundamental solution $\Phi'(\mathbf x')=\Phi_{\mathcal L'} (\mathbf x')$ for $\mathcal L'$ has the following form (for instance, see [9]):
$$
\begin{equation}
\Phi'(\mathbf x')=k_0\log|\mathbf x'|+\Psi_0(\mathbf x'),
\end{equation}
\tag{1.2}
$$
where $k_0=k_0(\mathcal L') \in \mathbb C \setminus \{0\}$ and $\Psi_0$ is a homogeneous function of order $0$ in the class $C^{\infty}(\mathbb R^2 \setminus \{\mathbf 0\})$, which is uniquely defined up to an additive constant. In what follows we fix the unique function $\Psi_0$ such that
$$
\begin{equation}
\|\Psi_0\|_{\mathbb R^2\setminus \{\mathbf 0\}} =\inf_{c \in \mathbb C}\bigl\{\|\Psi_0+c\|_{\mathbb R^2\setminus \{\mathbf 0\}}\bigr\},
\end{equation}
\tag{1.3}
$$
and fix the corresponding fundamental solution $\Phi'$. For $\delta>0$ we also need another fundamental solution, namely,
$$
\begin{equation*}
\Phi'_{\delta}(\mathbf x')=\Phi'\biggl(\frac{\mathbf x'}{\delta}\biggr) =\Phi'(\mathbf x')-k_0 \log(\delta).
\end{equation*}
\notag
$$
Let $B'=B'(\mathbf x'_0, R)$ be a disc and $E'$ be a nonempty open subset of the disc $(1/2) B'=B'(\mathbf x'_0, R/2)$. By definition the $C\mathcal L'$-capacity of the set $E'$ in $B'$ is the quantity
$$
\begin{equation}
\kappa'_{B'}(E')=\sup_{T'} \bigl\{|\langle T', 1\rangle| \colon \operatorname{Spt} (T') \subset E',\, \Phi'_R * T' \in C(\mathbb R^2), \,\|\Phi'_R * T'\|_{B'} \leqslant 1\bigr\}.
\end{equation}
\tag{1.4}
$$
It is easy to show (see Proposition 2.2) that for any objects in the above definition, given a translation $Q$ and a dilation $P$ in $\mathbb R^2$, we have
$$
\begin{equation}
\kappa'_{B'}(E')=\kappa'_{P(B')}(P(E'))=\kappa'_{Q(B')}(Q(E')).
\end{equation}
\tag{1.5}
$$
For example, if $\mathcal L'=\Delta_2$ in $\mathbb R^2$, then $k_0=1/(2\pi)$ and $\Psi_0 \equiv 0$. In this case (as we show in Proposition 2.1) $\kappa'_{B'(\mathbf 0', 1)}(E')=2\pi \kappa_2(E')$, where $\kappa_2(E')$ is the harmonic (Wiener) capacity in $\mathbb R^2$, which for $E'\subset B'(\mathbf 0', 1/2)$ is defined by
$$
\begin{equation}
\kappa_2(E')=\sup_{\mu} \biggl\{\int d\mu \colon \operatorname{Spt} (\mu) \subset E', \,\log|\mathbf x'| * \mu \in C(\mathbb R^2),\,-\log|\mathbf x'| * \mu \leqslant 1 \biggr\},
\end{equation}
\tag{1.6}
$$
where the supremum is taken over all the indicated nonnegative Borel measures $\mu$, and the last inequality holds in the whole of $\mathbb R^2$. This definition is equivalent to the definition of the Wiener capacity presented in [10], Ch. II, § 4, because we can drop the condition that the potential (1.6) is continuous: see the proof of Lemma 2.3 below. Recall that
$$
\begin{equation}
\kappa_2(B'(\mathbf 0', r))=\frac{1}{\log(1/r)}, \qquad r \in \biggl(0, \frac12\biggr)
\end{equation}
\tag{1.7}
$$
(see [10], Ch. 2, § 4). In Proposition 2.2 we prove that for all sets $E'\subset B'(\mathbf 0', 1/2)$) we have
$$
\begin{equation*}
\kappa'_{B'(\mathbf 0', 1)}(E')\geqslant A_0 \kappa_2(E'),
\end{equation*}
\notag
$$
where $A_0 > 0$ depends only on $\mathcal L'$. As a consequence of this and Corollary 3.1 in [11], in Proposition 2.3 we obtain sharp lower bounds for the capacities $\kappa'_{B'(\mathbf 0', 1)}(E')$ in terms of Hausdorff contents. The central result in this paper is as follows. Theorem 1.1. Let $E'\subset B'(\mathbf x'_0, R/2)$ and $E=E'\times [-R, R]_{x_3}$. Then where $\kappa(E)=\kappa_{\mathcal L_3}(E)$ for the operator $\mathcal L_3=\mathcal L$, and $A_1 > 1$ only depends on $\mathcal L'$. We prove this theorem in § 3. We also present there a consequence of it ( Theorem 3.1) for problems in the theory of uniform approximations of functions by solutions of the equation $\mathcal L'v=0$ on compact subsets of $\mathbb R^2$. § 2. Preliminaries and some estimatesLet $\mathcal L$ be the elliptic operator in $\mathbb R^3$ introduces above, let $\Phi=\Phi_\mathcal L$ be its fundamental solutions (in the class $C^{\infty}(\mathbb R^3\setminus \{\mathbf 0\})$ which is homogeneous of order $-1$) and let $\delta > 0$, $B=B(\mathbf a, \delta)$ and $\psi \in C^{\infty}_0(B)$. The operator
$$
\begin{equation*}
g \to V_{\psi}(g)=\Phi*(\psi \mathcal L g), \qquad g \in \mathrm{BC}(B),
\end{equation*}
\notag
$$
is called the Vitushkin-type localization operator corresponding to $\mathcal L$ and the index function $\psi$ (see [12] and [5]). Here the distribution $\psi \mathcal L g$ is set to vanish outside $\operatorname{Spt}(\psi)$. The following property of this operator is well known (for instance, see [6], Lemma 2.2). Lemma 2.1. In the above notation the operator $V_{\psi}$ acts continuously from $\mathrm{BC}(B)$ to $\mathrm{BC}(\mathbb R^3)$. More precisely, there exists a constant $A_2=A_2(\mathcal L)\in (0,+\infty)$ such that $V_{\psi} (g) \in \mathrm{BC}(\mathbb R^3)$ and for all $g \in \mathrm{BC}(B)$. Moreover, $\mathcal L V_{\psi}(g)=\psi \mathcal L g$ (that is, $V_{\psi}$ ‘localizes’ the $\mathcal L$-singularities of $g$ on the support $\operatorname{Spt}(\psi) \subset B$). We also need a similar definition for dimension two. For $\delta >0$ let $B'=B(\mathbf a', \delta)$, and let $\psi \in C^{\infty}_0(B')$. The localization operator
$$
\begin{equation*}
g \to V'_{\psi \delta}(g)=\Phi'_{\delta}*(\psi \mathcal L' g), \qquad g \in \mathrm{BC}(B'),
\end{equation*}
\notag
$$
corresponds to the operator $\mathcal L'$, the disc $B'$ and the index function $\psi$. Its continuity properties can be described as follows (see [9], Lemma 2.5). Lemma 2.2. The operator $V'_{\psi \delta}$ acts continuously from $\mathrm{BC}(B')$ to $\mathrm{BC}(B')$. More precisely, there exists a constant $A_3=A_3(\mathcal L')\in (0,+\infty)$ such that $V'_{\psi \delta} (g) \in C(\mathbb R^2)$ and for all $g \in \mathrm{BC}(B')$. Moreover, $\mathcal L' V'_{\psi \delta}(g)=\psi \mathcal L' g$. Proposition 2.1. Let $\mathcal L'\,{=}\,\Delta_2$ in $\mathbb R^2$. Then $k_0\,{=}\,1/(2\pi)$ and $\Psi_0 \equiv 0$. Furthermore, for each set $E'\subset B'(\mathbf 0', 1/2)$ Proof. It is well known that in the case under consideration $\Phi'(\mathbf x')=(2\pi)^{-1}\log|\mathbf x'|$. For $r>0$ set $B'_r=B'(\mathbf 0', r)$ and
First we prove that $\kappa'(E') \geqslant 2\pi \kappa_2(E')$. Let $\mu$ be some positive measure from the definition (1.6) and let $f=\Phi'*\mu$. Then it is obvious that $f \geqslant -1/(2\pi)$ on $\mathbb R^2$ (and therefore on $B'_1$). Since (because of (1.6) and (1.7)) $\|\mu\|\leqslant 1/\log 2$, for each $\mathbf x'\in \partial B'_1$ we have
$$
\begin{equation*}
f(\mathbf x')=(2\pi)^{-1}\int_{B'_{1/2}}\log|\mathbf x'-\mathbf y'|\,d\mu(\mathbf y') \leqslant \frac{1}{2\pi}.
\end{equation*}
\notag
$$
It remains to use the definition (1.4) for the disc $B'_1$.
Next we prove that $\kappa'(E') \leqslant 2\pi \kappa_2(E')$. Suppose that $\kappa'(E')>0$, for otherwise there is nothing to prove. Fix some $\varepsilon \in (0,1)$ and consider a distribution $T'$ in (1.4) such that $\alpha:=\langle T', 1\rangle=(1-\varepsilon)\kappa'(E')>0$. Here we can assume that $T'$ is real. The function $g=-\log|\mathbf x'|*T'=-2\pi \Phi'*T'$ is harmonic in a neighbourhood of $\mathbb R^2\setminus B'_1$ and does not exceed $2\pi$ on $B'_1$. Furthermore, as $|\mathbf x'|\to+\infty$, it has the negative principal part $-\alpha \log|\mathbf x'|$. Hence $-\log|\mathbf x'|*T' \leqslant 2\pi$ throughout $\mathbb R^2$. It remains to use Lemma 2.3 below. Unfortunately, we could not find a direct reference to its result, although it is known: authors usually prove it for measures, rather than for distributions, and for three or more dimensions. The idea of the proof of this lemma is due to M. Ya. Mazalov. Lemma 2.3. The supremum in (1.6) does not change if arbitrary real distributions are considered in place of positive measures $\mu$ and $\langle T', 1\rangle$ is taken in place of ${\displaystyle\int d\mu}$. Proof. It is clear that we can limit ourselves to compact sets $E'$. For a contradiction, suppose that there exists a real distribution $T'$ such that $\operatorname{Spt}(T')\subset E'$, $g=-\log|\mathbf x'|*T' \in C(\mathbb R^2)$, $g(\mathbf x')\leqslant 1$ in $\mathbb R^2$ and $\langle T', 1\rangle=\kappa_2(E')+\alpha$ for some $\alpha >0$. It follows from the Luzin $C$-property (see [10], Theorem 1.8) that in the definition (1.6) we can drop the condition that the potential $\log|\mathbf x'| * \mu$ is continuous. Hence it is easy to show that there exists a compact set $K$ with smooth boundary such that $E' \subset K \subset B'_{1/2}$ and $\kappa_2(K)<\kappa_2(E')+\alpha$. Let $\nu$ be an equilibrium measure on $K$ (see [10], Ch. 2, § 4, Theorem $2.6'$) and let $\Omega$ be the unbounded component of the set $\mathbb R^2\setminus K$, so that $\mathbb R^2 \setminus B'_{1/2} \subset \Omega$. It is well known that the potential $f=-\log|\mathbf x'|*\nu$ is continuous on $\overline{\Omega}$ and equal to $1$ on $\partial \Omega$ (this follows from the connection between $f$ and the Green’s function of $\Omega$: see [13], Ch. 7, § 3). As $|\mathbf x'| \to+\infty$, the function $f(\mathbf x')$ has the order of $-(\log|\mathbf x'|) \|\nu\|$, where $|\nu\|=\kappa_2(K)<\kappa_2(E')+\alpha$, while $g(\mathbf x')$ has the order of $-(\log|\mathbf x'|) \langle T', 1\rangle$, where $\langle T', 1\rangle=\kappa_2(E')+\alpha$, so it follows from the maximum principle in $\Omega \cap B'_R$ (where $R$ is sufficiently large) that
Now let $T'_1$ be a distribution in $B'_{1/2}$ with potential $g_1=-\log|\mathbf x'|*T'_1$, and let $r\in (1/2, 1)$. Then we claim that
$$
\begin{equation}
\langle T'_1, 1\rangle=\biggl(2\pi r \log\frac1r\biggr)^{-1}\int_{\partial B'_r} g_1(\mathbf x')\,d\ell_{\mathbf x'},
\end{equation}
\tag{2.4}
$$
where $d\ell$ is the arc length element on $\partial B'_r$.
To prove (2.4) it is convenient to express vectors in $\mathbb R^2$ in the complex form. First we establish (2.4) for $T'_1=\delta_a$, where $a \in B'_{1/2}$ and $\delta_a$ is the unit measure at $a$. In other words, $g_1 (z)=-\log|z-a|$. Let $b=r^2/{\overline a}$ be the point symmetric to $a$ relative to the circle $\partial B'_r$. Then for each $z=re^{i\theta} \in \partial B'_r$ we have
$$
\begin{equation*}
\frac{|z-a|}{|z-b|}=\frac{|re^{i\theta}-a|}{|re^{i\theta}-r^2/{\overline a}|}=\frac{|a|\,|re^{i\theta}-a|}{r|{\overline a} e^{i\theta}-r|}=\frac{|a|}{r},
\end{equation*}
\notag
$$
so that $g_1 (z)=-\log|z-a|=-\log|z-b|-\log|a|+\log(r)$, and by the mean value theorem for harmonic functions we obtain
$$
\begin{equation*}
(2\pi r)^{-1}\int_{\partial B'_r} g_1(z) d\ell_z=g_1(0)=-\log|b|-\log|a|+\log(r)=-\log(r),
\end{equation*}
\notag
$$
as required. Now, if $T'_1$ is a finite measure on $B'_{1/2}$, then to complete the proof of (2.4) it remains to use Fubini’s theorem on the right-hand side of (2.4), by substituting the appropriate integral (convolution) in place of $g_1$. In the general case we use the method of regularization first (see the reasoning around formulae (3.1) and (3.2) in § 3, but here we have to deal with $\mathbb R^2$ in a similar way).
Finally, applying (2.4) to the distributions $\nu$ and $T'$, in view of (2.3) we obtain which gives a contradiction.Proposition 2.2. For any operator $\mathcal L'$ under consideration the following hold: (1) for any $B'=B'(\mathbf x'_0, R)$, $E' \subset (1/2) B'$, and any dilation $P$ and translation $Q$ in $\mathbb R^2$,
$$
\begin{equation*}
\kappa'_{B'}(E')=\kappa'_{P(B')}(P(E'))=\kappa'_{Q(B')}(Q(E'));
\end{equation*}
\notag
$$
(2) for any set $E'\subset B'(\mathbf 0', 1/2)$ where $A_0 > 0$ depends only on $\mathcal L'$. Proof. We prove (1) for a dilation $P$ with coefficient $p>0$ (the part concerning the translation $Q$ is obvious). To each distribution $T'$ satisfying the assumptions of the definition (1.4) for $E'$ in $B'(\mathbf x'_0, R/2)$ we can bijectively assign the distribution $PT'$ acting by the formula
$$
\begin{equation*}
\langle PT'(\mathbf x'), \psi(\mathbf x') \rangle =\langle T'(\mathbf y'), \psi(P(\mathbf y'))\rangle, \qquad \psi \in C^{\infty}(\mathbb R^2),
\end{equation*}
\notag
$$
which satisfies the assumptions of the definition (1.4) for $PE'$ in $B'(P(\mathbf x'_0), pR/2)$. Furthermore, $\langle T', 1\rangle=\langle PT', 1\rangle$ and (1) is proved.
Now we prove (2). Given some $\Psi_0$ in (1.3), set $I=\|\Psi_0\|$. Let $\mu$ satisfy the assumptions of the definition (1.6). Then it follows from the proof of Proposition 2.1 that $\|\log(\mathbf x')*\mu\|_{B'_1}\leqslant 1$, which yields the inequality $\|\Phi'*\mu\|_{B'_1}\leqslant |k_0|+I\|\mu\|$ directly. Hence $\kappa'(E')=\kappa'_{B'_1}(E')\geqslant \|\mu\|/(|k_0|+I\|\mu\|)$. It remains to let $\|\mu\|$ approach $\kappa_2(E')$ and use the inequality $\kappa_2(E')\leqslant \kappa_2(B'_{1/2})=1/\log(2)$. Then we can take $(|k_0|+I/\log(2))^{-1}$ as $A_0$. The proposition is proved. If in (1.4) we take positive measures $\mu$ in place of distributions $T'$ (and limit ourselves to $\mathbf x'_0=\mathbf 0'$ and $R=1$); then we obtain another capacity useful in applications ($E' \subset B'_{1/2}$):
$$
\begin{equation}
\kappa'_+(E')=\sup_{\mu} \biggl\{\int d\mu \colon \operatorname{Spt} (\mu) \subset E',\,\Phi' * \mu \in C(\mathbb R^2),\,\|\Phi' * \mu\|_{B'_1} \leqslant 1\biggr\}.
\end{equation}
\tag{2.5}
$$
Proposition 2.3. There exist $r_0=r_0(\mathcal L')\in (0, 1/2]$ and $A'_0=A'_0(\mathcal L') > 0$ such that for all $E' \subset B'(\mathbf 0', r_0)=:B'_{r_0}$ If $\mathcal L'=\Delta_2$, then $r_0=1/2$ and $\kappa'_+(E')=2\pi \kappa_2(E')$. Proof. The left-hand estimate is proved similarly to part (2) of Proposition 2.2. We prove the right-hand estimate. Let $\mu$ be some positive measure from the definition (2.5). It is sufficient to show that $f=-\log(\mathbf x')*\mu$ satisfies $f(\mathbf x') \leqslant A'_0$ on $\mathbb R^2$. Moreover, as the maximum principle holds outside $B'_{1/2}$ (because $f(\infty)\leqslant 0$) it is sufficient to prove the inequality $f(\mathbf x') \leqslant A'_0$ on $B'_{1/2}$. As before, in (1.2), let $\Phi'(\mathbf x')=k_0\log|\mathbf x'|+\Psi_0(\mathbf x')$ and $I=\|\Psi_0\|$. Then, since we have $\|\mu\|\leqslant 1/(|k_0|\log(1/r_0)-I)$. Now, fixing $r_0< \exp(-I/|k_0|)$, for ${\mathbf x'\in B'_{1/2}}$ we obtain For $\mathcal L'=\Delta_2$, when $k_0=1/(2\pi)$ and $I=0$, we can take $r_0=1/2$; then $f(\mathbf x')\leqslant 2\pi$ in $B'_{1/2}$, as required. The proof is complete. Since the capacity $\kappa_2$ is semiadditive (that is, $\kappa_2(E'_1\cup E'_2)\leqslant \kappa_2(E'_1)+\kappa_2(E'_2)$ for all $E'_1$ and $E'_2$ in $B'_{1/2}$), from Proposition 2.3 we obtain that the capacities $\kappa'_+$ have the important property of subadditivity. More precisely, there exist $r_0$ (specified above) and $A=A(\mathcal L')\geqslant 1$ such that for all $E'_1$ and $E'_2$ in $B'_{r_0}$ we have
$$
\begin{equation*}
\kappa'_+(E'_1\cup E'_2)\leqslant A(\kappa'_+(E'_1)+\kappa'_+(E'_2)).
\end{equation*}
\notag
$$
The following problems are of particular interest in this connection. Problem 2.1. Is $\kappa'$ comparable with $\kappa'_+$? That is, does there exist $A=A(\mathcal L')\geqslant 1$ such that for all $E'$ in $B'_{1/2}$ (or at least in $B'_{r_0}$)? In applications (for example, in Theorem 3.1 below), in place of the $C\mathcal L'$-capacity (1.4) it is convenient to use the $L^{\infty}\mathcal L'$-capacity
$$
\begin{equation*}
\gamma'_{B'}(E')=\sup_{T'} \bigl\{|\langle T', 1\rangle| \colon \operatorname{Spt} (T') \subset E',\, \Phi'_R * T'\in L_{\mathrm{loc}}^{\infty}(\mathbb R^2),\, \|\Phi'_R * T'\|_{L^{\infty}(B')} \leqslant 1\bigr\},
\end{equation*}
\notag
$$
where the spaces $L^{\infty}$ involved are associated with the Lebesgue measure in $\mathbb R^2$. This capacity has the following important property (which is not yet known for $C\mathcal L'$-capacity!). It is established similarly to Proposition 3.1 in [6]. Proposition 2.4. For any disc $B'$ and any compact set $ K'\subset (1/2)B'$ where $U'_{\delta}(K')$ is the open $\delta$-neighbourhood of $K'$ in $\mathbb R^2$. In the case when $\mathcal L'=\Delta_2$ the capacities $\gamma'_{B'}$ and $\kappa'_{B'}$ coincide (as mentioned in the proof of Lemma 2.3). Using standard arguments we can show that for all $\mathcal L'$ and open sets $E' \subset (1/2)B'$ we have The following important question is also still open.Problem 2.3. Is $\gamma'_{B'}$ comparable with $\kappa'_{B'}$? More precisely, does there exist $A=A(\mathcal L')\geqslant 1$ such that for all $E'$ in $(1/2)B'$ (or at least in $B'_{r_0}$)? Recall the definition of the Hausdorff $p$-content ($p \in (0, 2]$) of a bounded subset $E'$ of $\mathbb R^2$: where the lower bound is taken over all covers $\{B'_{(j)}\}$ of $E$ by discs (each $\{B_{(j)}\}$ is an almost countable cover of $E$ by discs $B'_{(j)}$ of radii $r_j$ in $\mathbb R^2$).From Proposition 2.3 and Eiderman’s bounds for the Wiener capacity $\kappa_2$ in [11], Corollary 3.1 (their accuracy was also discussed in [11]), we obtain directly the following lower bounds for the capacities $\kappa'_+$ (and therefore for $\kappa'$). Proposition 2.5. There exists a constant $A=A(\mathcal L')>1$ with the following properties. For each $p \in (0, 2]$ and any set $E' \subset B'_{1/2}$ such that $\kappa'_+(E') \leqslant p/A$ the relation holds. § 3. Proof of Theorem 1.1. Corollaries Proof of Theorem 1.1. Let $P$ be a dilation with coefficient $p>0$ in $\mathbb R^3$. Since the fundamental solution $\Phi$ for the operator $\mathcal L$ is homogeneous of order $-1$, it is easy to show that $\kappa(P(E))=p\kappa(E)$ for each bounded set $E \subset \mathbb R^3$. Furthermore, the capacities $\kappa$ and $\kappa'_{B'}(E')$ are invariant under translations (when the sets ${B'=B'(\mathbf x'_0, R)}$ and $E'\subset (1/2) B'$ are translated likewise). Hence, in view of (1.8), in this proof we can assume without loss of generality that $\mathbf x'_0=\mathbf 0'$ and $R=1$.
For brevity let $B_{r}=B(\mathbf 0, r)$ and $B_r'=B'(\mathbf 0', r)$, $r>0$, and let $\kappa'(E')=\kappa'_{B_1'}(E')$. We prove the left-hand estimate in (1.8). We find the distribution $T'$ satisfying the assumptions in the definition (1.4) (the case when $\mathbf x'_0=\mathbf 0'$ and $R=1$) and such that $\langle T',1 \rangle=\kappa'(E')/2$. Set $f=\Phi'*T' \in C(\mathbb R^2)$, $\|f\|_{B'_1} \leqslant 1$ and $F(\mathbf x',x_3)=f(\mathbf x')$. Then $\|F\|_{B_1}\leqslant 1$. We take a function $\psi_{12} \in C^{\infty}_0(B'_{2/3})$ such that $0\leqslant\psi_{12}\leqslant 1$, $\psi_{12}=1$ in $B'_{1/2}$ and $\|\nabla^2\psi_{12}\|\leqslant A$. Here and below the parameter $A>1$, which only depends on $\mathcal L'$, can take different values in different relations. We also choose $\psi_3 \in C^{\infty}_0((-2/3, 2/3))$ such that $0\leqslant\psi_3\leqslant 1$, $\psi_3=1$ on $(-1/2, 1/2)$ and $\|\psi_3''\|\leqslant A$. Let $\psi (\mathbf x)=\psi(\mathbf x',x_3)=\psi_{12}(\mathbf x')\psi_3(x_3)$ for $\mathbf x\in\mathbb R^3$. Then $\psi \in C^{\infty}_0(B_1)$ and $\|\nabla^2 \psi\|\leqslant A$. We apply the Vitushkin localization operator in $\mathbb R^3$ by setting $F_{\psi} = V_{\psi}(F)$. Then by Lemma 2.1 we have
$$
\begin{equation*}
\|F_\psi\|\leqslant A \|\nabla^2 \psi\| \, \|F\|_{B_1} \leqslant A^2,
\end{equation*}
\notag
$$
and moreover,
$$
\begin{equation*}
\mathcal L F_{\psi}(\mathbf x',x_3)=\psi_{12}(\mathbf x')\psi_3(x_3) \mathcal L F(\mathbf x)=\psi_{12}(\mathbf x')\psi_3(x_3) \mathcal L' f(\mathbf x')
\end{equation*}
\notag
$$
in the sense of distributions. Let $T=\mathcal L F_{\psi}$. It is clear that $\operatorname{Spt} T\subset E$, $F_{\psi}=\Phi*T$ and $\|F_{\psi}\| \leqslant A^2$. Thus,
$$
\begin{equation*}
\begin{aligned} \, A^2 \kappa(E) &\geqslant \langle T, 1\rangle=\langle \psi_{12}(\mathbf x')\psi_3(x_3) \mathcal L' f(\mathbf x'),1\rangle \\ &=\langle T',1\rangle \int_{-1}^1\psi_3(x_3)\,dx_3 \geqslant 2^{-1} \kappa'(E'), \end{aligned}
\end{equation*}
\notag
$$
so that as required.
Next we prove the right-hand inequality in (1.8). We have $E \subset B'_{1/2}\times [-1,1] \subset B_{5/4}$. By the definition of $\kappa(E)$ there exists a distribution $T$ with (compact) support $\operatorname{Spt}(T)\subset K \subset E$ (where $K=K'\times [-R,R]$ is a compact set) such that $F=\Phi*T \in C(\mathbb R^3)$, $\|F\|\leqslant 1$ and $\langle \mathcal L F, 1\rangle=\langle T, 1\rangle=\kappa(E)/2$. Fix a function $\varphi_1 \in C^{\infty}_0(B_1)$ such that $0 \leqslant \varphi_1\leqslant 1$, $\displaystyle\int_{B_1}\varphi_1(\mathbf x)\, d\mathbf x=1$ and $\|\nabla^2\varphi_1\|\leqslant A$. For $\varepsilon>0$ set $\varphi_{\varepsilon}(\mathbf x)=\varepsilon^{-3}\varphi_1(\mathbf x/\varepsilon)$, so that $\displaystyle\int_{B_{\varepsilon}}\varphi_{\varepsilon}(\mathbf x)\, d\mathbf x=1$ and $\|\nabla^2\varphi_{\varepsilon}\|\leqslant A/\varepsilon^2$. Let $T_{\varepsilon}=T*\varphi_{\varepsilon}$, $F_{\varepsilon}=F*\varphi_{\varepsilon}=\Phi*T_{\varepsilon}$. Then $T_{\varepsilon}$ is a regular distribution, that is, it has the form
$$
\begin{equation}
\langle T_{\varepsilon}, \psi \rangle =\int h_{\varepsilon}(\mathbf x) \psi (\mathbf x)\,d \mathbf x, \qquad \psi \in C^{\infty}(\mathbb R^N),
\end{equation}
\tag{3.1}
$$
where $h_{\varepsilon} \in C^{\infty}_0(\mathbb R^3)$ is some (generally speaking, complex-valued) function. In addition,
$$
\begin{equation}
\int_{\mathbb R^3} h_{\varepsilon}(\mathbf x)\, d \mathbf x=\langle T_{\varepsilon}, 1 \rangle =\langle T, \varphi_{\varepsilon}*1 \rangle=\langle T, 1 \rangle=\frac{\kappa(E)}2.
\end{equation}
\tag{3.2}
$$
For $0<\varepsilon < 3/2-5/4=1/4$ set $G_{\varepsilon}(\mathbf x)=F_{\varepsilon}(\mathbf x)-F_{\varepsilon}(\mathbf x-(3,0,0))$. This is an $\mathcal L$-analytic function outside $U_{\varepsilon}(K\cup K_3)$, where $K_3=\{\mathbf x+(3,0,0)\mid \mathbf x \in K\}$ and $U_{\varepsilon}(Q)$ is the $\varepsilon$-neighbourhood of $Q$ in $\mathbb R^3$. Consider the function
$$
\begin{equation}
g_{\varepsilon}(\mathbf x)=\int_{-\infty}^{+\infty} G_{\varepsilon}(\mathbf x', x_3+t)\,dt.
\end{equation}
\tag{3.3}
$$
The uniform convergence of this integral on compact subsets of $\mathbb R^3$ follows from standard estimates (for instance, see [6], (2.6) and Lemma 2.3):
$$
\begin{equation*}
|G_{\varepsilon}(\mathbf x)|\leqslant\frac{A}{|\mathbf x|^2+1},
\end{equation*}
\notag
$$
moreover, $g_{\varepsilon}(\mathbf x)=g_{\varepsilon}(\mathbf x')$ is independent of $x_3$ and $\mathcal L'$-analytic outside the set $U'_{\varepsilon}(K'\cup K'_3)$ (the latter holds by Theorem 4.4.2 in [4] on the limit of $\mathcal L$-analytic functions, because the partial sums in the integral (3.3) converge locally uniformly), where $K'_3=\{\mathbf x'+(3,0)\mid \mathbf x' \in K'\}$. Hence
$$
\begin{equation*}
|g_{\varepsilon}(\mathbf x')|\leqslant \frac{A}{\sqrt{|\mathbf x'|^2+1}}\leqslant A.
\end{equation*}
\notag
$$
Moreover,
$$
\begin{equation*}
\begin{aligned} \, \mathcal L' g_{\varepsilon}(\mathbf x') &=\int_{-\infty}^{+\infty} \mathcal L G_{\varepsilon}(\mathbf x', x_3+t)\,dt \\ &=\int_{-\infty}^{+\infty} \bigl(h_{\varepsilon}(\mathbf x', x_3+t) -h_{\varepsilon}(\mathbf x'-(3,0), x_3+t)\bigr)\,dt \\ &=\int_{-\infty}^{+\infty} \bigl( h_{\varepsilon}(\mathbf x', t)- h_{\varepsilon}(\mathbf x'-(3,0), t) \bigr)\,dt =: h'_{\varepsilon}(\mathbf x')-h'_{\varepsilon}(\mathbf x'-(3,0)). \end{aligned}
\end{equation*}
\notag
$$
Consider now $\psi \in C^{\infty}_0(B'_1)$ such that $0 \leqslant \psi \leqslant 1$, $\psi=1$ in $B'(\mathbf 0', 3/4)$ and $\|\nabla^2 \psi\|\leqslant A$, and look at the localization
$$
\begin{equation*}
f_{\varepsilon}=V_{\psi 1} g_{\varepsilon}=\Phi'*(\psi \mathcal L' g_{\varepsilon})=\Phi'*(\psi (\mathbf x') (h'_{\varepsilon}(\mathbf x')-h'_{\varepsilon}(\mathbf x'-(3,0))= \Phi'*(h'_{\varepsilon}).
\end{equation*}
\notag
$$
From Lemma 2.2 we obtain $\|f_{\varepsilon}\|_{B'_1} \leqslant A$. Furthermore, $\operatorname{Spt}(h'_{\varepsilon}) \subset U'_{\varepsilon}(E')$ and
$$
\begin{equation*}
\langle h'_{\varepsilon}, 1 \rangle =\int_{\mathbb R^3} h_{\varepsilon}(\mathbf x)\,d\mathbf x=\frac{\kappa(E)}2.
\end{equation*}
\notag
$$
Finally, as $\varepsilon \to 0$, we have $G_{\varepsilon}(\mathbf x) \to G(\mathbf x)=F(\mathbf x)-F(\mathbf x-(3,0,0))$ uniformly on $\mathbb R^3$, $\displaystyle g_{\varepsilon}(\mathbf x') \to g(\mathbf x')=\int_{-\infty}^{+\infty} G(\mathbf x', t)\,dt $ uniformly on $\mathbb R^2$, and therefore ${f_{\varepsilon}(\mathbf x') \to f(\mathbf x')}$ uniformly on compact subsets of $\mathbb R^2$, where $f$ is a continuous function on $\mathbb R^2$ which is $\mathcal L'$-analytic outside $K'$ and satisfies $\|f\|_{B'_1} \leqslant A$. We fix $\psi_1 \in C^{\infty}_0(2B'_1)$ such that $\psi_1 \equiv 1$ on $B'_1$; then
$$
\begin{equation*}
\langle h'_{\varepsilon}, 1 \rangle=\langle \mathcal L'f_{\varepsilon}, 1\rangle=\langle \mathcal L'f_{\varepsilon}, \psi_1\rangle=\langle f_{\varepsilon}, \mathcal L'\psi_1\rangle\to \langle f, \mathcal L'\psi_1\rangle=\langle \mathcal L'f, 1\rangle \geqslant \frac{\kappa(E)}2.
\end{equation*}
\notag
$$
Therefore, by the definition of $\kappa'(E')$ we have
$$
\begin{equation*}
A \kappa'(E') \geqslant |\langle \mathcal L' f, 1 \rangle|= \frac{\kappa(E)}2,
\end{equation*}
\notag
$$
which completes the proof of Theorem 1.1. Theorem 1.1 and the estimate in Proposition 3.2 (1) in [6] yield directly the following upper bound for $\kappa'_{B'(\mathbf x'_0, R)}$. In [6], Corollary 1.1, we obtained the following (individual) criterion for uniform approximability by $\mathcal L'$-analytic functions. Let $X'$ be a compact set in $\mathbb R^2$, let $\rho >0$, and let $X=X' \times [-\rho, \rho]_{x_3}$. Then for each function $f \in C_0(\mathbb R^2)$ the following conditions are equivalent: (a) there exists a sequence of functions $\{f_n\}_{n=1}^{+\infty}$ each of which is $\mathcal L'$-analytic in an (own) neighbourhood of $X'$, such that $f_n \to f$ uniformly on $X'$; (b) there exist $\lambda \geqslant 1$ and a function $\omega (r) \to 0$ as $r \to 0$ such that for each open ball $B$ with centre $\mathbf a=(\mathbf a', 0)$ and radius $r$ in $\mathbb R^3$ (and for $\lambda B=B(\mathbf a, \lambda r)$)
$$
\begin{equation*}
\biggl| \frac{1}{\pi r^2} \int_{B'} f(\mathbf x') \frac{L'(\mathbf x'-\mathbf a')+(1+c_{22})(|\mathbf x'- \mathbf a'|^2-r^2)}{\sqrt{r^2-|\mathbf x'-\mathbf a' |^2}} \,d\mathbf x' \biggr| \leqslant\omega (r) \kappa (\lambda B \setminus X).
\end{equation*}
\notag
$$
If (a) is satisfied, then (b) holds for $\lambda=1$ and $\omega (r)=A\omega (f, r)$. Here $L'$ is the symbol of $\mathcal L'$ and the capacity $\kappa$ is defined in terms of $\mathcal L$ (see (1.1) for $N=3$ and $\mathcal L=\mathcal L_3$) and $A=A(\mathcal L) \in (0,+\infty)$. This result and Theorem 1.1 yield the following. Theorem 3.1. Let $X'$ be a compact set in $\mathbb R^2$. Then for each $f \in C_0(\mathbb R^2)$ the following conditions are equivalent: ($\mathrm a'$) there exists a sequence of functions $\{f_n\}_{n=1}^{+\infty}$ that are $\mathcal L'$-analytic in (own) neighbourhoods of $X'$ such that $f_n \to f$ uniformly on $X'$; ($\mathrm b'$) there exist $\lambda \geqslant 1$ and a function $\omega$, $\omega (r) \to 0$ as $r \to 0$, such that for each open ball $B'=B'(\mathbf a', r)$ (and for $\lambda B'=B(\mathbf a', \lambda r)$)
$$
\begin{equation*}
\biggl| \frac{1}{\pi r^2} \int_{B'} f(\mathbf x') \frac{L'(\mathbf x'-\mathbf a')+(1+c_{22})(|\mathbf x'- \mathbf a'|^2-r^2)}{\sqrt{r^2-|\mathbf x'-\mathbf a' |^2}} \,d\mathbf x' \biggr| \leqslant\omega (r)r\kappa'_{2\lambda B'} (\lambda B' \setminus X').
\end{equation*}
\notag
$$
If ($\mathrm a'$) holds, then ($\mathrm b'$) also holds for $\lambda\!=\!1$ and $\omega (r)\!=\!A\omega (f, r)$, ${A\!=\!A(\mathcal L') \!\in\! (0,+\infty)}$. Proof. Under the assumptions and in the notation of these two criteria we can limit ourselves to the case when $2\lambda r < \rho$. In this case
$$
\begin{equation*}
\lambda B \setminus X \subset (\lambda B' \setminus X')\times (-r, r) \subset 2\lambda B \setminus X,
\end{equation*}
\notag
$$
and the fact that condition $(\mathrm b')$ coincides with $(\mathrm b)$ (in the above criterion) is a consequence of Theorem 1.1. Theorem 3.1 is proved. Theorem 3.1 is an analogue of Vitushkin’s well-known criterion for uniform rational approximability from [12], Ch. IV, § 2, Theorem 2, and Mazalov’s criterion in [8], Theorem 4. In a standard way, from Theorem 3.1, similarly to Theorem 4 in [8], we obtain the following criterion for approximation in function classes. Corollary 3.2. Let $X'$ be a compact set in $\mathbb R^2$. Then the following conditions are equivalent: (1) for each function $f \in C(X')\cap \mathcal A_{\mathcal L'}((X')^{\circ})$ there exists a sequence of functions $\{f_n\}_{n=1}^{+\infty}$ which are $\mathcal L'$-analytic in (own) neighbourhoods of $X'$ such that $f_n \to f$ uniformly on $X'$; (2) $\kappa'_{2B'} (B' \setminus (X')^{\circ})=\kappa'_{2B'} (B' \setminus X')$ for each disc $B'$; (3) there exists $\lambda \geqslant 1$ such that for each open disc $B'=B'(\mathbf a', r)$ the relation $\kappa'_{2B'} (B' \setminus (X')^{\circ}) \leqslant A\kappa'_{2\lambda B'} (\lambda B' \setminus X')$ holds. The author is deeply grateful to the referees for their efforts to read this paper and for a number of useful comments they made. 
Citation in format AMSBIB
\Bibitem{Par22} |
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