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This article is cited in 1 scientific paper (total in 1 paper) Capacities commensurable with harmonic ones M. Ya. Mazalovab a National Research University "Moscow Power Engineering Institute", Smolensk, Russia b Saint Petersburg State University, St. Petersburg, Russia
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     § 1. IntroductionLet $\mathbf x=(x_1,\dots,x_N) \in \mathbb R^N$, where $N\geqslant2$, let $|\mathbf x|=(x_1^2+\dots+x_N^2)^{1/2}$, and let
$$
\begin{equation}
L(\mathbf x)=L_N(\mathbf x)=\sum_{i, j=1}^N c_{i j} x_{i} x_{j},
\end{equation}
\tag{1.1}
$$
be an arbitrary second-order homogeneous polynomial in $\mathbb R^N$ with constant complex coefficients $c_{i j}=c_{ji}$ which satisfies the ellipticity condition: $L(\mathbf x)\neq 0$ for $\mathbf x \neq \mathbf 0$. This polynomial is the symbol of the corresponding elliptic differential operator
$$
\begin{equation}
\mathcal L=\mathcal L_N =\sum_{i, j=1}^N c_{ij} \frac{\partial^2}{\partial x_{i} \,\partial x_{j}};
\end{equation}
\tag{1.2}
$$
for example, $\sum_{n=1}^N x_n^2$ corresponds to the Laplace operator $\Delta=\Delta_N$ in $\mathbb R^N$. The case $N=2$ has special features. Let
$$
\begin{equation}
\mathcal L=\mathcal L_2=c_{11}\frac{\partial^2}{\partial x_1^2} + 2c_{12}\frac{\partial^2}{\partial x_1 \,\partial x_2} + c_{22}\frac{\partial^2}{\partial x_2^2}
\end{equation}
\tag{1.3}
$$
be an elliptic operator in $\mathbb R^2$, and let $\lambda_1$ and $\lambda_2$ be the roots of the corresponding characteristic equation $c_{11}\lambda^2 + 2c_{12}\lambda + c_{22}=0$. It is obvious that $\mathcal L_2$ is elliptic if and only if $\lambda_1$ and $\lambda_2$ are not real. The operator $\mathcal L_2$ is strongly elliptic if the imaginary parts of $\lambda_1$ and $\lambda_2$ have different signs, and it is nonstrongly elliptic if these imaginary parts have the same sign. The properties of solutions of the equation $\mathcal L_2f=0$ are fundamentally different in the strong and nonstrong elliptic cases; in particular, this can be seen in the investigation of removable singularities, in conditions for the solvability of boundary value problems, or in the development of criteria for the uniform approximability of functions by solutions of these equations. The main reason for these differences consists in the structure of fundamental solutions. Recall that a fundamental solution $\Phi_\mathcal L(\mathbf x)=\Phi_{\mathcal L_2} (\mathbf x)$ of the operator $\mathcal L_2$ in $\mathbb R^2$ has the form
$$
\begin{equation}
\Phi_\mathcal L(\mathbf x)=k_0\log|\mathbf x| + \Psi_0(\mathbf x),
\end{equation}
\tag{1.4}
$$
where $\Psi_0$ is a homogeneous function of degree $0$ in the class $C^{\infty}(\mathbb R^2 \setminus \{0\})$, which is uniquely defined up to an additive constant, and $k_0$ is a complex constant; furthermore, $k_0\ne0$ if and only if the operator $\mathcal L_2$ is strongly elliptic. Hence only in the case of a nonstrongly elliptic operator is a fundamental solution bounded. The precise expression for $\Phi$ in its dependence on $\mathcal L_2$ was presented in Proposition 2.2 in [1]. For $N\geqslant3$ elliptic operators in $\mathbb R^N$ are closely connected with strongly elliptic operators in $\mathbb R^2$. If $L(x_1,x_2,\dots, x_N)$ is the symbol of an elliptic operator, then $L(x_1,x_2,0,\dots,0)$ is the symbol of a strongly elliptic operator in $\mathbb R^2$ (see [2], Lemma 2). For $N\geqslant3$ a fundamental solution $\Phi_\mathcal L(\mathbf x)=\Phi_{\mathcal L_N} (\mathbf x)$ of the operator $\mathcal L_N$ is a homogeneous function of degree $2-N$ in the class $C^{\infty}(\mathbb R^N \setminus \{0\})$; a precise expression for it in its dependence on $\mathcal L_N$ was presented in Theorem 1 in [2]. The main case considered here is $N\geqslant3$, when $\lim_{|\mathbf x|\to\infty}\Phi_{\mathcal L}(\mathbf x)=0$. Recall the definitions of the capacities $\gamma_{\mathcal L}$ and $\alpha_{\mathcal L}$, as introduced by Harvey and Polking [3] for the description of removable singularities of solutions of the equation $\mathcal Lf=0$ in the classes of bounded and continuous functions, respectively. Note that in terms of these capacities one can state a criterion for the uniform approximation of a function by solutions of the equation $\mathcal Lf=0$ on compact subsets of $\mathbb R^N$ for $N\geqslant3$ (see [4]). Let $K\subset\mathbb R^N$ be a compact set, where $N\geqslant3$, and let
$$
\begin{equation}
\gamma_{\mathcal L}(K)=\sup_T\bigl\{|\langle T\mid 1\rangle|\colon \operatorname{Spt}(T)\subset K,\,\|T*\Phi_{\mathcal L}\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1\bigr\},
\end{equation}
\tag{1.5}
$$
where $\operatorname{Spt}(T)$ is the support of the distribution (function or measure) $T$, $\sup$ is taken over all the distributions $T$ indicated, $*$ is the convolution operator, and $\langle T\mid \varphi\rangle$ is the action of $T$ on the function $\varphi$ in the class $C^{\infty}(\mathbb R^N)$; for $K=\varnothing$ we set $\gamma_{\mathcal L}(K)=0$. Note that $T*\Phi_{\mathcal L}\in C^{\infty}(\mathbb R^N\setminus \operatorname{Spt}(T))$. We give comments to Definition (1.5) in § 2; in particular, (1.5) is equivalent to the following definition:
$$
\begin{equation}
\gamma_{\mathcal L}(K)=\sup_f\Bigl\{|\langle \mathcal Lf\mid 1\rangle|\colon \operatorname{Spt}(\mathcal Lf)\subset K,\,\|f\|_{\mathrm L^{\infty}(\mathbb R^N\setminus\operatorname{Spt}(T))}\leqslant1,\,\lim_{\mathbf x\to\infty}f(\mathbf x)=0\Bigr\}.
\end{equation}
\tag{1.6}
$$
The capacity $\alpha_{\mathcal L}(K)$ is defined in accordance with (1.5), where we assume additionally that $T*\Phi_{\mathcal L}\in C(\mathbb R^N)$. The capacities $\gamma_{{\mathcal L,+}}$ and $\alpha_{{\mathcal L,+}}$ are defined similarly to $\gamma_{\mathcal L}$ and $\alpha_{\mathcal L}$, under the proviso that the distributions $T$ are nonnegative Borel measures with support on $K$; in particular,
$$
\begin{equation}
\gamma_{{\mathcal L,+}}(K)=\sup_{\mu}\bigl\{\|\mu\|\colon \operatorname{Spt}(\mu)\subset K,\, \mu\geqslant0,\,\|\mu*\Phi_{\mathcal L}\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1\bigr\},
\end{equation}
\tag{1.7}
$$
where $\|\mu\|$ is the total mass of $\mu$. In all above cases the capacity of a bounded (Borel) set is defined to be the supremum of the capacities of its compact subsets. It is obvious from the definitions that each of the capacities $\gamma_{\mathcal L}$, $\alpha_{\mathcal L}$, $\gamma_{{\mathcal L,+}}$ and $\alpha_{{\mathcal L,+}}$ is a nondecreasing function of the set and for each bounded set $U$ we have the inequalities
$$
\begin{equation}
\begin{gathered} \, \gamma_{\mathcal L}(U)\geqslant \alpha_{\mathcal L}(U), \qquad \gamma_{{\mathcal L,+}}(U)\geqslant \alpha_{{\mathcal L,+}}(U), \\ \gamma_{\mathcal L}(U)\geqslant\gamma_{{\mathcal L,+}}(U),\qquad \quad \alpha_{\mathcal L}(U)\geqslant\alpha_{{\mathcal L,+}}(U). \end{gathered}
\end{equation}
\tag{1.8}
$$
In potential theory harmonic capacity is proportional to the capacity $\gamma_{{\Delta,+}}$ in (1.7), where $\mathcal L=\Delta$ is the Laplace operator. Namely, for $N\geqslant3$
$$
\begin{equation}
\Phi_{\Delta}(\mathbf x)=-\frac{1}{(N-2)\sigma_N|\mathbf x|^{N-2}},
\end{equation}
\tag{1.9}
$$
where $\sigma_N$ is the area of the unit sphere in $\mathbb R^N$, so that (1.7) is $(N-2)\sigma_N$ times greater than the (classical) harmonic capacity. In addition, for each bounded set $U$ we have
$$
\begin{equation}
\alpha_{{\Delta,+}}(U)=\alpha_{\Delta}(U)=\gamma_{{\Delta,+}}(U)=\gamma_{\Delta}(U);
\end{equation}
\tag{1.10}
$$
the key equalities here, $\gamma_{{\Delta,+}}(U)=\gamma_{\Delta}(U)$ and $\alpha_{{\Delta,+}}(U)=\gamma_{{\Delta,+}}(U)$, were proved in Theorem 3.1 in [3] and Lemma 6 in [5], Ch. III (also see Lemma XII in [6]). It is important to note that (1.10) is closely connected with the solvability of the Dirichlet problem and the maximum principle for harmonic functions. In the case of arbitrary operators $\mathcal L$ with complex coefficients suitable machinery is not sufficiently well developed, which does not allow one to apply it to the investigations of significant properties of the capacities $\gamma_{\mathcal L}$, $\alpha_{\mathcal L}$ and $\gamma_{{\mathcal L,+}}$, $\alpha_{{\mathcal L,+}}$ in the general case. A natural question on the commensurability of the capacities $\gamma_{\mathcal L}$ and $\gamma_{\Delta,+}$ (up to a positive coefficient depending only on the polynomial $L$, the symbol of the operator $\mathcal L$). Note that an affirmative answer in combination with (1.10) yields easily the commensurability with $\gamma_{\Delta,+}$ of each of the capacities $\alpha_{\mathcal L}$, $\gamma_{{\mathcal L,+}}$ and $\alpha_{{\mathcal L,+}}$. Since for $N\geqslant3$ we have $\sup_{\mathbf x\ne0} |\Phi_{\mathcal L}(\mathbf x)|/|\mathbf x|^{2-N}<\infty$ for each $\mathcal L$, it is obvious that there exists a constant $A(L)\geqslant1$ such that for each bounded set $U\subset\mathbb R^N$ we have
$$
\begin{equation}
A(L)\gamma_{\mathcal L,+}(U)\geqslant \gamma_{\Delta,+}(U)\quad\text{and} \quad A(L)\alpha_{ \mathcal L,+}(U)\geqslant \alpha_{\Delta,+}(U).
\end{equation}
\tag{1.11}
$$
Taking (1.10) into account we obtain
$$
\begin{equation}
A(L)\gamma_{\mathcal L}(U)\geqslant A(L)\gamma_{\mathcal L,+}(U) \geqslant \gamma_{\Delta,+}(U),
\end{equation}
\tag{1.12}
$$
so the problem consists in reverting inequality (1.12), which can obviously be carried out for compact sets. This is done in the following statement, which is our main result in this paper. Theorem 1. For $N \geqslant 3$ let $\mathcal L$ be an arbitrary elliptic operator (1.2) with complex coefficients. Then there exists a constant $A \!=\! A(L) \!\geqslant\! 1$ such that ${A\gamma_{\Delta,+}(K)\!\geqslant\! \gamma_{\mathcal L}(K)}$ for each compact set $K\subset\mathbb R^N$. Some comments on $N=2$ are in order. If $\mathcal L$ is a nonstrongly elliptic operator in $\mathbb R^2$, then its fundamental solution (1.4) is bounded, and therefore the capacities $\gamma_{\mathcal L}$ and $\gamma_{\mathcal L,+}$ defined in accordance with (1.5) and (1.7) are positive even for singletons. One quite nontrivial consequence of this fact is the following result, which is the special case of Theorem 1 in [7]. Let $X\subset\mathbb R^2$ be an arbitrary (nonempty) compact set, let $X^o$ be its interior and $f$ be a function in $C(X)$ such that $\mathcal Lf=0$ in $X^o$; then $f$ can uniformly be approximated on $X$ to any accuracy by functions $F$ satisfying the equation $\mathcal LF=0$ in some (own) neighbourhood of $X$. If $\mathcal L$ is a strongly elliptic operator in $\mathbb R^2$, then $\lim_{|\mathbf x|\to\infty}\Phi_{\mathcal L}(\mathbf x)=\infty$, so that the capacities $\gamma_{\mathcal L}$, $\alpha_{\mathcal L}$, $\gamma_{{\mathcal L,+}}$ and $\alpha_{{\mathcal L,+}}$ are defined locally (for instance, see [8], § 1, (1.4)). In particular, this means that in (1.5) and (1.7) we replace the norm $\|(\,\cdot\,)\|_{\mathrm L^{\infty}(\mathbb R^2)}$ by $\|(\,\cdot\,)\|_{\mathrm L^{\infty}(B)}$, where $B$ is a disc such that $K\subset(1/2)B$. In terms of local capacities, a criterion for uniform approximation of functions by solutions of strongly elliptic equations on compact sets in $\mathbb R^2$ was obtained in [9]. For harmonic capacities in $\mathbb R^2$ there is a natural analogue of equality (1.10) (see [8], Proposition 2.1), and the corresponding capacities $\gamma_{\Delta,+}$ and $\gamma_{\mathcal L,+}$ are commensurable (see [8], Proposition 2.3). The last result is a direct consequence of the structure of fundamental solutions (1.4), where $k_0\ne0$. The central result of [8] is Theorem 1.1, which has the following meaning. Let $\mathcal L_2$ be an arbitrary strongly elliptic operator (1.3) in $\mathbb R^2$; then $\mathcal L_3=\mathcal L_2+c_{11}\partial^2/\partial x_3^2$ is an elliptic operator in $\mathbb R^3$, and the local capacity $\gamma_{\mathcal L_2}(K)$ of a compact set $K$ relative to a disc $B$ of radius $R$ such that $K\subset(1/2)B$ is commensurable with $R^{-1}\gamma_{\mathcal L_3}(K')$, where $K'=K\times[-R,R]$ is the Cartesian product in $\mathbb R^3$. Thus, by Proposition 2.1 and Theorem 1.1 in [8], our Theorem 1 for $\mathbb R^3$ also results in the commensurability of the local capacities $\gamma_{\mathcal L_2}$ in $\mathbb R^2$ with the harmonic capacities. In recent decades a number of results on the commensurability of different capacities were obtained (for instance, see the survey [10] by Volberg and Eiderman), where the authors usually dealt with convolutions of distributions and measures with odd kernels. Note that fundamental solutions $\Phi_{\mathcal L}$ of the operators $\mathcal L$ have a significantly different structure. For $N=3$ and $N=4$ there exists $\vartheta=\vartheta(L)\in(-\pi,\pi]$ such that $\operatorname{Re}(e^{i\vartheta}\Phi_{\mathcal L}(\mathbf x))>0$ (this implies easily the commensurability of $\gamma_{\Delta}$ and $\gamma_{\mathcal L,+}$, but does not yield the commensurability of $\gamma_{\Delta}$ and $\gamma_{\mathcal L}$ directly), while for $N\geqslant5$ there is no similar inequality in the general case (see [2], Corollaries 2 and 3), and the question of whether $\gamma_{\Delta}$ and $\gamma_{\mathcal L,+}$ are commensurable makes up Conjecture 1 in [2]. On the other hand (see [2], Theorem 2) the mean value of the function $\Phi_{\mathcal L_N}$ on the unit sphere in $\mathbb R^N$, $N\geqslant3$, is never zero. Recall that Tolsa [11] proved the commensurability of the analytic capacities $\gamma$ and $\gamma_+$ (with a consequence that $\gamma$ is countably subadditive). The capacities $\gamma$ and $\gamma_+$ are defined in accordance with (1.5) and (1.7) for $\mathcal L=(\partial/\partial x_1\!+\!i \partial/\partial x_2)/2$ being the Cauchy–Riemann operator with fundamental solution $\pi^{-1}/(x_1\!+\!ix_2)$. In our proof of Theorem 1 we use some ideas from [11], in particular, induction arguments. The elementary result of Lemma 3 in [2] also proves to be rather useful. The paper has the following structure. In § 2 we present auxiliary results. In § 3 we apply the energy approach to the study of capacities and prove Proposition 2, which is an analogue of Theorem 1 for the capacities $\gamma_{\mathcal L,+}$. In § 4 we prove Proposition 1, to which Theorem 1 reduces; the key result there is Lemma 9. In § 5 we present various corollaries to Theorem 1 which are related to removable singularities, the properties of capacities and uniform approximation (Theorems 2–5). Note that Lemma 9 is significantly different from the central lemmas in [11] (Lemmas 5.1 and 9.1). In contrast to the odd Cauchy kernel, the functions $\Phi_{\mathcal L}$ are even and have nontrivial means, so we cannot replicate the proof in [11], which is based on the $\mathrm L^2$-theory of singular integrals. We also introduce some technical improvements, which can apparently be useful in the investigations of other capacities. § 2. Auxiliary resultsIn what follows we denote by $A_1, A_2,\dots$ positive constants which can only depend on the polynomial $L$, the symbol of the operator $\mathcal L$ (and, in particular, on $N$). The values of each of these constants can be different in different relations. Let $f=T*\Phi_{\mathcal L}$, where $T$ is a distribution with compact support in $\mathbb R^N$. As $\Phi_{\mathcal L}$ is a fundamental solution, we have $\mathcal Lf=T$, and therefore $f=(\mathcal Lf)*\Phi_{\mathcal L}$ in the sense of distributions. If $f\in\mathrm L^{\infty}(\mathbb R^N)$, then using the notation (1.5)
$$
\begin{equation}
\langle T\mid 1\rangle=\langle \mathcal Lf\mid 1\rangle =\int f(\mathbf x)\mathcal L(\varphi(\mathbf x))\,d\mathbf x,
\end{equation}
\tag{2.1}
$$
where $\varphi\in C_0^{\infty}(\mathbb R^N)$ is an arbitrary function equal to 1 in some neighbourhood of $\operatorname{Spt}(\mathcal Lf)$, and the integral is taken against the Lebesgue measure in $\mathbb R^N$. Note (see, for instance, Lemma 3.1 in [12]) that to have a representation $f=(\mathcal Lf)*\Phi_{\mathcal L}$ it is sufficient that $\lim_{\mathbf x\to\infty}f(\mathbf x)=0$ and the support $\mathcal Lf$ be bounded; this and the fact that $\mathcal L\varphi=0$ on $\operatorname{Spt}(\mathcal Lf)$ imply the equivalence of (1.5) and (1.6). The functional $\langle T\mid 1\rangle$ is the leading coefficient $c_0$ of the Laurent expansion of ${T*\Phi_{\mathcal L}}$. We introduce further notation: $\alpha=(\alpha_1,\alpha_2,\dots,\alpha_N)\in\mathbb Z_+^N$ is a multi-index,
$$
\begin{equation*}
\begin{gathered} \, |\alpha|=\sum_{k=1}^N\alpha_k, \qquad \partial^{\alpha}=\frac{\partial^{|\alpha|}}{\partial x_1^{\alpha_1}\partial x_2^{\alpha_2} \dotsb\partial x_N^{\alpha_N}}, \\ \alpha!=\alpha_1!\,\alpha_2!\dotsb\alpha_N!\quad\text{and} \quad x^{\alpha}=x_1^{\alpha_1}x_2^{\alpha_2}\dotsb x_N^{\alpha_N}. \end{gathered}
\end{equation*}
\notag
$$
Let $\operatorname{Spt}(T)\subset B(\mathbf a,r)$, where $B(\mathbf a,r)$ is a ball in $\mathbb R^N$ with centre $\mathbf a$ and radius $r$. Then (see, for instance, [13], § 1.B, and [14], § 7.2) outside a ball $B(\mathbf a,A_1r)$, where $A_1 = A_1(L) > 1$, the function $T*\Phi_{\mathcal L}$ expands in a Laurent series converging in $C^{\infty}$:
$$
\begin{equation}
T*\Phi_{\mathcal L}(\mathbf x)=c_0\Phi_{\mathcal L}(\mathbf x-\mathbf a)+\sum_{\{\alpha\colon |\alpha|\geqslant1\}} c_{\alpha}\partial^{\alpha}\Phi_{\mathcal L}(\mathbf x-\mathbf a),
\end{equation}
\tag{2.2}
$$
and the coefficients are defined as the actions of $T$ on $C^{\infty}$-test functions:
$$
\begin{equation}
c_0=\langle T\mid 1\rangle, \qquad c_{\alpha}=c_{\alpha}(T,\mathbf a) =\frac{(-1)^{|\alpha|}}{\alpha!}\langle T\mid (\mathbf x-\mathbf a)^{\alpha}\rangle =\frac{(-1)^{|\alpha|}}{\alpha!}\langle (\mathbf x-\mathbf a)^{\alpha}T\mid 1\rangle.
\end{equation}
\tag{2.3}
$$
Recall several elementary properties of the capacities $\gamma_{\mathcal L}$. 1. The capacity $\gamma_{\mathcal L}$ is preserved by parallel translations. 2. Let $K$ be a compact set, let $\widehat{K}$ be the union of $K$ and all bounded components of the complement to $K$ and $\partial \widehat{K}$ be the boundary of $\widehat{K}$. Then $\gamma_{\mathcal L}(K)=\gamma_{\mathcal L}(\widehat{K})= \gamma_{\mathcal L}(\partial\widehat{K})$. (In fact, as we formally have $\gamma_{\mathcal L}(\partial\widehat{K})\leqslant\gamma_{\mathcal L}(K)\leqslant\gamma_{\mathcal L}(\widehat{K})$, it suffices to show that $\gamma_{\mathcal L}(\partial\widehat{K})\geqslant\gamma_{\mathcal L}(\widehat{K})$. To do this it is sufficient to set the function $f$, $\operatorname{Spt}(\mathcal Lf)\subset\widehat{K}$, equal to zero on $\widehat{K}\setminus\partial\widehat{K}$: the functional $\langle T\,|\, 1\rangle=\langle \mathcal Lf\,|\, 1\rangle=c_0$ is fully determined by the asymptotic behaviour (2.2) of $f$ at infinity, and its value does not change.) 3. Let $B$ be a ball of radius $r$ in $\mathbb R^N$, $N \geqslant 3$; then there exists a constant $A_2 > 1$ such that
$$
\begin{equation}
(A_2)^{-1}r^{N-2}\leqslant\gamma_{\mathcal L}(B)\leqslant A_2r^{N-2}
\end{equation}
\tag{2.4}
$$
(in fact, the upper bound for the capacity follows from (2.1), and for a lower bound we can take, for example, the convolution of $\Phi_{\mathcal L}$ with a measure spread uniformly over $B$). By a cube we mean a closed cube with edges parallel to coordinate axes. Given a cube $Q=Q(\mathbf a,s)$ with centre $\mathbf a\in\mathbb R^N$ and edge $s$, we let $\lambda Q$ denote the concentric cube with edge $\lambda s$. By dyadic cubes we mean cubes of the form
$$
\begin{equation*}
Q=[m_12^{-p},(m_1+1)2^{-p}]\times[m_22^{-p},(m_2+1)2^{-p}]\times\dots \times[m_N2^{-p},(m_N+1)2^{-p}],
\end{equation*}
\notag
$$
where $p, m_1,m_2,\dots, m_N$ are integers. In considering covers we always assume that dyadic cubes are (pairwise) nonoverlapping, that is, have no common interior points. In the work with dyadic cubes it is convenient to use partitions of unity from the following lemma, which was proved in [15]. Lemma 1. Let $\{Q_j\}$ be a finite family of nonoverlapping dyadic cubes with edges $s_j$. Then there exist (nonnegative) functions $\varphi_j\in C^{\infty}_0(\mathbb R^N)$ such that $\operatorname{Spt}(\varphi_j)\subset (17/16)Q_j$, $0\leqslant \varphi_j(\mathbf x)\leqslant1$, $\sum_j\varphi_j(\mathbf x)=1$ on the set $\bigcup_jQ_j$, and the inequalities $\|\partial^{\alpha}\varphi_j\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A(N)s_j^{-|\alpha|}$ hold for all $j$ and $\alpha$, $|\alpha|\leqslant2$. Note that the construction of the $\varphi_j$ is quite transparent. For each cube $Q_j$ consider a function $\phi_j\in C^{\infty}_0(\mathbb R^N)$ such that $\operatorname{Spt}(\phi_j)\subset (17/16)Q_j$, $0\leqslant \phi_j(\mathbf x)\leqslant1$ for all $\mathbf x$, $\|\partial^{\alpha}\phi_j\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A_0(N)s_j^{-|\alpha|}$ for all $|\alpha|\leqslant2$ and $\phi_j(\mathbf x)=1$ on $Q_j$. We number the cubes so that $s_1\geqslant\cdots\geqslant s_j\geqslant s_{j+1}\geqslant\cdots$. We set $\varphi_1=\phi_1$, and for $j\geqslant2$ set $\varphi_j=\phi_j\prod_{k=1}^{j-1}(1-\phi_k)$. Then for all $j$ we have $\sum_{k=1}^j\varphi_k=1-\prod_{k=1}^j(1-\phi_k)$, and estimates for derivatives of the $\varphi_j$ are obtained by summing geometric progressions (see [15], Lemma 3.1). In Lemma 3.1 in [15] the constant is $3/2$, rather than $17/16$; however, this is not essential (it is clear from the construction that any constant of the form $1+\varepsilon$, where $\varepsilon>0$, can be taken). In what follows, in dealing with Whitney cubes we would like to control the multiplicity of intersections ‘with a margin’. We use the principle of localization of singularities, which was in fact proposed by Vitushkin (see [16], Ch. 2). It involves partitions of unity and estimates for functions with localized singularities, where one uses the expansion (2.2), (2.3) and the asymptotic behaviour of the function at infinity. For $\varphi\in C^{\infty}_0(\mathbb R^N)$ the function is called the localization of $f$ with respect to $\varphi$. The following result on the properties of localization is well known (see, for instance, [14], § 14.3, or [7], Lemma 1.2).Lemma 2. Let $Q=Q(\mathbf a,s)$ be a cube in $\mathbb R^N$, and let $f\in \mathrm L^{\infty}(\mathbb R^N)$ and $\varphi\in C^{\infty}_0(Q)$. Then the following hold for the function $V_{\varphi}f$ in (2.5): (a) $\mathcal L(V_{\varphi}f)=\varphi\mathcal Lf$, so that $\operatorname{Spt}(\mathcal L(V_{\varphi}f))\subset(\operatorname{Spt}(\varphi)\cap \operatorname{Spt}(\mathcal Lf))$; (b) $\lim_{\mathbf x\to\infty}V_{\varphi}f(\mathbf x)=0$; (c) the following estimate holds, where $A_3=A_3(L)$: where $\omega_f(Q)$ is the oscillation of $f$ on $Q$.Note that assertions (a) and (b) are obvious, and (c) can be proved for $f\in C^{\infty}(\mathbb R^N)$ using integration by parts, where we must bear in mind that $\Phi_{\mathcal L}$ is a fundamental solution; in the general case we use regularization and a limiting procedure. Inequality (2.6) enables us to estimate Laurent coefficients of the above localizations in terms of the capacity $\gamma_{\mathcal L}$. The coefficient $c_0$ can be estimated directly on the basis of the definition of capacity:
$$
\begin{equation}
|c_0(V_{\varphi}f)|\leqslant A_3\gamma_{\mathcal L}(\operatorname{Spt}(\varphi)\cap\operatorname{Spt}(\mathcal Lf))\omega_f(Q)\|\nabla^2\varphi\|_{\mathrm L^{\infty}(Q)}s^2.
\end{equation}
\tag{2.7}
$$
To estimate the other coefficients $c_{\alpha}$, in (2.5) we must replace $\varphi$ by the product $(\mathbf x-\mathbf a)^{\alpha}\varphi$ and use in (2.6) the elementary estimate
$$
\begin{equation*}
\|\nabla^2(\varphi(\mathbf x)(\mathbf x-\mathbf a)^{\alpha})\|_{\mathrm L^{\infty}(Q)}\leqslant A(N)\|\nabla^2\varphi\|_{\mathrm L^{\infty}(Q)}(2s)^{|\alpha|}.
\end{equation*}
\notag
$$
As a result, taking (2.3) and (2.7) into account we obtain ($\mathbf a$ is the centre of the cube $Q$)
$$
\begin{equation}
|c_{\alpha}(V_{\varphi}f,\mathbf a)|\leqslant(\alpha!)^{-1}A_3\gamma_{\mathcal L}\bigl(\operatorname{Spt}(\varphi)\cap\operatorname{Spt}(\mathcal Lf)\bigr)\omega_f(Q) \|\nabla^2\varphi\|_{\mathrm L^{\infty}(Q)}(2s)^{2+|\alpha|}.
\end{equation}
\tag{2.8}
$$
As a consequence of Lemma 2, the expansion (2.2)–(2.3) and estimates (2.7)–(2.8) we obtain the following standard assertion, which is in fact a special case of Lemma 2.7 in [4]. Lemma 3. Let $f\in \mathrm L^{\infty}(\mathbb R^N)$, let $Q$ be a dyadic cube with centre $\mathbf a$ and edge $s$, and let $\varphi$ be a function such that $\varphi\in C^{\infty}_0(\mathbb R^N)$, $\operatorname{Spt}(\varphi)\subset (17/16)Q$ and $\|\partial^{\alpha}\varphi\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A(N)s^{-|\alpha|}$ for all $|\alpha|\leqslant2$, where $A(N)$ is the constant in Lemma 1. Let $V_{\varphi}f$ be the localization of $f$ in (2.5) relative to $\varphi$. Then the following estimates hold, where $A_4=A_4(L)$: (i)
$$
\begin{equation}
|c_{\alpha}(V_{\varphi}f,\mathbf a)|\le A_4s^{|\alpha|} \gamma_{\mathcal L}\biggl(\frac{17}{16}Q\cap\operatorname{Spt}(\mathcal Lf)\biggr)\omega_f \biggl(\frac{17}{16}Q\biggr), \qquad |\alpha|\le1;
\end{equation}
\tag{2.9}
$$
(ii) $\|V_{\varphi}f\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A_4\omega_f((17/16)Q)$; moreover, if $\mathbf x\not\in(9/8)Q$, then
$$
\begin{equation}
|V_{\varphi}f(\mathbf x)|\leqslant A_4\omega_f\biggl(\frac{17}{16}Q\biggr) \frac{\gamma_{\mathcal L}((17/16)Q\cap\operatorname{Spt}(\mathcal Lf))}{|\mathbf x-\mathbf a|^{N-2}};
\end{equation}
\tag{2.10}
$$
(iii) if $c_0(V_{\varphi}f)=0$, then for $\mathbf x\not\in(9/8)Q$ Since Lemma 3 is standard, we limit ourselves to some comments. Assertion (i) follows from (2.7) and (2.8). The first claim in (ii) follows from (2.6). Estimates (2.10) and (2.11) follow from (2.2)–(2.3) and (2.7)–(2.8) by taking the sum of a geometric progression (that is, in fact, in the same way as for holomorphic functions), but outside the cube $A'Q$ (rather than $(9/8)Q$), where $A'$ depends on a certain constant $A_1$ related to the expansion (2.2). We eliminate $A'$ by partitioning the cube $(9/8)Q$ into smaller dyadic cubes on a scale of $1/A_1$, representing $V_{\varphi}f$ as the sum of the corresponding localizations with $\varphi_j$ from Lemma 1, and expanding each of these localizations in a Laurent series. The next result is an elementary consequence of Lemma 3. Lemma 4. Let $Q=Q(\mathbf a,s)$ be a dyadic cube and $V_{\varphi}f$ be a localization of the function $f$ from Lemma 3. Then there exists a function $\nu_Q\in C_0^{\infty}((1/4)Q)$ such that (1) $\|\nu_Q\|_{\mathrm L^{\infty}((1/4)Q)}\leqslant A_4\gamma_{\mathcal L}((17/16)Q\cap\operatorname{Spt}(\mathcal Lf))\omega_f((17/16)Q)/s^N$; (2) if $h_Q=\nu_Q*\Phi_{\mathcal L}$, then $\langle\varphi\mathcal Lf\mid 1\rangle=\langle\nu_{Q}\mid 1\rangle$ (that is, $c_0(h_Q)=c_0(V_{\varphi}f)$); in addition, $\|h_Q-V_{\varphi}f\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A_4\omega_f((17/16)Q)$ and the following estimate holds for $\mathbf x\not\in(9/8)Q$: Proof. Consider a function $\nu_{Q,0}\in C_0^{\infty}((1/4)Q)$ such that
$$
\begin{equation}
0\leqslant\nu_{Q,0}(\mathbf x)\leqslant 8^N, \qquad \int_{(1/4)Q}\nu_{Q,0}(\mathbf x)\,d\mathbf x=s^N\quad\text{and} \quad \nu_{Q}=\nu_{Q,0}c_0\frac{(V_{\varphi}f)}{s^N}.
\end{equation}
\tag{2.13}
$$
Using the notation (2.3), for $h_{Q,0}=\nu_{Q,0}*\Phi_{\mathcal L}$ we have $c_0(h_{Q,0})=s^N$ and, moreover, $\|h_{Q,0}\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A_4s^2$. It remains to take $h_Q=\nu_{Q}*\Phi_{\mathcal L}=c_0(V_{\varphi}f)h_{Q,0}/s^N$ and use (2.9) and estimate (2.11) for the difference $h_Q(\mathbf x)-V_{\varphi}f(\mathbf x)$, taking the equality $c_0(h_Q-V_{\varphi}f)=0$ into account. Lemma 4 is proved. The next result is a slightly weaker version of Lemma 8.1 in [11] (bearing in mind that, in contrast to the Cauchy–Riemann operator, $\mathcal L$ has the second order). We use it in the proof of Lemma 9, our main lemma. Lemma 5. Let $X$ be a compact set in $\mathbb R^N$ such that $\gamma_{\mathcal L}(X)>0$, and let $f=(\mathcal Lf)*\Phi_{\mathcal L}$ be a function such that $\operatorname{Spt}(\mathcal Lf)\subset X$ and $\|f\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1$. Let $\{Q_j\}$ be a covering of $X$ by a finite family of nonoverlapping dyadic cubes such that there exists $M\geqslant1$ such that the following estimate holds for each dyadic cube $D$ containing at least one $Q_j$:
$$
\begin{equation}
\sum_{\{Q_j|Q_j\subset D\}}\gamma_{\mathcal L}\biggl(\frac{17}{16}Q_j\cap X\biggr)\leqslant (s(D))^{N-2}M.
\end{equation}
\tag{2.14}
$$
Let $\{\varphi_j\}$ be the partition of unity from Lemma 1 that corresponds to $\{Q_j\}$, let $V_{\varphi_j}f$ be the localizations of $f$ from (2.5) for $\varphi=\varphi_j$, and let $\nu_{Q_j}$ be the functions $\nu_Q$ from (2.13) corresponding to $V_{\varphi_j}f$ and $Q=Q_j$. Let $\nu=\sum_j\nu_{Q_j}$. Then the following estimates hold for any dyadic cube $Q=Q(\mathbf a,s)$ and a (nonnegative) function $\varphi\in C_0^{\infty}((17/16)Q)$ such that $0\leqslant \varphi(\mathbf x)\leqslant1$ and $\|\partial^{\alpha}\varphi\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A(N)s^{-|\alpha|}$ for all $|\alpha|\leqslant2$:
$$
\begin{equation}
\begin{aligned} \, &(1)\ \int_{(17/16)Q}|\nu(\mathbf x)|\,d\mathbf x\leqslant A_5 M s^{N-2}, \\ &(2)\ |\langle\nu\mid \varphi\rangle|=\biggl|\int_{(17/16)Q}\nu(\mathbf x)\varphi(\mathbf x)\,d\mathbf x\biggr|\leqslant A_5 M^{2/3} s^{N-2}. \end{aligned}
\end{equation}
\tag{2.15}
$$
Proof. Estimate (2.15), (1) is an immediate consequence of (2.13) and (2.14). Now we prove (2.15), (2), which is significantly harder.
We assume below that $(17/16)Q$ does not lie in any $Q_j$, for otherwise estimate (2.15) for $M=1$ follows from (2.13) and $\nu_{Q_{j'}}(\mathbf x)=0$ on $(17/16)Q$ for all $j'\ne j$. Using the partition of unity $\{\varphi_j\}$ we represent $\langle\mathcal Lf\mid \varphi\rangle$ as follows:
$$
\begin{equation*}
\langle\mathcal Lf\mid \varphi\rangle=\mathop{{\sum}'}_j\langle\varphi_j\mathcal Lf\mid \varphi\rangle+\mathop{{\sum}''}_j\langle \varphi_j\mathcal L f\mid \varphi\rangle,
\end{equation*}
\notag
$$
where we sum over all $j$ such that $(17/16)Q_j$ intersects $(17/16)Q$; in the first sum all the $Q_j$ lie in $16Q$, and the other indices $j$ are collected in the second sum.
We consider the second sum first. For all $j$ we have $s(Q_j)\geqslant4s(Q)$; furthermore, $Q_j$ does not contain $16Q$. Since $\operatorname{Spt}(\nu_{Q_j})\subset(1/4)Q_j$, it follows that $\nu_{Q_j}(\mathbf x)=0$ on $(17/16)Q$. Hence for $\varphi$, from (2.15), (2) we obtain
$$
\begin{equation}
\mathop{{\sum}''}_j\langle \nu_{Q_j}\mid \varphi\rangle=0.
\end{equation}
\tag{2.16}
$$
Let $\varphi_0=\mathop{{\sum}''}_j\varphi_j$. By construction $0\leqslant \varphi_0(\mathbf x)\leqslant1$, and taking the sum of a geometric progression with terms corresponding to the cubes $Q_j$ of the same size, it is easy to show that $\|\partial^{\alpha}\varphi_0\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant A_6s^{-|\alpha|}$ for $|\alpha|\leqslant2$. Hence from inequality (2.7) (for $\varphi$ replaced by $\varphi\varphi_0$) we obtain
$$
\begin{equation}
\biggl|\mathop{{\sum}''}_j\langle{\varphi_j\mathcal L}f\mid \varphi\rangle\biggr|\leqslant A_5s^{N-2}.
\end{equation}
\tag{2.17}
$$
Since we also have $|\langle\mathcal Lf\mid \varphi\rangle|\leqslant A_5s^{N-2}$ by (2.7), from (2.17) and (2.16) we obtain
$$
\begin{equation}
\biggl|\mathop{{\sum}'}_j\langle{\varphi_j\mathcal L}f\mid \varphi\rangle\biggr|\leqslant A_5s^{N-2}\quad\text{and} \quad \langle\nu\mid \varphi\rangle=\mathop{{\sum}'}_j\langle \nu_{Q_j}\mid \varphi\rangle.
\end{equation}
\tag{2.18}
$$
Let $Q_j=Q(\mathbf a_j,s_j)$. It is clear that
$$
\begin{equation*}
\mathop{{\sum}'}_j\langle \nu_{Q_j}\mid \varphi\rangle=\mathop{{\sum}'}_j\varphi(\mathbf a_j)\langle\nu_{Q_j}\mid 1\rangle +\mathop{{\sum}'}_j\langle\nu_{Q_j}\mid (\varphi-\varphi(\mathbf a_j))\rangle.
\end{equation*}
\notag
$$
By the definition of $\nu_{Q_j}$ and Lemma 4 we have $\varphi(\mathbf a_j)\langle\varphi_j\mathcal Lf\mid 1\rangle =\varphi(\mathbf a_j)\langle\nu_{Q_j}\mid 1\rangle$, so that
$$
\begin{equation*}
\mathop{{\sum}'}_j\langle \nu_{Q_j}\mid \varphi\rangle =\mathop{{\sum}'}_j\varphi(\mathbf a_j)\langle\varphi_j\mathcal Lf\mid 1\rangle+\mathop{{\sum}'}_j\langle\nu_{Q_j}\mid (\varphi-\varphi(\mathbf a_j))\rangle.
\end{equation*}
\notag
$$
Thus, by the obvious equality
$$
\begin{equation*}
\mathop{{\sum}'}_j\varphi(\mathbf a_j)\langle\varphi_j\mathcal Lf\mid 1\rangle =\mathop{{\sum}'}_j\langle\varphi_j \mathcal Lf\mid \varphi\rangle-\mathop{{\sum}'}_j\langle\mathcal Lf\mid (\varphi-\varphi(\mathbf a_j))\varphi_j\rangle
\end{equation*}
\notag
$$
and (2.18), the following estimate is sufficient for the proof of (2.15):
$$
\begin{equation}
\mathop{{\sum}'}_j\bigl(|\langle\mathcal Lf\mid (\varphi-\varphi(\mathbf a_j))\varphi_j\rangle| +|\langle\nu_{Q_j}\mid (\varphi-\varphi(\mathbf a_j))\rangle|\bigr)\leqslant A_7M^{2/3}s^{N-2}.
\end{equation}
\tag{2.19}
$$
Let $s_j/s=2^{-k}$, where $k\geqslant-4$ and $k\in\mathbb Z$. By (2.6), where we estimate the localization $((\varphi-\varphi(\mathbf a_j))\varphi_j\mathcal Lf)*\Phi_{\mathcal L}$, and by (2.7) and Lemma 4, (1) we have
$$
\begin{equation}
\bigl|\langle\mathcal Lf\mid (\varphi-\varphi(\mathbf a_j))\varphi_j\rangle\bigr| +\bigl|\langle\nu_{Q_j}\mid (\varphi-\varphi(\mathbf a_j))\rangle\bigr|\leqslant 2^{-k}A_8\gamma_{\mathcal L}\biggl(\frac{17}{16}Q_j\cap X\biggr).
\end{equation}
\tag{2.20}
$$
We denote the sum of all terms $A_8\gamma_{\mathcal L}((17/16)Q_j\cap X)$ for which $s_j/s=2^{-k}$ by $\lambda_k$ and note that the number of these terms is at most $2^{(k+5)N}$. The left-hand side of (2.19) can be estimated as follows using (2.14), (2.20) and Hölder’s inequality:
$$
\begin{equation*}
\begin{aligned} \, \sum_k\lambda_k2^{-k} &\leqslant \biggl(\sum_k\lambda_k\biggr)^{1/p} \biggl(\sum_k\lambda_k2^{-kq}\biggr)^{1/q} \\ &\leqslant A_9M^{1/p}s^{(N-2)/p}\biggl(\sum_k\frac{2^{kN}2^{-kq}}{2^{k(N-2)}}\biggr)^{1/q}s^{(N-2)/q}. \end{aligned}
\end{equation*}
\notag
$$
To bound the series on the right by $\sum_{k=1}^{\infty}2^{-k}$ we can set $q=3$ and therefore $p=3/2$. This completes the proof of (2.19) and, by implication, of estimate (2.15). Lemma 5 is proved. § 3. Energies, capacities $\gamma_{\mathcal L,+}$, Propositions 1 and 2Recall the following result established in [2], Lemma 3. Lemma 6. Let $L(\mathbf x)$ in (1.1) be the symbol of an elliptic operator for $N\geqslant3$ or of a strongly elliptic operator for $N=2$. Then there exist $\tau\in(0,1)$ and $\vartheta\in(-\pi,\pi]$ depending on $L$ such that the following estimate holds in $\mathbb R^N \setminus \{0\}$: Note (see [2], Lemma 1) that Lemma 6 does not extend to nonstrongly elliptic operators: the $L(\mathbf x)$-image of the circle $|\mathbf x|=1$ under the map is always an ellipse containing $\mathbf x=0$ inside. Recall that in [2], Theorem 1, fundamental solutions $\Phi_{\mathcal L}$ were explicitly described for all $N\geqslant3$. In particular, the following estimate holds for $N=3$ and $4$ (see [2], Corollary 2):
$$
\begin{equation}
\frac1{A_1|\mathbf x|^{N-2}}\leqslant \operatorname{Re} \bigl(e^{i\lambda}\Phi_{\mathcal L}(\mathbf x)\bigr) \leqslant|\Phi_{\mathcal L}(\mathbf x)|\leqslant\frac{A_1}{|\mathbf x|^{N-2}}
\end{equation}
\tag{3.2}
$$
(where $\lambda=\lambda(\mathcal L)\in(-\pi,\pi]$ and $A_1=A_1(L)>1$), which implies immediately that the capacities $\gamma_{\Delta}$ and $\gamma_{\mathcal L,+}$ are commensurable for $N=3$ and $N=4$. In the general case we cannot extend (3.2) to $N\geqslant5$. Let $(\mathbf y, \mathbf x)=y_1x_1+\cdots+y_Nx_N$ for $\mathbf y$, $\mathbf x\in\mathbb R^N$, let $\mathcal S(\mathbb R^N)$ be the Schwartz space of functions in $C^{\infty}(\mathbb R^N)$ decreasing at infinity, together with all partial derivatives, more rapidly than any negative power of $|\mathbf x|$. Recall the formulae for the direct and inverse Fourier transforms ($\psi\in\mathcal S(\mathbb R^N)$):
$$
\begin{equation*}
F[\psi](\mathbf x) =\int_{\mathbb R^N} e^{-i(\mathbf y, \mathbf x)}\psi(\mathbf y)\,d\mathbf y\quad\text{and} \quad F^{-1}[\psi](\mathbf x) =\frac{1}{(2\pi)^N}\int_{\mathbb R^N} e^{i(\mathbf y, \mathbf x)}\psi(\mathbf y)\,d\mathbf y.
\end{equation*}
\notag
$$
If $\psi\in\mathcal S(\mathbb R^N)$ and $\Psi\in\mathcal S'(\mathbb R^N)$, where $\mathcal S'(\mathbb R^N)$ is the space of tempered distributions in $\mathbb R^N$, then the Fourier transform acts on $\Psi$ by the formulae $\langle F[\Psi]\mid \psi\rangle=\langle \Psi\mid F[\psi]\rangle$ and $\langle F^{-1}[\Psi]\mid \psi\rangle=\langle \Psi\mid F^{-1}[\psi]\rangle$; in addition, $\langle \Psi\mid \psi\rangle=\langle F[\Psi]\mid F^{-1}[\psi]\rangle$. Recall that the distribution $F[\Phi_{\mathcal L}]$ coincides with the function $-1/L$, where $L=L(\mathbf x)$ in (1.1) is the symbol of $\mathcal L$. For $N\geqslant3$, $-1/L$ is locally integrable on $\mathbb R^N$. Using this fact we can give a simple short proof of Theorem 2 in [2] ($\sigma_{\mathbf x}$ is the surface measure on the unit sphere in $\mathbb R^N$). Lemma 7. For $N\geqslant 3$, $\displaystyle\int_{|\mathbf x|=1}\Phi_{\mathcal L}(\mathbf x)\,d\sigma_{\mathbf x}\ne0$ for any $\mathcal L$. Proof. Note that the function $\psi(\mathbf x)=\exp(-|\mathbf x|^2/2)$ belongs to $\mathcal S(\mathbb R^N)$, and we have $F[\psi]=c_N\psi$, where $c_N\ne0$, so that
$$
\begin{equation*}
c_N\langle\Phi_{\mathcal L}\mid \psi\rangle=\langle\Phi_{\mathcal L}\mid F[\psi]\rangle=\langle F[\Phi_{\mathcal L}]\mid \psi\rangle=-\biggl\langle \frac1{L} \biggm| \psi\biggr\rangle.
\end{equation*}
\notag
$$
Since $\operatorname{Re}(e^{i\vartheta} L(\mathbf x))\geqslant\tau|L(\mathbf x)|$ by Lemma 6 and $\psi(\mathbf x)$ is positive, it follows that $\operatorname{Re}(e^{-i\vartheta}\langle1/L\mid \psi\rangle)>0$, and therefore $\langle\Phi_{\mathcal L}\mid \psi\rangle\ne0$. Now, since $\Phi_{\mathcal L}$ is homogeneous of degree $2-N$, we obtain
Lemma 7 is proved. Let $K_{\varepsilon}$ be the closure of the $\varepsilon$-neighbourhood of the compact set $K$. Then (see [12], Proposition 3.1)
$$
\begin{equation}
\lim_{\varepsilon\to0}\gamma_\mathcal L(K_{\varepsilon})=\gamma_\mathcal L(K),
\end{equation}
\tag{3.3}
$$
and Theorem 1 reduces to the following result. Proposition 1. Let $\mathcal X$ be a union of a finite family of nonoverlapping dyadic cubes in $\mathbb R^N$, $N\geqslant3$. Then $A\gamma_{\Delta,+}(\mathcal X)\geqslant \gamma_{\mathcal L}(\mathcal X)$. The proof is based on the energy approach. Let $X$ be a union of a finite family of nonoverlapping dyadic cubes ($X$ is not necessarily equal to $\mathcal X$ in Proposition 1). Taking (1.9) into account, by the energy of a finite nonnegative measure $\mu$ such that $\operatorname{Spt}(\mu)\subset X$ we mean the integral
$$
\begin{equation}
I_{\mu, \Delta}=-(N-2)\sigma_N\int_{X}(\mu*\Phi_{\Delta})(\mathbf x)\,d\mu(\mathbf x)=\iint\frac1{|\mathbf x-\mathbf y|^{N-2}}\,d\mu(\mathbf x)\,d\mu(\mathbf y)
\end{equation}
\tag{3.4}
$$
(since $X$ is regular, there obviously exist measures with finite energy on $X$). Recall (for instance, see [17], Ch. II, § 1) one of the equivalent definitions of the harmonic capacity of a compact set $X$: this is $1/\inf(I_{\mu^0, \Delta})$, where $\inf$ is taken over all nonnegative measures $\mu^0$ such that $\operatorname{Spt}(\mu^0)\subset X$ and $\|\mu^0\|=1$. We require a generalization of (3.4) to arbitrary $\mathcal L$ only for functions of the form $f=(\mathcal Lf)*\Phi_{\mathcal L}$, where $\mathcal Lf\in C^{\infty}_0(X)$ is a complex function, which we denote by $\nu$ (note that, in contrast to the special case $\mathcal L=\Delta$, the relationship between energy and capacity is unknown). Let
$$
\begin{equation}
I_{\nu, \mathcal L} =-(N-2)\sigma_N\int_{X}(\nu*\Phi_{\mathcal L})(\mathbf x)\overline{\nu(\mathbf x)}\,d\mathbf x =-(N-2)\sigma_N\langle \nu*\Phi_{\mathcal L}\mid \overline{\nu}\,\rangle
\end{equation}
\tag{3.5}
$$
(a bar denotes complex conjugation). The Fourier transform yields
$$
\begin{equation*}
F[\nu](\mathbf x)=\int_{\mathbb R^N} e^{-i(\mathbf y, \mathbf x)}\nu(\mathbf y)\,d\mathbf y
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
F^{-1}[\overline{\nu}\,](\mathbf x) =\frac{1}{(2\pi)^N}\int_{\mathbb R^N} e^{i(\mathbf y, \mathbf x)}\overline{\nu(\mathbf y)}\,d\mathbf y =\frac{\overline{F[\nu](\mathbf x)}}{(2\pi)^N}.
\end{equation*}
\notag
$$
Hence, from the equality $F[\nu*\Phi_{\mathcal L}]=-F[\nu]/L$ we obtain
$$
\begin{equation}
\langle \nu*\Phi_{\mathcal L}\mid \overline{\nu}\, \rangle=-\biggl\langle \frac{F[\nu]}{L}\biggm| F^{-1}[\overline{\nu}\,]\biggr\rangle =-\frac{1}{(2\pi)^N}\int_{\mathbb R^N}|F[\nu](\mathbf x)|^2 \frac{1}{L(\mathbf x)}\,d\mathbf x.
\end{equation}
\tag{3.6}
$$
In our proofs we use the standard regularization of distributions. Consider a function $\varphi_1 \in C^{\infty}_0(B(\mathbf 0,1))$ (where $B(\mathbf 0,1)$ is the unit ball in $\mathbb R^N$) such that $\varphi_1\geqslant0$ and $\displaystyle\int_{B}\varphi_1(\mathbf x) \,d\mathbf x=1$. For an arbitrary $\varepsilon>0$ set $\varphi_{\varepsilon}(\mathbf x)=\varepsilon^{-N}\varphi_1(\mathbf x/\varepsilon)$, so that $\displaystyle\int_{|\mathbf x|\leqslant\varepsilon}\varphi_{\varepsilon}(\mathbf x) \,d\mathbf x =1$. Given a distribution $T$ with compact support, consider the convolution ${T_{\varepsilon}=T*\varphi_{\varepsilon}}$. It is obvious that $T_{\varepsilon}\in C^{\infty}_0(\mathbb R^N)$, $\operatorname{Spt}(T_{\varepsilon})$ lies in the $\varepsilon$-neighbourhood of $\operatorname{Spt}(T)$ and $\langle T_{\varepsilon}\mid 1\rangle=\langle T\mid 1\rangle$. As $T_{\varepsilon}*\Phi_{\mathcal L}=(T*\Phi_{\mathcal L})*\varphi_{\varepsilon}$, we have $\|T_{\varepsilon}*\Phi_{\mathcal L}\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant\|T*\Phi_{\mathcal L}\|_{\mathrm L^{\infty}(\mathbb R^N)}$. First we establish an analogue of Theorem 1 for the capacity $\gamma_{\mathcal L,+}$. Proposition 2. Let $N\geqslant3$, and let $\mathcal L$ be an elliptic operator (1.2) with complex coefficients. Then there exists a positive constant $A=A(L) $ such that $A\gamma_{\Delta,+}(K)\geqslant \gamma_{\mathcal L,+}(K)$ for each compact set $K\subset\mathbb R^N$. Proof. By (3.3), bearing in mind regularization it suffices to show the following ($X$ is the closure of a suitable neighbourhood of $K$).
Let $X$ be a union of a finite family of nonoverlapping dyadic cubes. Then the inequality $A\gamma_{\Delta,+}(X)\geqslant \langle \nu\mid 1\rangle$ holds for any function $f=(\mathcal Lf)*\Phi_{\mathcal L}$ such that $\|f\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1$, $\operatorname{Spt}(\mathcal Lf)\subset X$, $\nu=\mathcal Lf\in C^{\infty}_0(X)$ and $\nu\geqslant0$. Since $\|\nu*\Phi_{\mathcal L}\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1$ and $\nu=\mathcal Lf\geqslant0$, by (3.5) we have
$$
\begin{equation}
|I_{\nu, \mathcal L}|\leqslant(N-2)\sigma_N\langle \nu\mid 1\rangle.
\end{equation}
\tag{3.7}
$$
From (3.5) and (3.6) we obtain (for $\vartheta$ in (3.1))
$$
\begin{equation}
I_{\nu, \mathcal L} =e^{i\vartheta}\frac{(N-2)\sigma_N}{(2\pi)^N} \int_{\mathbb R^N}|F[\nu(\mathbf x)]|^2\frac{e^{-i\vartheta}}{L(\mathbf x)}\,d\mathbf x.
\end{equation}
\tag{3.8}
$$
By (3.1), in $\mathbb R^N\setminus\{0\}$ we have the estimate (where $A_2=A_2(L)>0$)
$$
\begin{equation}
A_2\operatorname{Re}\frac{e^{-i\vartheta}}{L(\mathbf x)}=A_2\frac{\operatorname{Re}(e^{i\vartheta} L(\mathbf x))}{|L(\mathbf x)|^2} \geqslant\frac{A_2\tau}{|L(\mathbf x)|}\geqslant\frac{1}{|\mathbf x|^{2}}.
\end{equation}
\tag{3.9}
$$
From (3.7), (3.8) and (3.9) we obtain
$$
\begin{equation}
I_{\nu, \Delta}\leqslant A_3\langle \nu\mid 1\rangle,
\end{equation}
\tag{3.10}
$$
where $I_{\nu, \Delta}$ is the energy integral (3.5) for $\mathcal L=\Delta$ (so that $L(\mathbf x)=|\mathbf x|^{2}$).
Hence for the function $\nu^o=\nu/\langle \nu\mid 1\rangle$ we have $\displaystyle\int_{\mathbb R^N}\nu^o(\mathbf x)\,d\mathbf x=1$ and
$$
\begin{equation*}
I_{\nu^o, \Delta}\leqslant \frac{A_3}{\langle \nu\mid 1\rangle},
\end{equation*}
\notag
$$
so that, by the ‘energy’ definition of harmonic capacity we have $A\gamma_{\Delta,+}(X)\geqslant \langle \nu\mid 1\rangle$. Proposition 2 is proved. In is important that in the general case, for complex-valued functions $\nu$, in contrast to nonnegative ones, instead of (3.7) we obtain directly only the much weaker estimate
$$
\begin{equation}
|I_{\nu,\mathcal L}|\leqslant(N-2)\sigma_N\int_{\mathbb R^N}|\nu(\mathbf x)|\,d\mathbf x,
\end{equation}
\tag{3.11}
$$
which, of course, does not yield Proposition 1. For $\mathcal L=\Delta$ this is not crucial because (since $X$ is regular), for a nonnegative measure $\mu$ delivering the minimum energy in the class of measures with the same total mass, the function $\mu*|\mathbf x|^{2-N}$ is a positive constant on the whole of $X$ (for instance, see [5], Ch. III, Theorem 3). In fact, let $\nu$ be a complex measure on $X$ such that $I_{|\nu|, \Delta}<\infty$, and let $\langle \nu\mid 1\rangle>0$. We represent $\nu$ in the form $\nu=\mu+\nu_1$, where $\mu$ is the above nonnegative measure and $\langle \nu_1\mid 1\rangle=0$ (and therefore $\langle \nu\mid 1\rangle=\langle \mu\mid 1\rangle$). Then
$$
\begin{equation*}
I_{\mu+\nu_1, \Delta}=I_{\mu, \Delta}+I_{\nu_1, \Delta}+\int_{X}\biggl(\mu*\frac1{|\mathbf x|^{N-2}}\biggr)(\mathbf x)\,d\bigl(\nu_1(\mathbf x)+\overline{\nu_1(\mathbf x})\bigr)=I_{\mu, \Delta}+I_{\nu_1, \Delta},
\end{equation*}
\notag
$$
where the energy $I_{\nu_1, \Delta}$ is defined according to (3.5) and is nonnegative by the equality $L(\mathbf x)=|\mathbf x|^{2}$, (3.5) and (3.6). In particular,
$$
\begin{equation}
I_{\nu, \Delta}=I_{\mu+\nu_1, \Delta}\geqslant I_{\mu, \Delta}.
\end{equation}
\tag{3.12}
$$
If, in addition, $\|\nu*|\mathbf x|^{2-N}\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1$, then by Fubini’s formula
$$
\begin{equation*}
\langle \mu\mid 1\rangle \geqslant\biggl|\int_{X}\biggl(\nu*\frac1{|\mathbf x|^{N-2}}\biggr) (\mathbf x)\,d\mu(\mathbf x)\biggr| =\biggl|I_{\mu, \Delta} +\int_{X}\biggl(\mu*\frac1{|\mathbf x|^{N-2}}\biggr)(\mathbf x)\,d\nu_1(\mathbf x)\biggr|=I_{\mu, \Delta},
\end{equation*}
\notag
$$
and after the normalization $\mu^o=\mu/\langle \mu\mid 1\rangle$ we obtain $I_{\mu^o, \Delta}\leqslant1/\langle \mu\mid 1\rangle=1/\langle \nu\mid 1\rangle$, so that $\gamma_{\Delta,+}(X)\geqslant\langle \nu\mid 1\rangle$. The case of arbitrary operators $\mathcal L$ with complex coefficients is much more difficult because the machinery for the solution of the Dirichlet problem for arbitrary compact set is poorly developed. So in the proof of Proposition 1 we use the induction arguments from [11], whose purpose is to improve (3.11), namely, to replace $\mathcal Lf$ by a function with the same local means but a significantly smaller variation. The final aim is our main lemma, Lemma 9. § 4. Proof of Proposition 1First we establish Lemma 8 on the ‘almost additivity’ property of harmonic capacity under certain special decompositions. This is an analogue of parts (a) and (b) of Lemma 5.1 in [11], which is a must more complicated result on a similar property of the capacity $\gamma_+$. Let $X$ be a union of a finite family of nonoverlapping dyadic cubes (for instance — but not necessarily — the compact set $\mathcal X$ in Proposition 1). Since $X$ is regular for the Dirichlet problem, by the definition of harmonic capacity (and taking the normalization in (1.9) into account) there exists a nonnegative measure $\mu$ such that $\operatorname{Spt}(\mu)\subset X$, $\|\mu\|=((N-2)\sigma_N)^{-1}\gamma_{\Delta,+}(X)$ and $\mu*|\mathbf x|^{2-N}=1$ everywhere on $X$. Consider the function $U_{\mu}=\mu*|\mathbf x|^{2-N}$. Fix a sufficiently small positive constant $\lambda=\lambda(N)\in(0,1/10)$, whose value will be specified in Lemma 8. Let $\Omega_{\lambda,X}=\{\mathbf x\colon U_{\mu}(\mathbf x)>\lambda\}$ be a bounded open subset of $\mathbb R^N$ and $\partial \Omega_{\lambda,X}$ be its boundary. By the maximum principle for harmonic functions (and the fact that the capacity of a compact set is equal to that of its boundary) $\gamma_{\Delta,+}(\Omega_{\lambda,X})=\gamma_{\Delta,+}(X)/\lambda$. We partition $\Omega_{\lambda,X}$ into Whitney cubes (for instance, see [18], Ch. 6, § 1). This is a countable family of nonoverlapping dyadic cubes $Q_j$ with the following properties:
We denote the set of dyadic cubes $Q$ in this family that satisfy $Q\cap X^o\ne \varnothing$ by $J(\lambda,X)$. Now we show that for all sufficiently small $\lambda=\lambda(N)>0$ each cube $D$ in the set of dyadic cubes forming $X$ lies in a cube $Q$ in $J(\lambda,X)$ such that $32s(D)\leqslant s(Q)$. In particular, $J(\lambda,X)$ is a finite set. Indeed, let $Q$ and $D$ be cubes as above with a common interior point. Then $U_{\mu}(\mathbf x)=1$ on $D$, and if the claim fails (that is, $s(Q)<32s(D)$), then it follows from simple estimates for harmonic measure that at a distance of at most $4\sqrt{N}s(Q)$ to $Q$ we have $U_{\mu}(\mathbf x)\gg\lambda$, in contradiction to property (1) of Whitney cubes (recall that $4\sqrt{N}s(Q)\geqslant \operatorname{dist}(Q,\partial \Omega_{\lambda,X})$). For an arbitrary cube $Q \in J(\lambda,X)$ consider the functions ($n\in \mathbb N$)
$$
\begin{equation}
U_{\mu,Q}=\mu\big|_{(5/4)Q}*|\mathbf x|^{2-N}, \ \ \widetilde{U}_{\mu,Q}=\mu\big|_{X\setminus (5/4)Q}*|\mathbf x|^{2-N},\ \ U_{\mu,Q}^n=\mu\big|_{(2^n5/4)Q}*|\mathbf x|^{2-N}.
\end{equation}
\tag{4.1}
$$
Let $\mathbf y$ be a point in $(17/16)Q\cap X$ and $\mathbf z$ be a point in $\partial \Omega_{\lambda,X}$ closest to $Q$. We show that (here $A_{1}=A_{1}(N)$)
$$
\begin{equation}
|\widetilde{U}_{\mu,Q}(\mathbf y)-\widetilde{U}_{\mu,Q}(\mathbf z)|\leqslant A_{1}\lambda.
\end{equation}
\tag{4.2}
$$
In fact, we have
$$
\begin{equation}
U_{\mu,Q}^n(\mathbf z)\leqslant U_{\mu}(\mathbf z)=\lambda\quad\text{and} \quad U_{\mu,Q}(\mathbf z)\leqslant U_{\mu}(\mathbf z)=\lambda,
\end{equation}
\tag{4.3}
$$
where $U_{\mu,Q}^n$ and $U_{\mu,Q}$ are from (4.1), and since $ \operatorname{dist}(Q,\mathbf z)\leqslant 4\sqrt{N}s(Q)$, for $n\in\mathbb N$ we clearly have
$$
\begin{equation*}
\mu\biggl(\frac{2^n5}4Q\setminus \frac{2^{n-1}5}4Q\biggr) \leqslant (2^ns(Q))^{N-2}A_{2}\lambda.
\end{equation*}
\notag
$$
From this, taking a sum over $n$, in view of (4.3) and since $\mathbf y \in (17/16)Q$ and $|\mathbf y-\mathbf z|\leqslant8\sqrt{N}s(Q)$, we obtain (4.2):
$$
\begin{equation*}
|\widetilde{U}_{\mu,Q}(\mathbf y)-\widetilde{U}_{\mu,Q}(\mathbf z)|\leqslant A_{3}\lambda\sum_{n=1}^{\infty}(2^ns(Q))^{N-2}\frac{s(Q)}{(2^ns(Q))^{N-1}}\leqslant A_{1}\lambda.
\end{equation*}
\notag
$$
Since $\mathbf y\subset X$ and $\mathbf z\subset \partial \Omega_{\lambda,X}$, it follows that
$$
\begin{equation*}
\widetilde{U}_{\mu,Q}(\mathbf y)-\widetilde{U}_{\mu,Q}(\mathbf z)+U_{\mu,Q}(\mathbf y)-U_{\mu,Q}(\mathbf z)=U_{\mu}(\mathbf y)-U_{\mu}(\mathbf z)=1-\lambda,
\end{equation*}
\notag
$$
and, moreover, by (4.3) we have $0<U_{\mu,Q}(\mathbf z)\leqslant\lambda$. Hence we obtain from (4.2) that for sufficiently small $\lambda=\lambda(N)>0$ and $\mathbf y\in (17/16)Q\cap X$ we have $U_{\mu,Q}(\mathbf y)\geqslant1/2$. Hence, from the definition of harmonic capacity and the maximum principle for harmonic functions (where $2U_{\mu,Q}$ plays the role of a majorant), taking account of the normalization in (1.9) and the definition of $U_{\mu,Q}$ in (4.1) we obtain
$$
\begin{equation}
\gamma_{\Delta,+}\biggl(\frac{17}{16}Q\cap X\biggr) \leqslant2(N-2)\sigma_N\mu\biggl(\frac54Q\biggr).
\end{equation}
\tag{4.4}
$$
Since property (2) of Whitney cubes bounds the multiplicities of intersections of the cubes $(5/4)Q$ for $Q\in J(\lambda,X)$, by construction, (4.4), and the equality $\|\mu\|=((N-2)\sigma_N)^{-1}\gamma_{\Delta,+}(X)$ we have the following result (where the union of all cubes in $J(\lambda,X)$ is denoted by $X'$). Lemma 8. Let $X$ be a compact set formed by a finite system of nonoverlapping dyadic cubes $D_k$. Then for all sufficiently small positive $\lambda=\lambda(N) $ there exists a compact set $X'$ with the following properties: (1) $X\subset X'$; in addition, $\gamma_{\Delta,+}(X')\leqslant\gamma_{\Delta,+}(X)/\lambda$; (2) $X'$ consists of a finite number of nonoverlapping dyadic cubes $Q_j$, each of the cubes $D_k$ forming $X$ lies in some $Q_j$, and $s(Q_j)\geqslant32s(D_k)$; (3) if $(5/4)Q_j$ intersects $(5/4)Q_{j'}$, then $1/4\leqslant s(Q_j)/s(Q_{j'})\leqslant4$; (4) the following inequality holds: Consider an ‘almost maximum possible’ positive $\lambda$ such that Lemma 8 holds. Consider an arbitrary cube $Q\in J(\lambda,X)$. By property (1) of cubes in Lemma 8 we have $\gamma_{\Delta,+}(X)/(s(Q))^{N-2}>\lambda$, which is equivalent to the inequality
$$
\begin{equation}
(s(Q))^{N-2}<(\operatorname{diam}(X))^{N-2} \frac{\gamma_{\Delta,+}(X)}{\lambda(\operatorname{diam}(X))^{N-2}}.
\end{equation}
\tag{4.6}
$$
To use the induction argument from [11] it is important that the cubes $Q_j$ in $X'$ have diameters at least twice less than that of $X$. This can easily be attained if $\gamma_{\Delta,+}(X)$ is significantly less than $(\operatorname{diam}(X))^{N-2}$. Otherwise, that is, if $\gamma_{\Delta,+}(X)$ and $(\operatorname{diam}(X))^{N-2}$ are commensurable, Theorem 1 and Proposition 1 follow easily from (2.4) as consequences. In view of (4.6), assuming that the ratio $\gamma_{\Delta,+}(X)/(\operatorname{diam}(X))^{N-2}$ is sufficiently small, we suppose in what follows that the cubes $Q_j$ in Lemma 8 satisfy $\operatorname{diam}(Q_j)=\sqrt{N}s(Q_j)\leqslant(1/10)\operatorname{diam}(X)$, which is in fact condition (c) in Lemma 5.1 in [11]. Now we can use the induction arguments analogous to § 7.1 in [11]. First let $X$ be the original compact set $\mathcal X$ from Proposition 1. We have an alternative: either for all $j$ we have
$$
\begin{equation}
\frac{\gamma_\mathcal L(X)}{\gamma_{\Delta,+}(X)}\geqslant \frac{\gamma_{\mathcal L}(X\cap (17/16)Q_j)}{2\gamma_{\Delta,+}(X\cap (17/16)Q_j)}
\end{equation}
\tag{4.7}
$$
(Case 1), or for some $j$ we have the reverse inequality (Case 2). It is obvious from the structure of $X$ and $Q_j$ that in all fractions in (4.7) the numerators and denominators are distinct from zero. Case 1. Below, as a consequence of Lemmas 5 and 8 and inequalities (4.7) we establish the following result (here $A_5=1/\lambda$, where the ‘almost maximum’ $\lambda=\lambda(N)$ from Lemma 8 is taken). Lemma 9 (main lemma). Let $X$ be a compact set such that $\gamma_{\mathcal L}(X)>0$, and let $f=(\mathcal Lf)*\Phi_{\mathcal L}$ be a function such that $\operatorname{Spt}(\mathcal Lf)\subset X$, $\|f\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1$ and ${\langle \mathcal Lf\mid 1\rangle}\geqslant(1/2)\gamma_{\mathcal L}(X)$. In this case there exists a compact set $X'$ consisting of a finite number of nonoverlapping dyadic cubes $Q_j$ and such that $X\subset X'$ and the following hold:
Then for each $M\geqslant2^N$ there exist $\alpha_M>0$, a compact set $\mathbf X$ formed by a finite number of nonoverlapping dyadic cubes and a complex function $\nu\in C^{\infty}_0(\mathbf X)$ such that
From Lemma 9 we can easily deduce Proposition 1 for the compact set $X$. In fact, as in (3.12), there exists a nonnegative measure $\mu_\mathbf X$ such that $\operatorname{Spt}(\mu_\mathbf X) \subset \mathbf X$, $\langle \mu_{\mathbf X}\mid 1\rangle=\langle \nu\mid 1\rangle$ and $I_{\mu_\mathbf X, \Delta}\leqslant I_{\nu, \Delta}$. Hence, by property (c), $\gamma_{\Delta,+}(\mathbf X)>\gamma_{\mathcal L}(X)/(4\alpha_M M)$. In combination with (b) and condition (1), this yields
$$
\begin{equation}
A_5\gamma_{\Delta,+}(X)\geqslant\gamma_{\Delta,+}(X')\geqslant\gamma_{\Delta,+}(\mathbf X)-\frac{A_7}{M}\gamma_{\mathcal L} (X)\geqslant\frac{1}{M}\biggl(\frac{1}{4\alpha_M}-A_7\biggr)\gamma_{\mathcal L}(X).
\end{equation}
\tag{4.8}
$$
Now using property (a) we can select $M=M(L)$ so that $M^{1/9}/(4A_6)>2A_7$ and therefore $A\gamma_{\Delta,+}(X)\geqslant \gamma_{\mathcal L}(X)$. Case 2. Consider $j$ such that (4.7) fails. Let $\gamma_{\mathcal L}(X)=C\gamma_{\Delta,+}(X)$, where $C$ is a positive constant of which we assume nothing. In Case 2, for $j$ under consideration we have $\gamma_\mathcal L(X\cap (17/16)Q_j)\geqslant2C\gamma_{\Delta,+}(X\cap (17/16)Q_j)$, and moreover, ${\operatorname{diam}(X\cap (17/16)Q_j)<(1/5)\operatorname{diam}(X)}$. Recall that the compact set $X$ in Lemma 8 is an arbitrary finite family of nonoverlapping dyadic cubes (which does not necessarily coincide with the original compact set $\mathcal X$ of the construction). Setting $X:={\mathcal X}^{(1)}=\mathcal X\cap (17/16)Q_j$ (where (1) is the ordinal number of the iteration) we repeat the construction leading to Lemma 8 and consider the alternative arising in connection with (4.7). If inequality (4.7) holds for $X=\mathcal X^{(1)}$, then using our main lemma (Lemma 9) we obtain $A\gamma_{\Delta,+}(\mathcal X^{(1)})\geqslant \gamma_\mathcal L(\mathcal X^{(1)})$, which yields
$$
\begin{equation*}
A\gamma_{\Delta,+}\biggl(\mathcal X\cap \frac{17}{16}Q_j\biggr)\geqslant 2C\gamma_{\Delta,+}\biggl(\mathcal X\cap \frac{17}{16}Q_j\biggr),
\end{equation*}
\notag
$$
so that $C\leqslant A/2$. Hence $\gamma_\mathcal L(\mathcal X)\leqslant(A/2)\gamma_{\Delta,+}(\mathcal X)$, which completes the proof of Proposition 1. If inequality (4.7) does not hold for $X=\mathcal X^{(1)}$, then arguing as at the previous step we can assume that for some $j$
$$
\begin{equation*}
\operatorname{diam}\biggl(\mathcal X^{(1)}\cap \frac{17}{16}Q_j^{(1)}\biggr)<\frac15\operatorname{diam}(\mathcal X^{(1)})
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\frac{\gamma_\mathcal L(\mathcal X^{(1)}\cap (17/16)Q_j^{(1)})}{\gamma_{\Delta,+}(\mathcal X^{(1)}\cap (17/16)Q_j^{(1)})}\geqslant2\frac{\gamma_\mathcal L(\mathcal X^{(1)})}{\gamma_{\Delta,+}(\mathcal X^{(1)})}\geqslant4C.
\end{equation*}
\notag
$$
Now we continue this procedure if necessary, and the occurrence of Case 1 corresponds to its termination. If we terminate at the $k$th iteration, then $C\leqslant A/2^k$. As a result, $\gamma_{\mathcal L}(\mathcal X)\leqslant(A/2^k)\gamma_{\Delta,+}(\mathcal X)$, which proves Proposition 1. On the other hand Case 2 cannot occur infinitely long, because condition (2) in Lemma 8 is satisfied at each iteration, so that the dyadic cubes $D_k$ forming the original compact set $\mathcal X$ do not decrease (the cubes $(17/16)Q_j$ do not ‘cut’ them since the dyadic cubes $Q_j$ are quite large); in addition the linear sizes of the cubes $(17/16)Q_j$ decrease with coefficient at least 5, and in the case when $X$ consists of one dyadic cube inequality (4.7) is trivial. Thus, for the proof of Proposition 1 and Theorem 1 it remains to establish Lemma 9. Proof of Lemma 9. The role of condition (4.7) consists in transferring estimate (4.5) from $\gamma_{\Delta,+}(X)$ to $\gamma_\mathcal L(X)$. By (4.5) and (4.7) we have
Now we perform the (main and additional) constructions, prove Lemmas 10–12 and complete the proof of Lemma 9. It is convenient to construct for each cube $Q_j$ a (nonnegative) function $\mu_{\mathcal L,j}$ with support on $(1/4)Q_j$ such that
$$
\begin{equation}
(1)\ \mu_{\mathcal L,j}(\mathbf x)=\mathrm{const}, \quad\text{and} \quad (2) \ \int_{(1/4)Q_j}\mu_{\mathcal L,j}(\mathbf x)\,d\mathbf x=\gamma_{\mathcal L}\biggl(X\cap \frac{17}{16}Q_j\biggr);
\end{equation}
\tag{4.10}
$$
let $ \mu_\mathcal L=\sum_j\mu_{\mathcal L,j}$. Consider some $M\geqslant2^{N}$. Since the capacity $\gamma_\mathcal L$ does not change under parallel translations, we assume without loss of generality that $\bigcup_jQ_j$ lies in a sufficiently large dyadic cube $D=D_0$ and the inequality
$$
\begin{equation}
0<\int_D\mu_{\mathcal L}(\mathbf x)\,d\mathbf x< 2^{-N}(s(D))^{N-2}M
\end{equation}
\tag{4.11}
$$
is satisfied. We perform the Calderón–Zygmund dyadic decomposition recursively, starting with $D=D_0$. If inequality (4.11) holds for a cube $D$, and $D$ is distinct from all the $Q_j$, then $D$ is partitioned into $2^N$ dyadic cubes with half the edge length. Otherwise we fix $D$, provided that it contains at least one cube $Q_j$, and leave it without consideration if it contains no $Q_j$ (so that the integral is zero). Thus, for a fixed cube $D$ at least one of two conditions is satisfied: either $D=Q_j$ for some $j$ or
$$
\begin{equation}
2^{-N}(s(D))^{N-2}M\leqslant\int_D\mu_{\mathcal L}(\mathbf x)\,d\mathbf x< (s(D))^{N-2}M.
\end{equation}
\tag{4.12}
$$
We repeat this procedure for each cube obtained by subdivision. The process terminates after a finite number of steps, once all the cubes obtained have been fixed. At the end of our construction we obtain the compact set $\mathbf X$ required in Lemma 9, which is formed by all cubes fixed in the construction. These are either cubes $Q_j$ in $X'$, which we denote by $Q_j^{\mathbf s}$, or cubes $D$ fixed in accordance with (4.12) and distinct from all the $Q_j$. We call them new $Q_j$ and denote them (after ordering) by $Q_j^{\mathbf n}$. In addition, we exclude from our consideration all the (old) $Q_j$ lying inside the (new) $D$, keeping (provisionally, till we start an additional construction) the corresponding functions $\mu_{\mathcal L,j}$ with $\operatorname{Spt}(\mu_{\mathcal L,j})\subset D$, and the function $\mu_\mathcal L$. As a result, we have the following lemma. Lemma 10. The following estimates hold: (1) $\quad\displaystyle \sum_j(s(Q_j^{\mathbf n}))^{N-2}\leqslant \frac{A_8}M\gamma_{\mathcal L}(X), \qquad A_8=A_8(N);$ (2) if $D$ is a dyadic cube containing at least one cube $Q_j^{\mathbf s}$ or $Q_j^{\mathbf n}$, then
$$
\begin{equation*}
\int_D\mu_{\mathcal L}(\mathbf x)\,d\mathbf x< (s(D))^{N-2}M;
\end{equation*}
\notag
$$
(3) if the dyadic cube $D$ contains at least one $Q_j^{\mathbf n}$, then the sum $(s(Q_j^{\mathbf n}))^{N-2}$ over all cubes $Q_j^{\mathbf n}$ in $D$ does not exceed $2^N(s(D))^{N-2}$; (4) $\gamma_{\Delta,+}(\mathbf X)\leqslant \gamma_{\Delta,+}(X')+(A_7/M)\gamma_{\mathcal L}(X)$. Proof. Estimate (1) follows from the first inequality in (4.12), (4.9) and (4.10). Estimate (2) follows from the construction: otherwise, by (4.12) $D$ is a proper subset of some $Q_j^{\mathbf n}$. Estimate (3) follows from (2) and the first inequality in (4.12) for all the $Q_j^{\mathbf n}$:
Estimate (4) follows from (1) and the semiadditivity of harmonic capacity:
$$
\begin{equation*}
\gamma_{\Delta,+}(\mathbf X)\leqslant \gamma_{\Delta,+}\biggl(\bigcup_jQ_j^{\mathbf s}\biggr) +\sum_j\gamma_{\Delta,+}(Q_j^{\mathbf n}) \leqslant\gamma_{\Delta,+}(X')+\frac{A_7}{M}\gamma_{\mathcal L}(X).
\end{equation*}
\notag
$$
Lemma 10 is proved. Additional construction. Now for each cube $Q_j^{\mathbf n}$ we exclude from consideration all functions $\mu_{\mathcal L,j}$ such that $\operatorname{Spt}(\mu_{\mathcal L,j})\subset Q_j^{\mathbf n}$; instead, we construct a single function $\mu_{\mathcal L,j}$ in accordance with (4.10) for $Q_j=Q_j^{\mathbf n}$. Remark 1 (preservation of estimates). By estimate (1) in Lemma 10 and the elementary estimate (2.4), inequality (4.9) is preserved in the transition from $X'$ in $\mathbf X$, although the constant $A_4$ can slightly increase. By estimate (3) in Lemma 10 and by (2.4), for the corresponding function $\mu_\mathcal L=\sum_j\mu_{\mathcal L,j}$ we still have estimate (2) in Lemma 10 for $M$ replaced by $M+A_8$ on the right-hand side. Now using the covering $\{Q_j\}=\{Q_j^{\mathbf s}\}\cup\{Q_j^{\mathbf n}\}$ we construct a partition of unity $\{\varphi_j\}$ from Lemma 1 and represent the original function $f$ from Lemma 9 as a finite sum of localizations (2.5). For each localization $V_{\varphi_j}f$ we construct the functions $\nu_{Q_j}$ and $h_{Q_j}=\nu_{Q_j}*\Phi_{\mathcal L}$ from Lemma 4, where $\operatorname{Spt}(\nu_{Q_j})\subset(1/4)Q_j$, and we set $\nu=\sum_j\nu_{Q_j}$. The following result holds by our construction, Lemmas 5 and 10 and the assumptions of Lemma 9. Lemma 11. (i) The following estimates hold: (ii) Estimates (2.15), (1) and (2) hold for an arbitrary dyadic cube $Q$, for a perhaps slightly greater constant $A_5=A_5(L)$. (iii) If $Q'$ and $Q''$ are distinct cubes in $\{Q_j^{\mathbf s}\}$ and $(5/4)Q'$ intersects $(5/4)Q''$, then $1/4\leqslant s(Q')/s(Q'')\leqslant4$; if $Q'$ is a cube in $\{Q_j^{\mathbf s}\}$ and $Q''$ is a cube in $\{Q_j^{\mathbf n}\}$, then $(5/4)Q'$ is disjoint from $(1/4)Q''$, the support of the function $\nu_{Q''}$ in (2.13). Proof. In (i), part (a) follows from Lemma 4 and the conditions on $f$ imposed in Lemma 9, and part (b) follows from the inequalities $|\nu_{Q_j}(\mathbf x)|\leqslant A_4\mu_{\mathcal L,j}(\mathbf x)$ (see (4.10) and Lemma 4) and the fact that the transition from $X'$ to $\mathbf X$ preserves estimates (4.9). Estimates (2.15) follows from the fact that the transition from $X'$ to $\mathbf X$ preserves estimate (2) in Lemma 10 (for $M$ replaced by $M+A_8$), (4.10) and Lemma 5.
Assertion (iii) for the cubes $Q_j^{\mathbf s}$ follows from the hypotheses of Lemma 9. Since each cube contains by construction at least one cube in $X'$ that is distinct from $X'$ (so that the edge length is at least twice as large), from condition (2) in Lemma 9 we also obtain the required result for pairs of cubes from $Q_j^{\mathbf s}$ and $Q_j^{\mathbf n}$. Lemma 11 is proved. The following key estimate is a consequence of Lemmas 4, 5 and 11. We need the complex conjugate of $\nu$ to calculate the energy $I_{\nu, \mathcal L}$ in accordance with (3.5). Lemma 12. Let $\{Q_j\}=\{Q_j^{\mathbf s}\}\cup\{Q_j^{\mathbf n}\}$ and $g_j=h_{Q_j}-V_{\varphi_j}f$, where $h_Q$ is from Lemma 4. Then Proof. First let $Q_j=Q_j^{\mathbf n}$. By (2.12), (2.4) and (2.15), 1) we have
$$
\begin{equation}
\int_\mathbf X|g_j(\mathbf x)\nu(\mathbf x)|\,d\mathbf x\leqslant A_{11} (s(Q_j))^{N-1}\sum_{n=1}^{\infty}\frac{(2^ns(Q_j))^{N-2}M}{(2^ns(Q_j))^{N-1}}=A_{11} (s(Q_j))^{N-2}M,
\end{equation}
\tag{4.14}
$$
so that it follows from estimate (1) in Lemma 10 that
For the cubes $Q_j=Q_j^{\mathbf s}$ the proof of (4.13) is much more complicated and uses estimate (2.15), (2). Let $Q_j=Q(\mathbf a_j,s_j)$ be some cube $Q_j^{\mathbf s}$ and $Q=Q(\mathbf a,s)$ be a dyadic cube such that $(9/8)Q_j$ and $(17/16)Q$ are disjoint, and let $\varphi\in C_0^{\infty}((17/16)Q)$ be a nonnegative function from Lemma 5 corresponding to $Q$. Then
$$
\begin{equation*}
\begin{aligned} \, &\int_{(17/16)Q}g_j(\mathbf x)\varphi(\mathbf x)\overline{\nu(\mathbf x)}\,d\mathbf x \\ &\qquad = g_j(\mathbf a)\int_{(17/16)Q}\varphi(\mathbf x)\overline{\nu(\mathbf x)}\,d\mathbf x +\int_{(17/16)Q}(g_j(\mathbf x)-g_j(\mathbf a))\varphi(\mathbf x)\overline{\nu(\mathbf x)}\,d\mathbf x. \end{aligned}
\end{equation*}
\notag
$$
For the first integral on the right-hand side we use (2.15), (2), and also (2.12). Then we obtain
$$
\begin{equation}
\begin{aligned} \, \notag & \biggl|\int_{(17/16)Q}g_j(\mathbf x)\varphi(\mathbf x)\overline{\nu(\mathbf x)}\,d\mathbf x\biggr| \\ &\qquad \leqslant \frac{A_{13}s_j\gamma_{\mathcal L}((17/16)Q_j\cap X)}{|\mathbf a_j-\mathbf a|^{N-1}}\biggl(M^{2/3} s^{N-2}+\frac{s}{|\mathbf a_j-\mathbf a|}\int_{(17/16)Q}|\nu(\mathbf x)|\,d\mathbf x\biggr). \end{aligned}
\end{equation}
\tag{4.15}
$$
Using a construction due to Whitney (see [18], Ch. 6, § 1) we partition the complement to $\mathbf a_j$ into a countable family of nonoverlapping dyadic cubes. Subdividing each Whitney cube into the same number (depending on $M$) of dyadic cubes it is easy to obtain the following. Let $\{D_m\}$ be the family of those cubes obtained that are disjoint from $((5/4)Q_j)^o$. Then the following assertions hold. 1. For each $m$ the cube $(17/16)D_m$ is disjoint from $(9/8)Q_j$. 2. If $(5/4)D_m$ and $(5/4)D_{m'}$ intersect, then $1/4\leqslant s(D_m)/s(D_{m'})\leqslant4$. 3. There exists a constant $A_{14}>1$ which only depends on $N$ and such that
$$
\begin{equation}
A_{14}^{-1}\operatorname{dist}(D_m,\mathbf a_j)M^{-1/9}\leqslant s(D_m)\leqslant A_{14}\operatorname{dist}(D_m,\mathbf a_j)M^{-1/9}.
\end{equation}
\tag{4.16}
$$
The following results are consequences of this construction. (A) The intersections of the cubes $(9/8)D_m$ have multiplicity bounded by a constant depending only on $N$. (B) By (4.16) the number of cubes $D_m$ intersecting a layer $U_k=\{s_j2^{k-1}\leqslant \operatorname{dist}(\mathbf x,\mathbf a_j)\leqslant s_j2^{k}\}$, where $k\in\mathbb N$, is at most $A_{15}M^{N/9}$. Correspondingly, the sum of the terms $(s(D_m))^{N-2}$ over all such cubes is at most $A_{15}M^{N/9}(2^ks_j/M^{1/9})^{N-2}=A_{15}M^{2/9}(2^ks_j)^{N-2}$. Hence, taking (2.15), 1) into account, for nonoverlapping dyadic cubes with edge length $2^ks_j$ that intersect $U_k$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\sum_{\{m|D_m\cap U_k\ne\varnothing\}}\biggl(M^{2/3} (s(D_m))^{N-2}+\frac{s(D_m)}{2^ks_j}\int_{(17/16)D_m}|\nu(\mathbf x)|\,d\mathbf x\biggr) \\ &\qquad \leqslant A_{16}\bigl(M^{2/3}M^{2/9}+M^{-1/9}M\bigr)(2^ks_j)^{N-2}. \end{aligned}
\end{equation}
\tag{4.17}
$$
It is clear that only finitely many cubes $D_m$ intersect the compact set $\mathbf X$. Consider a partition of unity $\{\varphi_m\}$ from Lemma 1 on $\bigcup_mD_m$ such that $\varphi_m\in C_0^{\infty}((17/16)D_m)$. Summing over all the $U_k$ similarly to (4.14) and using estimate (4.15) for ${Q\!=\!D_m}$, taking (4.17) into account we obtain
$$
\begin{equation*}
\sum_m\biggl|\int_{(17/16)D_m}g_j(\mathbf x)\varphi_m(\mathbf x)\overline{\nu(\mathbf x)}\,d\mathbf x\biggr|\leqslant 2A_{16}M^{8/9}\gamma_{ \mathcal L}\biggl(\frac{17}{16}Q_j\cap X\biggr).
\end{equation*}
\notag
$$
Summing these inequalities over all $j$, since (4.9) remains valid after going over from $X'$ to $\mathbf X$, we have
$$
\begin{equation*}
\sum_{\{j|Q_j=Q_j^{\mathbf s}\}}\biggl|\int_{\mathbf X\setminus(5/4)Q_j}g_j(\mathbf x)\overline{\nu(\mathbf x)}\,d\mathbf x\biggr|\leqslant A_{17}M^{8/9}\gamma_\mathcal L(X).
\end{equation*}
\notag
$$
In the case of integration over $(5/4)Q_j$ for $M=1$ a similar estimate follows from Lemma 4, (2) and Lemma 11 (part (i), (b) and part (iii), which ensures that the intersections of the supports of the relevant $\nu$ have a finite multiplicity). This establishes (4.13). Lemma 12 is proved. We complete the proof of our main lemma (Lemma 9). As $\|f\|_{\mathrm L^{\infty}(\mathbb R^N)}\leqslant1$, integrating the equality
$$
\begin{equation*}
\sum_jg_j=\sum_j\bigl(\nu_{Q_j}*\Phi_{\mathcal L}-V_{\varphi_j}f\bigr)=\nu*\Phi_{\mathcal L}-f
\end{equation*}
\notag
$$
with respect to $\overline{\nu(\mathbf x)}d\mathbf x$, in view of Lemma 12 and part (i), (b) of Lemma 11, using the notation (3.5), for the resulting function $\nu$ we obtain
$$
\begin{equation*}
|I_{\nu, \mathcal L}|\leqslant (N-2)\sigma_N(A_9+A_{10}M^{8/9})\gamma_{\mathcal L}(X).
\end{equation*}
\notag
$$
Now, just as in the proof of Proposition 2, using the Fourier transform, (3.1), (3.6), (3.8) and (3.9) we obtain
$$
\begin{equation*}
I_{\nu, \Delta}\leqslant A_{18} M^{8/9}\gamma_{\mathcal L}(X);
\end{equation*}
\notag
$$
in combination with part (i), a) of Lemma 11 this proves assertions (a) and (c) of Lemma 9. Assertion (b) of Lemma 9 was established in part (4) of Lemma 10. Lemma 9, and therefore also Proposition 1 and Theorem 1, are proved. § 5. Several corollaries to Theorem 1Theorem 2. Let $N\geqslant3$, and let $\operatorname{Cap}$ be any of the capacities $\gamma_{\mathcal L}$, $\alpha_{\mathcal L}$, $\gamma_{{\mathcal L,+}}$ and $\alpha_{{\mathcal L,+}}$. Then the following hold: (1) there exists a constant $A=A(L)>1$ such that for each bounded (Borel) set $U$
$$
\begin{equation*}
A^{-1}\gamma_{\Delta,+}(U)\leqslant\operatorname{Cap}(U)\leqslant A\gamma_{\Delta,+}(U);
\end{equation*}
\notag
$$
(2) these capacities are countably subadditive, namely, if $U=\bigcup_jU_j$, then Proof. Part (1) is a direct consequence of Theorem 1, equality (1.10) and the elementary estimates (1.8) and (1.11). Part (2) follows from (1) and the semiadditivity of harmonic capacity:
Theorem 2 is proved. Theorem 2 extends to strongly elliptic equations in $\mathbb R^2$. This case is special because fundamental solutions are unbounded at infinity, and the capacities $\gamma_{\mathcal L}$, $\alpha_{\mathcal L}$, $\gamma_{{\mathcal L,+}}$ and $\alpha_{{\mathcal L,+}}$ are defined locally. Namely, the function $\Psi_0$ in (1.4) is defined up to an additive constants; we fix this constant as in [8], § 1.3; in this case $|\Phi_{\mathcal L}(\mathbf x)|\leqslant A_1(L)$ for $|\mathbf x|=1$. We define capacities in accordance with [8], (1.4), for $R=1$. Let $U\subset B(\mathbf a,1/2)$. Then
$$
\begin{equation}
\gamma_{\mathcal L}(U)=\gamma_{\mathcal L}^1(U)=\sup_T\{|\langle T\mid 1\rangle|\colon \operatorname{Spt}(T)\subset U,\,\|T*\Phi_{\mathcal L}\|_{\mathrm L^{\infty}(B(\mathbf a,1))}\leqslant1\};
\end{equation}
\tag{5.1}
$$
for the capacity $\gamma_{{\mathcal L,+}}$ the distributions $T$ in (5.1) are nonnegative measures with support in $U$, and for $\alpha_{\mathcal L}$ and $\alpha_{{\mathcal L,+}}$ we assume additionally in (5.1) that $T* \Phi_{\mathcal L}\in C(B(\mathbf a,1))$. Recall that in $\mathbb R^2$ we have $\Phi_{\Delta}=(1/{2\pi})\log|\mathbf x|$. Taking the sign of the logarithm into account, we can give an alternative definition of $\gamma_{{\Delta,+}}$ (see [17], Ch. II, § 4):
$$
\begin{equation}
\gamma_{{\Delta,+}}(U)=\sup_{\mu}\{\|\mu\|\colon \operatorname{Spt}(\mu)\subset U,\, \mu\geqslant0,\,\mu*\Phi_{\Delta}\geqslant-1\};
\end{equation}
\tag{5.2}
$$
note that the capacity in (5.2) is $2\pi$ times greater than the harmonic (Wiener) capacity. Recall that for harmonic capacities in $\mathbb R^2$ we have equality (1.10) (see [8], Proposition 2.1), and the capacities $\gamma_{\Delta,+}$ and $\gamma_{\mathcal L,+}$ are commensurable up to a positive constant $A(L)$ (see [8], Proposition 2.3). Theorem 1.1 in [8] reduces the question of the commensurability of the capacities $\gamma_{\mathcal L}(U)$ and $\gamma_{{\Delta,+}}(U)$ and $\mathbb R^2$ to the commensurability of the capacities of Cartesian products $\gamma_{\mathcal L}(U\times[-1,1])$ and $\gamma_{{\Delta,+}}(U\times[-1,1])$ in $\mathbb R^3$. Hence, by Theorem 1 for $U\times[-1,1]$ in $\mathbb R^3$ we have the following result. Theorem 3. Let $U$ be a subset of the disc $B(\mathbf a,1/2)$ and $\operatorname{Cap}$ be any of the capacities $\gamma_{\mathcal L}$, $\alpha_{\mathcal L}$, $\gamma_{\mathcal L,+}$ and $\alpha_{{\mathcal L,+}}$ defined in accordance with (5.1). Let $\gamma_{{\Delta,+}}$ be defined by (5.2). Then parts (1) and (2) of Theorem 2 hold in this notation. Note that the case $U\subset B(\mathbf a,R/2)$, where $B=B(\mathbf a,R)$ is arbitrary, is treated by means of a dilation (see (1.4), (1.5) and Proposition 2.2 in [8]). Namely, the fundamental solution $\Phi_\mathcal L$ in (1.4) fixed for $R=1$ is replaced by $\Phi_\mathcal L^R(\mathbf x) = k_0\log|\mathbf x/R| + \Psi_0(\mathbf x)$ (that is, changes by an additive constant), and local capacities are introduced similarly to
$$
\begin{equation}
\gamma_{\mathcal L}^B(U)=\sup_T\{|\langle T\mid 1\rangle|\colon \operatorname{Spt}(T)\subset U,\,\|T*\Phi_{\mathcal L}^R\|_{\mathrm L^{\infty}(B)}\leqslant1\}.
\end{equation}
\tag{5.3}
$$
Here for each dilation $P$ in $\mathbb R^2$ we have the equality
$$
\begin{equation}
\gamma_{\mathcal L}^B(U)=\gamma_{\mathcal L}^{P(B)}(P(U)).
\end{equation}
\tag{5.4}
$$
Recall (see [3]) that in terms of the capacity $\gamma_{\mathcal L}$ we can describe removable singularities of solutions of the equation $\mathcal Lf=0$ in the class of bounded function, and in terms of $\alpha_{\mathcal L}$ we can do this for removable singularities in the class of continuous functions. By Theorem 1, for all $\mathcal L$ under consideration we have the following result. Theorem 4. A compact set $K\subset\mathbb R^N$, $N\geqslant2$, is removable for solutions of the equation $\mathcal Lf=0$ in the classes of bounded and continuous functions if and only if $\gamma_{{\Delta,+}}(K)=0$. In combination with Theorem 1 in [5], Ch. IV, Theorem 4 provides estimates for the dimensions of removable sets for solutions of the equation $\mathcal Lf=0$. Note that no such estimates whatsoever were previously known for $N=2$ and arbitrary $\mathcal L$. Here we present just one new result. Corollary 1. Let $K\subset\mathbb R^2$ be a compact set. If for each $\varepsilon>0$ $K$ can be covered by a finite family of open discs $B_j$ of radii $r_j<1$ such that $\sum_j(\log(1/r_j))^{-1}<\varepsilon$, then $K$ is removable for solutions of the equations $\mathcal Lf=0$ in the classes of bounded and continuous functions for all $\mathcal L$. Recall that a criterion for the uniform approximability of continuous functions by solutions of the equations $\mathcal Lf=0$ on compact sets in $\mathbb R^N$ was obtained in [4] for $N\geqslant3$, in terms of the capacities $\gamma_{\mathcal L}$, and in [9] for $N=2$, in terms of the capacities $\gamma_{\mathcal L}^B$ from (5.3). Such a criterion can be stated in various equivalent forms; Theorem 5 is a consequence of our Theorem 1 and Theorems 1 and 3 in [4] for $N\geqslant3$. Let $N\geqslant3$, and let $X\subset\mathbb R^N$ be a nonempty compact set. Given a function $f\in C(X)$, we assume that it is extended outside $X$ (by the Urysohn–Brouwer lemma) as a continuous function with compact support; let $\omega_f$ be its modulus of continuity. We let $H_\mathcal L(X)$ denote the class of functions $f\in C(X)$ such that for each $\varepsilon>0$ there exists a function $F$ satisfying $\mathcal LF=0$ in a neighbourhood and such that $\|F-f\|_{\mathrm L^{\infty}(X)}<\varepsilon$. In approximation problems the notion of the $\mathcal L$-oscillation $\mathcal O^\mathcal L_B(f)$ of a function $f$ on the ball $B=B(\mathbf a,r)$ was introduced by Paramonov in [19], § 2. Namely, using the notation (1.1), for the polynomial $L$, the symbol of an operator $\mathcal L$, we have ($m(B)$ is the volume of $B$ in $\mathbb R^N$)
$$
\begin{equation*}
\mathcal O^\mathcal L_B(f)=\frac1{{\sigma(\partial B)}}\int_{\partial B}f(\mathbf x) \frac{L(\mathbf x-\mathbf a)}{r^2}\,d\sigma_{\mathbf x}-\frac{\sum_{j=1}^Nc_{jj}}{m(B)N}\int_B f(\mathbf x)\,d\mathbf x.
\end{equation*}
\notag
$$
In particular,
$$
\begin{equation*}
\mathcal O^{\Delta}_B(f) =\frac1{{\sigma(\partial B)}}\int_{\partial B}f(\mathbf x)\,d\sigma_\mathbf x-\frac1{m(B)}\int_B f(\mathbf x)\,d\mathbf x
\end{equation*}
\notag
$$
is the difference between the mean values of the function on the sphere and in the ball (which can be regarded as a measure of the deviation from the mean value theorem). Theorem 5. If there exist a constant $k\geqslant1$ and a function $\epsilon\colon\mathbb R_+\to\mathbb R_+$, $\epsilon(t)\to0$ as $t\to0$, such that for each ball $B$ of radius ${r}$ By Theorems 1.1 and 1.4 in [9] and by (5.4), Theorem 5 extends to $N=2$. Then condition (5.5) assumes the form
$$
\begin{equation*}
|\mathcal O^L_B(f)|\leqslant \epsilon({r})\gamma_{{\Delta,+}}(P(k B\setminus X)),
\end{equation*}
\notag
$$
where $P$ is the composition of a translation and a dilation that takes the disc $k B$ to $B(\mathbf 0,1/2)$, and $\gamma_{{\Delta,+}}$ is the capacity in (5.2).
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