M. Ya. Mazalov, “Commensurability of some capacities with harmonic capacities”, Russian Math. Surveys, 78:5 (2023), 964

For ${\mathbf x}=(x_1,\dots,x_N) \in {\mathbb R}^N$, $N\geqslant3$, let р $L({\mathbf x})$ be a second-degree homogeneous polynomial in ${\mathbb R}^N$ with constant complex coefficients which satisfies the condition of ellipticity $L({\mathbf x})=0 \Leftrightarrow{\mathbf x}=0$. The following result was established in [1], Lemma 3.

The polynomial $L({\mathbf x})$ is the symbol of an elliptic differential operator ${\mathcal L}$; for example, $|{\mathbf x}|^2=\sum_{n=1}^N x_n^2$ corresponds to the Laplace operator ${\mathcal L}=\Delta$ in ${\mathbb R}^N$. The fundamental solution $\Phi({\mathbf x})=\Phi_{{\mathcal L}}({\mathbf x})$ of ${\mathcal L}$ is a homogeneous function of order $2-N$ in the class $C^{\infty}({\mathbb R}^N \setminus \{0\})$; an explicit expression can be found in [1], Theorem 1.

The capacity $\gamma_{{\mathcal L,+}}$ can be defined similarly to the classical harmonic capacity $\gamma_{\Delta}$ in potential theory. Let $K\subset{\mathbb R}^N$ be a compact set; then

Since $|\Phi_{{\mathcal L}}({\mathbf x})|\leqslant A_1(L)|{\mathbf x}|^{2-N}$ for each ${\mathcal L}$, it follows that $A(L)\gamma_{{\mathcal L},+}(K)\geqslant \gamma_{\Delta}(K)$ for all $K$. Consider the question of whether this inequality is reversible (that is, whether $\gamma_{\Delta}$ and $\gamma_{{\mathcal L},+}$ are commensurable). This question is non-trivial in the general case because ${\mathcal L}$ has complex coefficients.

For $N=3$ and $N=4$ the fact that $\gamma_{\Delta}$ and ш $\gamma_{{\mathcal L},+}$ are commensurable was established in [1], Corollary 3, but this proof cannot be extended to $N>4$. Below (in Theorem 1) we prove the corresponding result for all $N\geqslant3$; for $N>4$ this is in fact Conjecture 1 in [1]. The case $N=2$ was consudered in [2], Proposition 2.3.

Proof. By (2) there exists a non-negative measure $\mu$ such that $\operatorname{Spt}(\mu)\subset K$, $\|\mu*\Phi_{{\mathcal L}}\|_{{\rm L}^{\infty}({\mathbb R}^N)}\leqslant1$, and $\|\mu\|\geqslant(1/2)\gamma_{{\mathcal L},+}(K)$. Let us regularize $\mu$ in the standard way. Fix a function $\varphi_1 \in C^{\infty}_0(B)$ (where $B$ is a unit ball in ${\mathbb R}^N$) such that $\varphi_1\geqslant0$ and $\displaystyle\int_{B}\varphi_1({\mathbf x})\,d{\mathbf x}=1$. For $\varepsilon>0$ set $\varphi_{\varepsilon}({\mathbf x})= \varepsilon^{-N}\varphi_1({\mathbf x}/\varepsilon)$, so that $\displaystyle\int_{|{\mathbf x}|\leqslant\varepsilon} \varphi_{\varepsilon}({\mathbf x})\,d{\mathbf x}= 1$. Let $h_{\varepsilon}=\mu*\varphi_{\varepsilon}$; then $h_{\varepsilon}\geqslant0$ and $h_{\varepsilon}\in C^{\infty}_0(K_{\varepsilon})$, where $K_{\varepsilon}$ is the closure of the $\varepsilon$-neighbourhood of $K$ and $\displaystyle\int_{{\mathbb R}^N}h_{\varepsilon}({\mathbf x}) \,d{\mathbf x}=\|\mu\|$.

We look at the following (generally speaking, complex-valued) energy integral:

$$ \begin{equation} I_{h_{\varepsilon},{\mathcal L}}=-(N-2)\sigma_N\int_{K_{\varepsilon}} (h_{\varepsilon}*\Phi_{{\mathcal L}})({\mathbf x})h_{\varepsilon} ({\mathbf x})\,d{\mathbf x}=-(N-2)\sigma_N \langle h_{\varepsilon}*\Phi_{{\mathcal L}}|h_{\varepsilon}\rangle, \end{equation} \tag{3} $$

where $\sigma_N$ is the area of the unit sphere in ${\mathbb R}^N$. We have chosen the normalization bearing in mind that for ${\mathcal L}=\Delta$ we obtain the (positive) energy integral associated with the kernel $1/|{\mathbf x}|^{N-2}$ (see [3], Chap. I, § 4). We write the action of a distribution on a test function in corner brackets.

Since $\Phi_{{\mathcal L}}$ is a locally integrable function in ${\mathbb R}^N$ and $h_{\varepsilon}\in C^{\infty}_0({\mathbb R}^N)$, the integral in (3) is absolutely convergent. By the relations $h_{\varepsilon}*\Phi_{\mathcal L}=(\mu*\Phi_{{\mathcal L}})*\varphi_{\varepsilon}$ and $\|\mu* \Phi_{{\mathcal L}}\|_{{\rm L}^{\infty}({\mathbb R}^N)}\leqslant 1$ we have $\|h_{\varepsilon}*\Phi_{{\mathcal L}}\|_{{\rm L}^{\infty} ({\mathbb R}^N)}\leqslant1$, and therefore

$$ \begin{equation} |I_{h_{\varepsilon},{\mathcal L}}|\leqslant(N-2)\sigma_N\|\mu\|. \end{equation} \tag{4} $$

Let $({\mathbf y},{\mathbf x})=y_1x_1+\cdots+y_Nx_N$ for ${\mathbf y},{\mathbf x}\in{\mathbb R}^N$. Since $h_{\varepsilon}$ is a real-valued function in $C^{\infty}_0({\mathbb R}^N)$, its direct and inverse Fourier transforms

$$ \begin{equation*} F[h_{\varepsilon}]({\mathbf x})=\int_{{\mathbb R}^N} e^{-i({\mathbf y},{\mathbf x})}h_{\varepsilon}({\mathbf y})\, d{\mathbf y} \end{equation*} \notag $$

and

$$ \begin{equation*} F^{-1}[h_{\varepsilon}]({\mathbf x})=\frac{1}{(2\pi)^N}\int_{{\mathbb R}^N} e^{i({\mathbf y},{\mathbf x})}h_{\varepsilon}({\mathbf y})\,d{\mathbf y}= \frac{\overline{F[h_{\varepsilon}]({\mathbf x})}}{(2\pi)^N} \end{equation*} \notag $$

belong to the Schwarz space ${\mathcal S}({\mathbb R}^N)$ of functions rapidly decreasing at infinity. 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 Fourier transformations act on $\Psi$ by the formulae $\langle F[\Psi]|\psi\rangle=\langle \Psi|F[\psi]\rangle$ and $\langle F^{-1}[\Psi]|\psi\rangle=\langle \Psi|F^{-1}[\psi]\rangle$, and we have $\langle \Psi|\psi\rangle=\langle F[\Psi]|F^{-1}[\psi]\rangle$.

For $N\geqslant3$ the distribution $F[\Phi_{{\mathcal L}}]$ coincides with the function $-1/L$, which is locally integrable in ${\mathbb R}^N$, where $L=L({\mathbf x})$ is the symbol of ${\mathcal L}$. As $F[h_{\varepsilon}*\Phi_{{\mathcal L}}]=-F[h_{\varepsilon}]/L$, we have

$$ \begin{equation} \langle h_{\varepsilon}*\Phi_{{\mathcal L}}|h_{\varepsilon}\rangle= -\frac{1}{(2\pi)^N}\int_{{\mathbb R}^N}|F[h_{\varepsilon}({\mathbf x})]|^2 \frac{1}{L({\mathbf x})}\,d{\mathbf x}. \end{equation} \tag{5} $$

From (3) and (5) we obtain (here $\vartheta$ is the quantity from (1))

$$ \begin{equation} I_{h_{\varepsilon},{\mathcal L}}=e^{i\vartheta}\, \frac{(N-2)\sigma_N}{(2\pi)^N} \int_{{\mathbb R}^N}|F[h_{\varepsilon}({\mathbf x})]|^2\, \frac{e^{-i\vartheta}}{L({\mathbf x})}\,d{\mathbf x}. \end{equation} \tag{6} $$

By (1) the following estimate holds in ${\mathbb R}^N\setminus\{0\}$ (where $A_2=A_2(L)>0$):

$$ \begin{equation*} A_2\operatorname{Re}\dfrac{e^{-i\vartheta}}{L({\mathbf x})}= A_2\,\dfrac{\operatorname{Re}(e^{i\vartheta} L({\mathbf x}))} {|L({\mathbf x})|^2}\geqslant\dfrac{A_2\tau}{|L({\mathbf x})|} \geqslant \dfrac{1}{|{\mathbf x}|^{2}}\,. \end{equation*} \notag $$

Hence it follows from (4) and (6) that

$$ \begin{equation} I_{h_{\varepsilon},\Delta}\leqslant A_3(L)\|\mu\|, \end{equation} \tag{7} $$

where $I_{h_{\varepsilon},\Delta}$ is the energy integral (3) for ${\mathcal L}=\Delta$ (and, accordingly, for $L({\mathbf x})=|{\mathbf x}|^{2}$).

Let $h_{\varepsilon}^0=h_{\varepsilon}/\|\mu\|$. Then

$$ \begin{equation*} \int_{{\mathbb R}^N}h_{\varepsilon}^0({\mathbf x})\,d{\mathbf x}=1, \end{equation*} \notag $$

while (3) and (7) yield $I_{h_{\varepsilon}^0,\Delta}\leqslant A_3(L)/\|\mu\|$.

Recall (for instance, see [3], Chap. II, § 1), that one equivalent version of the definition of the harmonic capacity of a compact set $K_{\varepsilon}$ is $1/\inf(I_{\mu^0,\Delta})$, where the infimum is taken over all non-negative measures $\mu^0$ such that $\operatorname{Spt}(\mu^0)\subset K_{\varepsilon}$ and $\|\mu^0\|=1$.

Hence (bearing in mind that $\|\mu\|\geqslant(1/2)\gamma_{{\mathcal L},+}(K)$) we see that

$$ \begin{equation*} A\gamma_{\Delta}(K_{\varepsilon})\geqslant 2\|\mu\|\geqslant\gamma_{{\mathcal L},+}(K). \end{equation*} \notag $$

It remains to let $\varepsilon$ tend to zero and use the fact that

$$ \begin{equation*} \lim_{\varepsilon\to0}\gamma_{\Delta}(K_{\varepsilon})=\gamma_{\Delta}(K) \end{equation*} \notag $$

(for instance, see [3], Chap. II, § 1.5, or [4], Proposition 3.1). $\Box$

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