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This article is cited in 2 scientific papers (total in 2 papers) Regularization of distributions A. L. Pavlovab a Donetsk State University, Donetsk, Russia b Institute of Applied Mathematics and Mechanics, Donetsk, Russia
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     § 1. IntroductionMany problems in analysis and its applications lead to the need to extend a distribution from a domain to a wider domain. One approach to the solution of some of these problems is based on constructing a regularisation of a distribution (see [1] and [2]). Let $N$ be a relatively closed subset of a domain $\Omega \subset \mathbb{R}^n$, and let $E'(\Omega)$ be a space of distributions over a test-function space $E(\Omega) \supset C_0^\infty(\Omega)$. By a regularization of a distribution $f\in E'(\Omega\setminus N)$ we mean extending it to a distribution $\widetilde{f}\in E'(\Omega)$. Thus, for each $\varphi\in C_0^\infty(\Omega\setminus N)$ we have $(f,\varphi)=(\widetilde{f},\varphi)$. In the general case the problem of regularization is solved in the negative (see relevant examples in [1]). Its solution depends on the choice of the space of distributions $E'(\Omega)$ and the behaviour of the distribution in question in a neighbourhood of the set $N$. Distributions that have, roughly speaking, power-like singularities, can be regularized in rather wide distribution spaces. In this paper we take as $E'(\Omega)$ the spaces $D'(\Omega)$, $\mathcal{E}'(\Omega)$ and subspaces of them. We obtain results similar to the ones in [3], where $E'(\Omega)$ was the space of tempered distributions $S'(\mathbb{R}^n)$ or a subspace of it. The main aim of our paper is to give a functional description of distributions in $E'(\Omega\setminus N)$ that can be extended to a distribution in $E'(\Omega)$. To solve this problem we construct special scales of distribution spaces and consider their properties. We interpret distributions in $D'(\Omega\setminus N)$ that can be extended to distributions in $D'(\Omega)$ in terms of these spaces. The description of these scales and multipliers in them are of independent interest in our opinion. There are various ways to regularize a distribution (see [1] and [2]). In this paper we use the method of subtraction. It is based on assigning to each test function $\varphi\in E(\Omega)$ a function $\varphi_N^j$ vanishing on the prescribed set $N\subset\Omega$, together with its derivatives up to order $j$ inclusive. If this map is continuous and the distribution $f\in E'(\Omega\setminus N)$ has a finite order of singularity, then in some cases, for sufficiently large $j$ a regularization $[f]_j$ of $f$ can be defined by
$$
\begin{equation}
([f]_j, \varphi)=(f, \varphi_N^j), \qquad \varphi \in E(\Omega).
\end{equation}
\tag{1.1}
$$
One case important for applications is when $f=a(\sigma)g$ and $g\in E'(\Omega)$, and the function $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfies the inequalities
$$
\begin{equation}
|\partial^\alpha a(\sigma)| \leqslant c_\alpha [d(\sigma,N)]^{q_\alpha}, \qquad \alpha\in \mathbb{Z}_+^n,
\end{equation}
\tag{1.2}
$$
in a neighbourhood of $N$, where $d(\sigma,N)$ is the distance from the point $\sigma$ to $N$ and $c_\alpha>0$ and $q_\alpha\leqslant 0$ are some numbers. In this case (1.1) assumes the form
$$
\begin{equation}
([a(\sigma)g]_j, \varphi)=(g, a(\sigma)\varphi_N^j), \qquad \varphi \in E(\Omega).
\end{equation}
\tag{1.3}
$$
The construction of $\varphi_N^j$ is related to subtracting from $\varphi$ a function coinciding with $\varphi$ on $N$, together with its derivatives up to a certain order. This explains the name for the method. If $N=\{0\}$, then the construction of $\varphi^j_N$ reduces to the use of Taylor’s formula
$$
\begin{equation*}
\varphi_N^j(\sigma)=\varphi(\sigma)-\mu(\sigma) \sum_{|\alpha|\leqslant j-1} \varphi^\alpha(0) \frac{\sigma^\alpha}{\alpha!},
\end{equation*}
\notag
$$
where $\mu(\sigma)\in C_0^\infty(\mathbb{R}^n)$ and $\mu(\sigma)=1$ in a neighbourhood of the origin. In the general case one can use Whitney’s extension theorem (for instance, see [4]) to extend $\varphi_N^j$. One of the main results in this paper (Theorem 4.3) is the existence proof for a regularization $a(\sigma)f$ of the distribution, where $a(\sigma)$ is a function satisfying (1.2) and $f$ is a distribution of finite order. Regularizations of distributions can be used to construct fundamental solutions of differential equations and solve boundary-value or Cauchy problems for equations not solved with respect to the time derivative. For example, in [5] the construction of a solution of a model boundary value problem in a half-space in the class of tempered distributions is reduced to constructing a regularization for a combination of the Fourier transforms of the boundary data with coefficients equal to partial Fourier transforms of Poisson kernels and constructing the solution of a boundary value problem such that the Fourier transform of its boundary data is concentrated on a finite set of points and reduces to the solution of a system of linear equations. In [6] this approach was used to obtain necessary and sufficient conditions for the solvability of the Cauchy problem for a Sobolev-type equation in the class of tempered distributions. In those papers the authors considered regularization for distributions obtained by multiplying some other distributions by functions infinitely smooth outside a finite set and whose derivatives have power-like singularities on this set. One special case of such a function is ${1}/{P(\sigma)}$, where $P(\sigma)$ is a polynomial. The problem of multiplication of a distribution by this function is equivalent to finding a solution of the equation The problem of the solvability of (1.4) is well known as the division problem. It is connected directly with the solvability of the equation $P(D)v=g$, where $D=(D_1,\dots, D_n)$ and $D_k=({1}/{i}){d}/{dx_k}$ in certain function spaces, because the Fourier transformation takes this equation to the problem of division by $P(\sigma)$. Many papers are concerned with this problem (see a brief overview in [2]). Its solution depends on the choice of the space of distributions in which we look for a solution. In the space of tempered distribution $S'$ the division problem was independently solved by Hörmander [7] and Łojasiewicz [8]. They proved that each distribution in $S'$ can be divided by a polynomial (and even by an analytic function, as Łojasiewicz showed) in this space. However, the proofs of this result and its generalizations in [9] do not allow one to track down the connection between the properties of the right-hand side and the solution. An effective construction of the solution of (1.4), which would allow one to track down this connection, can be of interest. The paper [10] contains results of this type in the case when the set of real zeros of the polynomial $P(\sigma)$ is discrete. The construction of the solution was reduced there to constructing a regularization of the distribution $({1}/{P(\sigma)})f \in D'(\mathbb{R}^n\setminus N)$, where $N$ is the set of real zeros of $P(\sigma)$, and then constructing a distribution concentrated on $N$ such that its sum with this regularization solves (1.4). In [11] this scheme was implemented in the case when $N$ is a smooth manifold of dimension $n-1$. Using the smoothness of the algebraic manifold $N$ and the localization property of equation (1.1), which means that local solution of the equation can be glued together using a partition of unity, we can find a solution of (1.4) by solving this equation in neighbourhoods of points in $N$ which cover $\Omega\cap N$. To do this, using Whitney’s extension theorem we construct regularizations $[(1/P(\sigma))\varphi_i f ]$ of the distributions $(1/P(\sigma))\varphi_i f \in D'(\Omega \setminus N)$, $\varphi_i \in D(\Omega)$, and we seek a solution of (1.4) in the following form:
$$
\begin{equation}
u_i=\biggl[\frac{1}{P(\sigma)}\varphi_i f\biggr] + \upsilon_i.
\end{equation}
\tag{1.5}
$$
Then the distribution $\upsilon_i$ is a solution of the equation where $h_i(f)=\varphi_if - P(\sigma) [(1/P(\sigma))\varphi_i f ]$ and $\operatorname{supp} h_i(f)\subset N \cap \Omega$. The solution of (1.6), by taking an appropriate partition of unity and making changes of variables, reduces to the case when $N$ is locally a linear submanifold. Using a theorem on the structure of a distribution concentrated on a linear submanifold [12], the construction of a solution of (1.6) in such a neighbourhood reduces to constructing a solution of a system of linear equations with coefficients in the ring of analytic functions. We find the required solution of (1.4) by adding the solution (1.5) and solutions in domains disjoint from the zeros of $P(\sigma)$, using the fact that the covering associated with a partition of unity is locally finite. In our paper we implement this approach for an effective construction of a solution of (1.4) under certain assumptions on the structure of the set of real zeros of the polynomial $P(\sigma)$. Namely, if the set $N$ of real zeros of $P(\sigma)$ is a smooth manifold and all points in $N$ satisfy a certain condition (condition (5.2)), then for each distribution $f$ of finite order we can construct a solution of (1.4) of order depending on the order of $f$ and the properties of $P(\sigma)$. § 2. Function spacesGiven an open set $\Omega\subset \mathbb{R}^n$, we let $D(\Omega)$ denote the set of infinitely differentiable functions with compact support in $\Omega$, and let $D'(\Omega)$ denote the space of linear forms on $D(\Omega)$ such that for each compact subset $K$ of $ \Omega$ there exist constants $C(K)$ and $p(K)$ such that
$$
\begin{equation}
|(f,\varphi)|\leqslant C(K) \sum_{|\alpha|\leqslant p(K)} \sup_{\sigma \in K}|\partial^\alpha\varphi(\sigma) | \quad\text{for } \varphi \in D(\Omega), \qquad \operatorname{supp}\varphi\subset K.
\end{equation}
\tag{2.1}
$$
The expressions on the right-hand side of (2.1) define seminorms on the space $D(\Omega)$ of functions with support in $K\subset\Omega$. In $D(\Omega)$ we consider the topology on it induced by these seminorms. A distribution $f \in D'(\Omega)$ has order at most $p$ if in (2.1) we can use the same integer $p$ for all compact sets $K\Subset\Omega$. The set of distributions of order at most $p$ is denoted by $D'^p (\Omega)$. Let $D'_F(\Omega)$ denote the union of the spaces $D'^p(\Omega)$, $p \in \mathbb{Z}_+$. We call it the space of distributions of finite order (see [12]). We denote the space of $p$-fold differentiable functions in the domain $\Omega$ by $C^p(\Omega)$; its subspace of functions with compact support is denoted by $C^p_0(\Omega)$. The topology in $C^p_0(\Omega)$ is introduced in terms of seminorms, as in $D(\Omega)$, but now these are considered for $\alpha\in \mathbb{Z}^n_+$, $|\alpha|\leqslant p$. A sequence of functions $\varphi_n \in C^p_0(\Omega)$ converges to $\varphi \in C^p_0(\Omega)$ if there exists a compact set $K \Subset \Omega$ such that $\operatorname{supp}\varphi_n\subset K$, $\operatorname{supp}\varphi\subset K$, and
$$
\begin{equation*}
\lim_{n\to\infty}\sum_{|\alpha|\leqslant p}\sup_{\sigma \in K} |\partial^\alpha (\varphi - \varphi_n ) |=0.
\end{equation*}
\notag
$$
The space $C^p_0(\Omega)$ is sequentially complete and $D(\Omega)$ is a dense subspace of it. Each element of $D'^p(\Omega)$ can uniquely be extended to a continuous map of $C^p_0(\Omega)$ into $\mathbb{R}$, and (2.1) remains true for all $\varphi \in C^p_0 ( \Omega)$ (see [12], Theorem 2.1.6). Let $\mathcal{E}(\Omega)$ denote the space of infinitely differentiable functions $C^\infty( \Omega)$ with the topology defined in terms of the seminorms
$$
\begin{equation*}
\| \varphi\|^j_K=\sum_{|\alpha|\leqslant j}\sup_{\sigma \in K}|\partial^\alpha\varphi(\sigma) |,
\end{equation*}
\notag
$$
where $K$ is an arbitrary compact subset of $\Omega$, $j \in \mathbb{Z}_+$. The space $\mathcal{E}'(\Omega)$ consists of distributions with compact support in $\Omega$. For each $p \in \mathbb{Z}_+$ and any compact set $K \Subset \Omega$ let $\mathcal{E}'^p(K)$ denote the set of distributions $f \in \mathcal{E}'(\Omega)$ such that
$$
\begin{equation}
|(f,\varphi)|\leqslant C \sup_{|\alpha|\leqslant p,\,\sigma \in K} | \partial^\alpha\varphi(\sigma) |, \qquad\varphi \in C^\infty(\Omega).
\end{equation}
\tag{2.2}
$$
Lemma 2.1. The set $\mathcal{E}'^p(K)$ is a linear subspace of $\mathcal{E}'(\Omega)$, which is a Banach space with respect to the norm Proof. It follows from the definition of $\mathcal{E}'^p(K)$ that it is a linear subspace of $\mathcal{E}'(\Omega)$, and for each function in $\mathcal{E}'^p(K)$ we have a well-defined quantity (2.3). That $\|f\|_{\mathcal{E}'^p(K)}$ has the properties of a norm is straightforward. We prove that the normed space $\mathcal{E}'^p(K)$ is complete.
Let $\{f_n \in \mathcal{E}'^p(K),\, n \in \mathbb{N} \}$ be a Cauchy sequence in $\mathcal{E}'^p(K)$. Since from the definition (2.3) of the norm we obtain the inequality
$$
\begin{equation}
|(f,\varphi)|\leqslant \|f\|_{\mathcal{E}'^p(K)}\sup_{|\alpha|\leqslant p,\,\sigma \in K}|\partial^\alpha\varphi(\sigma)|, \qquad \varphi \in C^\infty(\Omega),
\end{equation}
\tag{2.4}
$$
for each $\varphi \in C^\infty(\Omega)$ the sequence $(f_n,\varphi )$ is convergent. Hence a linear form $f$ is defined on $C^\infty(\Omega)$:
$$
\begin{equation*}
(f,\varphi)=\lim _{n\to +\infty}(f_n,\varphi), \qquad \varphi \in C^\infty(\Omega).
\end{equation*}
\notag
$$
It satisfies inequality (2.4), where $\|f\|_{\mathcal{E}'^p(K)}=\lim_{n\to\infty}\|f_n\|_{\mathcal{E}'^p(K)}$. This inequality means that $(f,\varphi)=0$ for $\varphi \in C^\infty_0(\Omega\setminus K)$. We show that inequality (2.1) holds for the linear form $f$, so that $f \in D'(\Omega)$. For each compact set $K_1 \Subset\Omega$ disjoint from $K$ inequality (2.1) holds because $(f,\varphi)=0$ for $\varphi \in C^\infty_0(\Omega\setminus K)$. If $K_1\cap K\neq\varnothing$ and $\varphi \in C^\infty_0(\Omega),\operatorname{supp} \varphi\subset K_1$, then, as (2.4) holds, we obtain
$$
\begin{equation*}
\begin{aligned} \, |(f,\varphi)| &\leqslant\|f\|_{\mathcal{E}'^p(K)}\sup_{|\alpha|\leqslant p,\,\sigma \in K}|\partial^\alpha\varphi(\sigma)|=\|f\|_{\mathcal{E}'^p(K)}\sup_{|\alpha|\leqslant p,\,\sigma \in K\cap K_1}|\partial^\alpha\varphi(\sigma)| \\ &\leqslant\|f\|_{\mathcal{E}'^p(K)}\sup_{|\alpha|\leqslant p,\,\sigma \in K_1}|\partial^\alpha\varphi(\sigma)|\leqslant C \|f\|_{\mathcal{E}'^p(K)}\sum_{|\alpha|\leqslant p}\sup_{\sigma \in K_1}|\partial^\alpha\varphi(\sigma)|. \end{aligned}
\end{equation*}
\notag
$$
Therefore, $f \in D'^p(\Omega)$. Since $\operatorname{supp} f\subset K$, it follows that $f \in \mathcal{E}'(\Omega)$ and (2.4) hods, that is, $f \in \mathcal{E}'^p(K)$ and $\|f\|_{\mathcal{E}'^p(K)}$ is the norm of $f$. The proof is complete. Each element of $\mathcal{E}'^p(K)$ can uniquely be extended to a continuous map of $C^p(\Omega)$ in $\mathbb{R}$, and (2.2) remains valid for all $\varphi \in C^p(\Omega)$. The corresponding proof is similar to the proof of Theorem 2.1.6 in [12]. Lemma 2.2. If $f$ is a linear form on $C^p_0(\Omega)$ and for each $\varphi \in C^p_0(\Omega)$ where $K\Subset\Omega$, then $f \in \mathcal{E}'^p(K)$. Proof. It follows from (2.5) that $f \in D'^p(\Omega)$ and $\operatorname{supp}f \subset K$. For an arbitrary function $\varphi \in C^p(\Omega)$ consider a sequence of functions $\varphi_n \in D(\Omega)$ converging to $\varphi$ in $C^p(\Omega)$. It exists because $D(\Omega)$ is dense in $C^p(\Omega)$.
By inequality (2.5) the sequence $(f,\varphi_n)$ is Cauchy. Hence this sequence has a limit as $n\to +\infty$. It is independent of the choice of the sequence $\varphi_n$, and we denote it by $(f,\varphi)$. Thus, the linear form $f$ extends to $C^\infty(\Omega)$. Inequality (2.5) holds for this extension, and therefore $f\in \mathcal{E}'^p(K)$. The proof is complete. Lemma 2.3. If $f \in D'^p(\Omega)$, then for each compact set $K \subset \Omega$ there exists a constant $C(K, f)$ such that for each function $\varphi \in D(\Omega)$ with $\operatorname{supp}\varphi\subset K$ the distribution $\varphi f$ belongs to $\mathcal{E}'^p(K)$ and Proof. By assumption $\operatorname{supp}\varphi f\subset K$. Hence $\varphi f \in \mathcal{E}'(\Omega)$. From (2.1) and Leibniz’s formula, for any function $\psi \in C^\infty(\Omega)$ we have
$$
\begin{equation*}
\begin{aligned} \, |(\varphi f,\psi)| &=|(f,\varphi \psi)|\leqslant C_1(k,f)\sum_{|\alpha|\leqslant p}\sup_{\sigma \in K}|\partial^\alpha(\varphi\psi)(\sigma)| \\ &\leqslant C_2(k,f)\sum_{|\alpha|\leqslant p}\sum_{\beta\leqslant\alpha}\sup_{\sigma \in K}|\partial^{\alpha - \beta}\varphi(\sigma)|\sup_{\sigma \in K}|\partial^\beta\psi(\sigma)| \\ &\leqslant C(k,f)\sup_{|\alpha|\leqslant p,\,\sigma \in K}|\partial^\alpha\varphi(\sigma)|\sup_{|\beta|\leqslant p,\,\sigma \in K}|\partial^\beta\psi(\sigma)|. \end{aligned}
\end{equation*}
\notag
$$
It follows from the definition of $\mathcal{E}'^p(K)$ that $\varphi f \in \mathcal{E}'^p(K)$ and (2.6) holds. The lemma is proved. For each space of test functions considered above we can consider its subspace of elements $\varphi$ satisfying the condition
$$
\begin{equation*}
\partial^\alpha\varphi|_N=0, \qquad |\alpha|\leqslant j,
\end{equation*}
\notag
$$
where $N$ is a relatively closed subset of $\Omega$. For $C^j_0(\Omega)$ we denote this subspace by $C^j_{0N}(\Omega)$, and for $C^j(\Omega)$ we denote it by $C^j_N(\Omega)$. The condition that the support of a distribution lie in a compact set $K$ is necessary for this distribution to belong to some $\mathcal{E}'^p(K)$. For some compact sets it is also sufficient. It follows from Theorem 2.3.10 in [12] that, for compact sets with finitely many connected components such that any two points in a connected components can be joined by a smooth curve lying in the same component, the space $\mathcal{E}'^p(K)$ consists of all distributions of order at most $p$ with support in $K$. If we represent $\Omega$ as a union of a nested sequence of compact sets $K_i$, $i\in \mathbb{N}$, satisfying this condition and exhausting $\Omega$, then
$$
\begin{equation*}
\mathcal{E}'(\Omega)=\bigcup_{i \in \mathbb{N},\,p \in \mathbb{Z}_+}\mathcal{E}'^p(K_i).
\end{equation*}
\notag
$$
In fact, if $f \in \mathcal{E}'(\Omega)$, then there is a compact set $K_i$ such that $\operatorname{supp}f\subset K_i$. It follows from (2.2) that there exists $p \in \mathbb{Z}_+$ such that $f \in \mathcal{E}'^p(K_i)$. It is easy to show that the above equality holds for each domain with smooth boundary. For a relatively closed subset $N$ of $\Omega$ consider the linear space
$$
\begin{equation*}
D^\infty_N(\Omega)=\bigl\{\varphi \in D(\Omega)\colon \partial^\alpha\varphi|_N=0,\,\alpha \in \mathbb{Z}^n_+ \bigr\}.
\end{equation*}
\notag
$$
The set $D^\infty_N(\Omega)$ is a closed subspace of $D(\Omega)$, which is invariant under differentiation and multiplication by infinitely smooth functions. For what follows we need functions $\mu_\mathcal{E}(\sigma) \in C^\infty(\mathbb{R}^n)$, $0 < \varepsilon < 1$, such that
$$
\begin{equation}
\begin{aligned} \, &1)\quad \operatorname{supp}\mu_\varepsilon\subset N^\varepsilon=\{\sigma \in \mathbb{R}^n\colon d(\sigma, N)\leqslant \varepsilon \}, \quad\text{where } d(\sigma, N)=\inf_{\eta \in N}|\eta - \sigma|; \nonumber \\ &2)\quad \mu_\varepsilon(\sigma)=1 \quad\text{for } \sigma \in N^{\varepsilon/4}; \nonumber \\ &3)\quad |\partial^\alpha\mu_\varepsilon(\sigma)|\leqslant c_\alpha\varepsilon^{-|\alpha|} \quad\text{for }\alpha \in \mathbb{Z}^n_+. \end{aligned}
\end{equation}
\tag{2.7}
$$
The construction of such a family of functions is similar to the construction for compact sets in [12], § 1.4; it is based on considering the convolution of the characteristic function of $N$ with a function $\chi_\varepsilon \in C^\infty_0(B_\varepsilon)$ such that $\displaystyle\int\chi_\varepsilon d\sigma=1$, where $B_\varepsilon$ is a ball of radius $\varepsilon$. Let $D(\Omega, N)$ denote the closure in $D(\Omega)$ of the subspace $C^\infty_0(\Omega\setminus N)$. Proof. The embedding $D(\Omega, N)\subset D^\infty_N(\Omega)$ follows from the definitions of the spaces involved. It is obvious that $C^\infty_0(\Omega\setminus N)\subset D^\infty_N(\Omega)$. Let $\varphi \in D(\Omega, N)$, and let $\varphi=\lim_{n\to+\infty}\varphi_n$, $\varphi_n \in C^\infty_0(\Omega\setminus N)$, in the space $D(\Omega)$. Hence there exists a compact set $K\subset\Omega$ containing the supports of the $\varphi_n$, and for each $\alpha \in \mathbb{Z}^n_+$ we have Then for each point $\sigma_0 \in N$ and any $\alpha \in \mathbb{Z}^n_+$ we have because $\operatorname{supp}\varphi_n\subset \Omega\setminus N$. Therefore, $\varphi \in D^\infty_N(\Omega)$.
We prove that $D^\infty_N(\Omega)\subset D(\Omega, N)$. Let $\varphi \in D^\infty_N(\Omega)$, and let $\mu_{1/m}$, $m \in \mathbb{N}$, be a sequence of functions in the family indicated above. Then the sequence $\psi_m=(1-\mu_{1/m})\varphi$ lies in $ C^\infty_0(\Omega\setminus N)$. We will show that the sequence of functions $\mu_{1/m}\varphi$ converges to zero in $D(\Omega)$. Then it will follow from the equality $\varphi=(1-\mu_{1/m})\varphi+\mu_{1/m}\varphi$ that the sequence $(1-\mu_{1/m})\varphi\in C^\infty_0(\Omega\setminus N)$ converges to $\varphi$ in $D(\Omega)$. Thus, $\varphi\in D(\Omega, N)$. Using Leibniz’s formula and (2.7) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &\sup_{\sigma \in \operatorname{supp}\varphi} |\partial^\alpha(\mu_{1/m}(\sigma)\varphi(\sigma)) | < c'_\alpha\sup_{\sigma \in N^{1/m} \cap\operatorname{supp}\varphi} \biggl[\sum_{\beta\leqslant\alpha}|\partial^\beta\mu_{1/m}(\sigma) |\,|\partial^{\alpha-\beta}\varphi(\sigma) |\biggr] \\ &\qquad \leqslant c''_\alpha\sum_{\beta\leqslant\alpha}m^{|\beta|}\sup_{\sigma \in N^{1/m}\cap\operatorname{supp}\varphi}|\partial^{\alpha-\beta}\varphi(\sigma) |. \end{aligned}
\end{equation}
\tag{2.8}
$$
We claim that for any $\gamma\in \mathbb{Z}_+^n$ and $k\in \mathbb{N}$ we have
$$
\begin{equation}
\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\partial^\gamma \varphi(\sigma)| \leqslant c_k m^{-k} \sum_{|\alpha|=k}\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\partial^{\gamma+\alpha} \varphi(\sigma)|.
\end{equation}
\tag{2.9}
$$
Because $\varphi(\sigma)$ and its derivatives of any order vanish on $N$, for all $\eta\in N$ and $k$ we have
$$
\begin{equation*}
\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\varphi(\sigma)| =\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} \biggl|\varphi(\sigma)- \sum_{|\alpha|\leqslant k-1} \varphi^\alpha(\eta) \frac{(\sigma-\eta)^\alpha}{\alpha!} \biggr|.
\end{equation*}
\notag
$$
Using Taylor’s formula with Lagrange remainder we obtain
$$
\begin{equation*}
\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\varphi(\sigma)| =\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} \biggl|\sum_{|\alpha|=k} \varphi^\alpha(\sigma+t(\eta-\sigma)) \frac{(\sigma-\eta)^\alpha}{\alpha!} \biggr|,
\end{equation*}
\notag
$$
where $t=t(\sigma)$, $0\leqslant t(\sigma)\leqslant 1$. By this equality
$$
\begin{equation}
\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\varphi(\sigma)|\leqslant c_km^{-k}\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi}\sum_{|\alpha|=k} \frac{1}{\alpha!}\varphi^\alpha(\sigma+t(\eta-\sigma))|,
\end{equation}
\tag{2.10}
$$
where we choose $\eta=\eta(\sigma)$ so that $d(\sigma,N)=|\sigma-\eta(\sigma)|$ and $\sigma\in N^{1/m}\cap\operatorname{supp}\varphi$.
This can be done for sufficiently large $m$ because the set $N^{1/m}\cap\operatorname{supp}\varphi$ is compact for large $m$. In the derivation of (2.10) we used the inequality
$$
\begin{equation*}
\frac{|(\sigma-\eta)^\alpha|}{|\sigma-\eta|^k} \leqslant \frac{c_k|(\sigma-\eta)^\alpha|}{(\sum_{i=1}^n |\sigma_i-\eta_i|)^k}= \frac{c_k|\sigma_1-\eta_1|^{\alpha_1} \dotsb |\sigma_n-\eta_n|^{\alpha_n}} {(|\sigma_1-\eta_1|+\dots+|\sigma_n-\eta_n|)^{\alpha_1+\dots+\alpha_n}} \leqslant c_k.
\end{equation*}
\notag
$$
If $\sigma\in N^{1/m}\cap\operatorname{supp}\varphi$ and $|\sigma-\eta(\sigma)|=d(\sigma,N)$, then
$$
\begin{equation*}
d(\sigma+t(\eta(\sigma)-\sigma),N) \leqslant d(\sigma+t(\eta(\sigma)-\sigma),\eta(\sigma))=(1-t) |\sigma-\eta(\sigma)|\leqslant \frac {1} {m}.
\end{equation*}
\notag
$$
Hence $\sigma+t(\eta(\sigma)-\sigma)\in N^{1/m}$.
Since the function $\varphi$ and its derivatives vanish outside $\operatorname{supp}\varphi$, we can write (2.10) as
$$
\begin{equation}
\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\varphi(\sigma)| \leqslant c_km^{-k} \sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} \biggl[\sum_{|\alpha|=k} \frac{1}{\alpha!} |\varphi^\alpha(\sigma)|\biggr].
\end{equation}
\tag{2.11}
$$
Thus (2.9) is proved for $\gamma=(0,\dots,0)$. Its proof for derivatives of $\varphi(\sigma)$ can be obtained by substituting these derivatives into (2.11) in place of the function. To complete the proof of the lemma it remains to estimate the right-hand side of (2.8) using (2.9). Then we obtain
$$
\begin{equation*}
\begin{aligned} \, &\sup_{\sigma\in \operatorname{supp}\varphi} |\partial^\alpha (\mu_{1/m}(\sigma)\varphi(\sigma)) | \leqslant c_\alpha'' \sum_{i=0}^{|\alpha|} m^i \sum_{\beta\leqslant\alpha,\,|\beta|=i} \sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\partial^{\alpha-\beta}\varphi(\sigma)| \\ &\qquad \leqslant \widetilde{c}_\alpha \sum_{i=0}^{|\alpha|} m^i \sum_{|\beta|=i,\,\beta\leqslant\alpha} m^{-i-1} \sum_{|\gamma|=i+1} \sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\partial^{\alpha-\beta+\gamma}\varphi(\sigma)|. \end{aligned}
\end{equation*}
\notag
$$
These inequalities imply that
$$
\begin{equation}
\sup_{\sigma\in\operatorname{supp}\varphi} |\partial^\alpha (\mu_{1/m}(\sigma)\varphi(\sigma)) | \leqslant \frac{1}{m}\widetilde{c}_\alpha \sum_{|\beta|\leqslant 2|\alpha|+1}\sup_{\sigma\in N^{1/m}\cap\operatorname{supp}\varphi} |\partial^\beta\varphi(\sigma)|.
\end{equation}
\tag{2.12}
$$
It follows from this inequality that the sequence of functions $\mu_{1/m}\varphi$ converges to zero in $D(\Omega)$ as $m\to +\infty$. Hence This equality means that $\varphi \in D(\Omega, N)$. The proof is complete.For further constructions, given a compact set $K \Subset \Omega$ which has a nonempty intersection with a relatively closed set $N\subset\Omega$, consider the set
$$
\begin{equation*}
(K\cap N^\varepsilon)_N=\bigl\{ \eta \in N\colon \exists\,\sigma \in K\cap N^\varepsilon,\, d(\sigma,N)=|\eta - \sigma|\bigr\},
\end{equation*}
\notag
$$
where $N^\varepsilon=\{\sigma \in \mathbb{R}^n\colon d(\sigma, N)\leqslant \varepsilon \}$. This set $(K\cap N^\varepsilon)_N$ is the ‘metric projection’ of $K\cap N^\varepsilon$ onto $N$. Proof. For an arbitrary sequence $\eta_n\in (K\cap N^\varepsilon)_N$, $n \in \mathbb{N}$, consider the corresponding sequence $\sigma_n \in K\cap N^\varepsilon$ such that $|\eta_n - \sigma_n|=d(\sigma_n,N)$.
Since $N$ is relatively closed in $\Omega$ and $K$ is a compact subset of $\Omega$, $K\cap N^\varepsilon$ is compact too. Hence there exists a subsequence $\sigma_{n_k}$ tending to $\sigma_0 \in K\cap N^\varepsilon$. Then
$$
\begin{equation*}
\lim_{k\to+\infty}|\sigma_{n_k}-\eta_{n_k}|=\lim_{k\to+\infty}d(\sigma_{n_k},N)=d(\sigma_0,N).
\end{equation*}
\notag
$$
As $\lim_{k\to+\infty}\sigma_{n_k}=\sigma_0$, it follows from the above equalities that the limit $\lim_{k\to +\infty}\eta_{n_k}=\eta_0$ exists and $d(\sigma_0,N)=|\sigma_0 - \eta_0|$. If $\varepsilon$ is small enough, then $\eta_0 \in (K\cap N^\varepsilon)_N$. Hence the set $(K\cap N^\varepsilon)_N$ is compact for sufficiently small $\varepsilon$. The proof is complete. Consider the set of line segments connecting $\sigma$ with $\eta(\sigma)$, where $\sigma \in K\cap N^\varepsilon$ and $\eta(\sigma)$ is an arbitrary ‘metric projection’ of $\sigma$ onto $N$. We denote this set by $(K\cap N^\varepsilon)_\varepsilon$. Lemma 2.5. For sufficiently small $\varepsilon$ the set $(K\cap N^\varepsilon)_\varepsilon$ is compact and lies in $\Omega$. Proof. Since for sufficiently small $\varepsilon$ the set $(K\cap N^\varepsilon)_N$ is compact and lies in $\Omega$ by Lemma 2.4, its $\varepsilon$-neighbourhood $(K\cap N^\varepsilon)_N^\varepsilon$ lies in $\Omega$ for perhaps even smaller $\varepsilon$.
If $\sigma \in K\cap N^\varepsilon$, then $\sigma \in (K\cap N^\varepsilon)^\varepsilon_N$ because $d(\sigma,N)\leqslant \varepsilon$ and $d(\sigma,N)=|\sigma - \eta(\sigma)|$, $\eta(\sigma) \in (K\cap N^\varepsilon)_N$. Thus the points $\sigma+t(\eta(\sigma)-\sigma)$, $0\leqslant t \leqslant 1$, on the line segment connecting $\sigma$ with $\eta(\sigma)$ belong to $\Omega$. We show that their union is a compact set. Consider any sequence of points $\sigma_n+t_n(\eta_n(\sigma)-\sigma_n)$, $0 \leqslant t_n \leqslant 1$, in these segments. Since $\sigma_n \in K\cap N^\varepsilon$ and the set $K\cap N^\varepsilon$ is compact, there exists a subsequence $\sigma_{n_k}$ tending to $\sigma_0 \in K\cap N^\varepsilon$ as $k\to +\infty$. From the sequence $\{t_{n_k}\}$ we extract a subsequence $t_{n_{kj}}$ tending to $t_0$ as $j\to +\infty$. Then the points $\sigma_{n_{kj}}+t_{n_{kj}}(\eta_{n_{kj}}- \sigma_{n_{kj}})$ tend to $\sigma_0+t_0(\eta_0(\sigma_0)-\sigma_0)\in(K\cap N^\varepsilon)_\varepsilon$. Therefore, $(K\cap N^\varepsilon)_\varepsilon$ is a compact set for $\varepsilon$ small enough. The proof is complete. Similarly to $D^\infty_N(\Omega)$, we can introduce and examine the space $\mathcal{E}^\infty_N(\Omega)=\{\varphi \in \mathcal{E}(\Omega)\colon \partial^\alpha\varphi|_N=0,\, \alpha \in \mathbb{Z}^n_+\}$. It is a closed subspace of $\mathcal{E}(\Omega)$, which is invariant with respect to differentiation and multiplication by an infinitely smooth function. Theorem 2.2. The subspace $D(\Omega\setminus N)$ of $\mathcal{E}^\infty_N(\Omega)$ is dense in this space. Proof. Consider a sequence of compact sets $K_m$, $m \in \mathbb{N}$, that exhaust the domain $\Omega$: For bounded $\Omega$ these compact sets are formed by the points in $\Omega$ lying at a distance at least $1/m$ from its boundary. Unbounded domains can be exhausted by bounded ones.
Consider functions $\varphi_m(\sigma) \in D(\Omega)$ such that $0\leqslant\varphi_m\leqslant1$, $\varphi_m|_{K_m}=1$ and $\varphi_m|_{\Omega\setminus \mathring{K}_{m+1}}=0$. They can be constructed using Theorem 1.4.1 in [12]. Let $\varphi\in \mathcal{E}^\infty_N (\Omega)$. The sequence of functions $\psi_m=\varphi_m\varphi$ lies in $ D^\infty_N(\Omega)$ and tends to $\varphi$ in $\mathcal{E}(\Omega)$, because for each compact set $K\Subset\Omega$ there exists $m_0$ such that $K\subset K_{m_0}$, and for $m>m_0$ and any $j \in \mathbb{Z}_+$ we have
$$
\begin{equation*}
\sum_{|\alpha|\leqslant j}\sup_{\sigma \in K}|\partial^\alpha(\varphi-\psi_m)|=0.
\end{equation*}
\notag
$$
Consider the sequence of functions $\nu_m=(1-\mu_{1/m})\psi_m$, where $\mu_{1/m}$ is a sequence of functions satisfying (2.7). By construction $\nu_m(\sigma)\in D(\Omega \setminus N)$. We show that the sequence $\mu_{1/m}\varphi_m\in D^\infty_N(\Omega)$ tends to zero in the space $\mathcal{E}^\infty_N(\Omega)$ as $m\to +\infty$. Using the arguments from the proof of Theorem 2.1 and the fact that for each compact set $K\Subset\Omega$ there exists $m_0$ such that $\mu_{1/m}\psi_m|_K=\mu_{1/m}\varphi|_K$ for $m>m_0$, for any $\alpha \in \mathbb{Z}^n_+$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &\sup_{\sigma \in K}|\partial^\alpha(\mu_{1/m}\psi_m)(\sigma) |\leqslant c'_\alpha\sum_{\beta\leqslant\alpha}m^{|\beta|}\sup_{\sigma \in K\cap N^{1/m}}|\partial^{\alpha-\beta}\varphi(\sigma) | \\ &\qquad \leqslant c''_\alpha\sum^{|\alpha|}_{i=0}m^i\sum_{|\beta|=i,\, \beta\leqslant\alpha}m^{-i-1}\sum_{|\gamma|=i+1}\sup_{\sigma\in K\cap N^{1/m}}| \partial^{\alpha-\beta+\gamma}\varphi(\sigma+t(\eta(\sigma)-\sigma))|, \end{aligned}
\end{equation*}
\notag
$$
where $\eta=\eta(\sigma)$ is selected so that $|\eta(\sigma)-\sigma|=d(\sigma, N)$, $0\leqslant t\leqslant 1$.
It follows from Lemma 2.5 that the set of points $\sigma+t(\eta(t)-\sigma)$ lies in a compact subset of $\Omega$ for $m$ large enough. The above inequalities show that the sequence of functions $\mu_{1/m}\psi_m$ converges to zero as $m\to+\infty$ in the space $\mathcal{E}^\infty_N(\Omega)$. The equality $\psi_m=\varphi_m\varphi=\nu_m+\mu_{1/m}\psi_m$ means that the sequence $\nu_m\in D(\Omega\setminus N)$ converges to $\varphi$ in $\mathcal{E}^\infty_N(\Omega)$. The proof is complete. Consider the space ${D^\infty_N}'(\Omega)$ dual to $D^\infty_N(\Omega)$. By definition the linear form $f$ on $D^\infty_N(\Omega)$ belongs to ${D^\infty_N}'(\Omega)$ if for each compact set $K\Subset\Omega$ there exist constants $j(K)$ and $C(K)$ such that
$$
\begin{equation}
|(f,\varphi)|\leqslant C(K)\sum_{|\alpha|\leqslant j(K)}|\partial^\alpha\varphi(\sigma)|, \qquad \varphi \in D^\infty_N(\Omega), \qquad\operatorname{supp}\varphi\subset K.
\end{equation}
\tag{2.13}
$$
Condition (2.13) is equivalent to sequential continuity: a linear form $f$ on $D^\infty_N(\Omega)$ belongs to ${D^\infty_N}'(\Omega)$ if and only if $(f, \varphi_m)\to 0$ as $m\to+\infty$ for each sequence $\varphi_m\in D^\infty_N(\Omega)$ converging to zero (so that for each $\alpha\in \mathbb{Z}^n_+$ we have $\sup_{\sigma \in K}|\partial^\alpha\varphi_m(\sigma)|\to 0$ as $m\to+\infty$, and $\operatorname{supp}\varphi_m\subset K\Subset\Omega)$. Since the embedding is continuous, there exists a dual map
$$
\begin{equation*}
i^*_\infty\colon D'(\Omega)\to {D^\infty_N}'(\Omega).
\end{equation*}
\notag
$$
Let $E^\infty_N(\Omega)$ denote the kernel of $i^*_\infty$ and $D'(\Omega, N)$ denote the set of distributions in $D'(\Omega)$ whose supports lie in $N$. Proof. If $f \in E^\infty_N(\Omega)$, then $(f,\varphi)=0$ for each function $\varphi \in D^\infty_N(\Omega)$. Since ${D(\Omega\setminus N)\subset D^\infty_N(\Omega)}$, it follows that $\operatorname{supp}f\subset N$ and we have the embedding $E^\infty_N(\Omega)\subset D'(\Omega, N)$.
If $f \in D'(\Omega, N)$, then $(f,\varphi)=0$ for each function $\varphi \in D(\Omega\setminus N)$. It follows from Theorem 2.1 that this equality holds for each function $f \in D^\infty_N(\Omega)$, that is, $D'(\Omega, N)\subset E^\infty_N(\Omega)$. Therefore, $D'(\Omega, N)=E^\infty_N(\Omega)$. The proof is complete. Because of the continuous embedding
$$
\begin{equation*}
j_\infty \colon D^\infty_N(\Omega)\to \mathcal{E}^\infty_N(\Omega),
\end{equation*}
\notag
$$
we have an embedding
$$
\begin{equation*}
j^*_\infty\colon {\mathcal{E}^\infty_N}'(\Omega)\to {D^\infty_N}'(\Omega).
\end{equation*}
\notag
$$
The space ${\mathcal{E}^\infty_N}'(\Omega)$ is similar to ${D^\infty_N}'(\Omega)$ in its properties: 1) $f \in {\mathcal{E}^\infty_N}'(\Omega)$ if there exist a compact set $\Subset\Omega$ and $p \in \mathbb{Z}_+$ such that for each function $\varphi \in \mathcal{E}^\infty_N(\Omega)$ we have
$$
\begin{equation}
|(f,\varphi)|\leqslant C(K)\sum_{|\alpha|\leqslant p}\sup_{\sigma\in K}|\partial^\alpha\varphi(\sigma)|,
\end{equation}
\tag{2.14}
$$
where $C(K)>0$ is a number depending on $K$; 2) the kernel of the map $\nu_\infty^*\colon \mathcal{E}'(\Omega) \to {\mathcal{E}^\infty_N}'(\Omega)$ dual to $\nu_\infty\colon\mathcal{E}^\infty_N(\Omega)\to \mathcal{E}(\Omega)$ coincides with the set of distributions in $\mathcal{E}'(\Omega)$ whose supports lie in $N$. The set of distributions in ${\mathcal{E}^\infty_N}'(\Omega)$ that satisfy (2.14) for fixed $K$ and $p$ will be denoted by ${\mathcal{E}^{\infty}_N}'^p(K)$. It is a closed subspace of ${\mathcal{E}^\infty_N}'(\Omega)$. The following embeddings hold by definition:
$$
\begin{equation*}
\begin{gathered} \, {\mathcal{E}^\infty_N}'^p(K)\subset {\mathcal{E}^{\infty}_N}'^{p'}(K), \qquad p\leqslant p', \\ {\mathcal{E}^\infty_N}'^p(K)\subset {\mathcal{E}^{\infty}_N}'^ {p}(K'), \qquad K\subset K'\Subset\Omega. \end{gathered}
\end{equation*}
\notag
$$
The space ${\mathcal{E}^\infty_N}'(\Omega)$ is the strict direct limit of the spaces ${\mathcal{E}^\infty_N}'^p(K)$:
$$
\begin{equation*}
{\mathcal{E}^\infty_N}'(\Omega)=\bigcup_{K\Subset \Omega,\, p\in \mathbb{Z}_+}{\mathcal{E}^\infty_N}'^p(K).
\end{equation*}
\notag
$$
In ${\mathcal{E}^\infty_N}'^p(K)$ we can introduce a norm similarly to Lemma 2.1. Consider the subspaces ${D^\infty_N}'(\Omega)$ similar to the $D'^p(\Omega)$ and $D'_F(\Omega)$. The subspace ${D^{\infty}_N}'^p(\Omega)$ consists of the elements of ${D^\infty_N}'(\Omega)$ such that (2.13) holds for all compact sets $K\Subset\Omega$ for the same $j=p$. We denote the union of the spaces ${D^\infty_N}'^p(\Omega)$ by ${D^\infty_{NF}}'(\Omega)$. Consider another family of subspaces of $D(\Omega)$:
$$
\begin{equation*}
D^k_N(\Omega)=\{\varphi \in D(\Omega)\colon \partial^\alpha\varphi(\sigma)|_N=0,\, |\alpha|\leqslant k\}.
\end{equation*}
\notag
$$
It follows from the definition of the spaces $D^k_N(\Omega)$ that
$$
\begin{equation*}
D^\infty_N(\Omega)\subset\dots\subset D^{k+1}_N(\Omega)\subset D^k_N(\Omega)\subset\dots\subset D(\Omega)
\end{equation*}
\notag
$$
and $D^\infty_N(\Omega)=\bigcap_{k \in \mathbb{Z}_+} D^k_N(\Omega)$. The space $D^\infty_N(\Omega)$ is the strict inverse limit of the separated locally convex spaces $D^k_N(\Omega)$, while ${D^\infty_N}'(\Omega)$ is the strict direct limit of the spaces ${D^k_N}'(\Omega)$. We need the following results for constructions below. Proof. We can represent an arbitrary function $\varphi\in C^p_{0N}(\Omega)$ in the form $\varphi=(1- \mu_{4/n})\varphi+\mu_{4/n}\varphi$, where the functions $\mu_{4/n}(\sigma)$ satisfy (2.7).
Now we prove that the sequence of functions $\mu_{4/n}\varphi$ converges to zero in $C^p_{0N}(\Omega)$ as $n\to +\infty$. Using Leibniz’s formula and (2.7) we obtain
$$
\begin{equation}
\begin{aligned} \, \notag \sup_{\sigma\in\operatorname{supp}\varphi}|\partial^\alpha(\mu_{4/n}\varphi)(\sigma) | &\leqslant c_\alpha\sup_{\sigma \in \operatorname{supp}\varphi}\sum_{\beta\leqslant\alpha}|\partial^\beta\mu_{4/n}(\sigma) |\,|\partial^{\alpha-\beta}\varphi(\sigma)| \\ &\leqslant c'_\alpha\sum_{\beta\leqslant\alpha}n^{|\beta|}\sup_{\sigma\in\operatorname{supp}\varphi\cap N^{4/n}}| \partial^{\alpha-\beta}\varphi(\sigma)|. \end{aligned}
\end{equation}
\tag{2.15}
$$
Since in the proof of relation (2.9) we only used the derivatives of $\varphi(\sigma)$ up to a certain order, it follows from it that
$$
\begin{equation}
\begin{aligned} \, \notag &\sup_{\sigma\in\operatorname{supp}\varphi\cap N^{4/n}}|\partial^{\alpha-\beta}\varphi(\sigma) | \\ &\qquad \leqslant c_{\alpha\beta}n^{-p+|\alpha|-|\beta|}\sum_{|\gamma|=p-|\alpha|+|\beta|,\,\beta\leqslant\alpha} \sup_{\sigma\in\operatorname{supp}\varphi\cap N^{4/n}}|\partial^{\alpha-\beta+\gamma}\varphi(\sigma) |. \end{aligned}
\end{equation}
\tag{2.16}
$$
Using this estimate on the right-hand side of (2.15) we obtain
$$
\begin{equation*}
\sup_{\sigma\in\operatorname{supp}\varphi}|\partial^\alpha(\mu_{4/n}\varphi)(\sigma) |\leqslant c''_\alpha\sum_{\beta\leqslant\alpha}n^{-p+|\alpha|}\sum_{|\gamma|=p-|\alpha|+|\beta|} \sup_{\sigma\in\operatorname{supp}\varphi\cap N^{4/n}}|\partial^{\alpha-\beta+\gamma}\varphi(\sigma)|.
\end{equation*}
\notag
$$
Here all terms on the right-hand side tend to zero as $n\to+\infty$ because the sequence $n^{-p+|\alpha|}$ is bounded for $|\alpha|\leqslant p$ and we have
$$
\begin{equation}
\lim_{n\to+\infty}\sup_{\sigma\in\operatorname{supp}\varphi\cap N^{4/n}}| \partial^\gamma\varphi(\sigma)|=0, \qquad \partial^\gamma\varphi|_N=0\quad\text{and} \quad |\gamma|\leqslant p.
\end{equation}
\tag{2.17}
$$
As each function $\partial^\gamma\varphi(\sigma)$ for $|\gamma|\leqslant p$ is uniformly continuous, for each $\varepsilon>0$ there exists $\delta(\varepsilon)>0$ such that $|\partial^\gamma\varphi(\sigma) |=|\partial^\gamma\varphi(\sigma)-\partial^\gamma\varphi(\eta) |<\varepsilon$ for $|\sigma - \eta|\leqslant\delta(\varepsilon)$, $\eta\in N$. Hence the above equality holds, and for all $\alpha$ such that $|\alpha|\leqslant p$ the sequence $\sup_{\sigma\in\operatorname{supp}\varphi}|\partial^\alpha(\mu_{4/n}\varphi)(\sigma) |$ tends to zero as $n\to +\infty$, that is, the sequence $\mu_{4/n}\varphi$ converges to zero in $C^p_{0N}(\Omega)$.
Consider the sequence of functions $\varphi_n=(1-\mu_{4/n})\varphi$. By construction $K_n=\operatorname{supp}\varphi_n\Subset\Omega\setminus N^{1/n}$ and $\varphi_n\in C^p_0(\Omega\setminus N^{1/n})$. The functions $\psi_n=\varphi_n*\chi_{1/(4n)}$ belong to $ D(\Omega \setminus N)$, where $\chi_\varepsilon\in C^\infty_0(B_\varepsilon)$, $\displaystyle\int\chi_\varepsilon d(\sigma)=1$, $B_\varepsilon$ is the ball of radius $\varepsilon$ (see [12], Theorem 1.4.1) and $\operatorname{supp}\psi_n\subset K^{1/(2n)}_n\subset\Omega\setminus N^{1/(2n)}$. We show that the sequence $\nu_n=\varphi_n-\psi_n$ tends to zero in the space $C^p_0(\Omega)$ as ${n\to +\infty}$. By construction $\operatorname{supp}\varphi_n\subset K_0=(\operatorname{supp}\varphi)^{\varepsilon_0}\Subset \Omega$ for sufficiently large $n$ and some $\varepsilon_0>0$. For $|\alpha|\leqslant p$ we have
$$
\begin{equation*}
\begin{aligned} \, &\sup_{\sigma\in K_0}| \partial^\alpha\nu_n(\sigma)|=\sup_{\sigma\in K_0}|\partial^\alpha\varphi_n-\partial^\alpha\varphi_n*\chi_{1/(4n)} | \\ &\qquad =\sup_{\sigma\in K_0}\biggl|\int(\partial^\alpha\varphi_n(\sigma)-\partial^\alpha\varphi_n(\sigma-\eta)) \chi_{1/(4n)}(\eta)\,d\eta\biggr| \\ &\qquad\leqslant\sup_{\sigma\in K_0,\,|\eta|\leqslant1/(2n)}|\partial^\alpha\varphi_n(\sigma) -\partial^\alpha\varphi_n(\sigma-\eta)| \\ &\qquad\leqslant\sup_{\sigma\in K_0,\,|\eta|\leqslant1/(2n)}|\partial^\alpha ((1-\mu_{4/n}(\sigma))\varphi(\sigma) -(1-\mu_{4/n}(\sigma-\eta))\varphi(\sigma-\eta))| \\ &\qquad\leqslant \sup_{\sigma\in K_0,\,|\eta|\leqslant1/(2n)}|\partial^\alpha\varphi(\sigma) -\partial^\alpha\varphi(\sigma-\eta)| \\ &\qquad\qquad +\sup_{\sigma\in K_0,\,|\eta|\leqslant1/(2n)}|\partial^\alpha(\mu_{4/n}(\sigma-\eta) -\mu_{4/n}(\sigma))\varphi(\sigma-\eta)| \\ &\qquad\qquad+\sup_{\sigma\in K_0,\,|\eta|\leqslant1/(2n)}|\partial^\alpha\mu_{4/n}(\sigma) (\varphi(\sigma-\eta)-\varphi(\sigma))|. \end{aligned}
\end{equation*}
\notag
$$
Each term of the resulting expression in the estimate for the derivatives $\partial^\alpha\nu_n(\sigma)$, $|\alpha|\leqslant p$, can be made less than any prescribed $\varepsilon>0$ for all $n>n(\varepsilon)$. The first term satisfies this because the functions $\partial^\alpha\varphi(\sigma)$, $|\alpha|\in p$, are uniformly continuous in $K_0$. For an estimate of the second term we use Leibniz’s formula and the inequalities (2.7) and (2.16):
$$
\begin{equation*}
\begin{aligned} \, &\sup_{\sigma\in K_0,\,|\eta|\leqslant1/(2n)}|\partial^\alpha(\mu_{4/n}(\sigma-\eta) -\mu_{4/n}(\sigma))\varphi(\sigma-\eta)| \\ &\qquad \leqslant c_\alpha\sup_{\sigma\in K_0,\,|\eta|\leqslant1/(2n)}\sum_{\beta<\alpha} \bigl(|\partial^\beta\mu_{4/n}(\sigma-\eta)|+|\partial^\beta\mu_{4/n}(\sigma)|\bigr) |\partial^{\alpha-\beta}\varphi(\sigma-\eta)| \\ &\qquad \leqslant c'_\alpha\sum_{\beta\leqslant\alpha}n^{|\beta|}\sup_{\sigma\in\operatorname{supp}\varphi\cap N^{1/n}|\eta|\leqslant1/(2n)}|\partial^{\alpha-\beta} \varphi(\sigma-\eta)| \\ &\qquad\leqslant c''_\alpha\sum_{\beta\leqslant\alpha}n^{|\beta|}n^{-p+|\alpha|-|\beta|} \sum_{|\gamma|=p-|\alpha|+|\beta|}\sup_{\sigma\in\operatorname{supp}\varphi\cap N^{1/n}|\eta|\leqslant1/(2n)}|\partial^{\alpha-\beta+\gamma} \varphi(\sigma-\eta)| \\ &\qquad\leqslant \widetilde{C}_\alpha n^{|\alpha|-p}\sum_{\beta\leqslant\alpha,\,|\gamma|=p-|\alpha|+\beta} \sup_{\sigma\in\operatorname{supp}\varphi\cap N^{1/n}|\eta|\leqslant1/(2n)}|\partial^{\alpha-\beta+\gamma} \varphi(\sigma-\eta)|. \end{aligned}
\end{equation*}
\notag
$$
Using arguments similar to the ones in the proof of (2.17) we can conclude that the second term tends to zero as $n\to+\infty$. Estimating the third term in a similar way we can assert that the sequence of functions $\nu_n(\sigma)$ tends to zero in $C^p_0(\Omega)$.
It follows from the equality $\varphi=\psi_n+\varphi_n-\psi_n+\mu_{1/n}\varphi$ that the sequence of functions $\psi_n\in D(\Omega\setminus N)$ converges to $\varphi$ in $C^p_0(\Omega)$. Lemma 2.7 is proved. Lemma 2.8. Each element of ${D^\infty_N}'^j(\Omega)$ can uniquely be extended to a continuous map of $C^j_{0N}(\Omega)$ in $\mathbb{R}$, and inequality (2.13) remains valid for all $\varphi\in C^j_{0N}(\Omega)$. The proof of the lemma is similar to the proof of Theorem 2.1.6 in [12]. It follows from Lemma 2.7 that for each function $\varphi\in C^j_{0N}(\Omega)$ we can find a sequence of functions $\varphi_n\in D^\infty_N(\Omega)$ with supports in a fixed compact neighbourhood $K$ of the support of $\varphi$ such that
$$
\begin{equation*}
\sum_{|\alpha|\leqslant j}\sup_{\sigma\in K}|\partial^\alpha(\varphi-\varphi_n)(\sigma) |\to 0 \quad \text{as } n\to +\infty.
\end{equation*}
\notag
$$
Let $f\in {D^\infty_N}'^j(\Omega)$. It follows from (2.13) that $(f,\varphi_n)$ is a Cauchy sequence. Hence the limit $\lim_{n\to +\infty}(f,\varphi_n)$ exists; we denote it by $(f,\varphi)$. It does not depend on the choice of a sequence converging to $\varphi$ because of (2.13). Hence on $C^j_{0N}(\Omega)$ we obtain a well-defined linear form $(f,\varphi)$. It is continuous in view of an inequality holding for all $\varphi\in C^j_{0N}(\Omega)$ with $\operatorname{supp}\varphi\subset K$ that can be obtained by taking the limit in (2.13). Consider the regularized distance $\triangle(\sigma, N)$ of the point $\sigma$ to the set $N$, which is a function in $C^\infty(\Omega\setminus N)$ satisfying the following conditions:
$$
\begin{equation}
c_1d(\sigma,N)\leqslant\triangle(\sigma,N)\leqslant c_2d(\sigma,N), \qquad \sigma\in\Omega\setminus N,
\end{equation}
\tag{2.18}
$$
and
$$
\begin{equation}
|\partial^\alpha\triangle(\sigma, N)|\leqslant c_\alpha[d(\sigma, N)]^{1-|\alpha|}, \qquad \alpha\in \mathbb{Z}^n_+,
\end{equation}
\tag{2.19}
$$
where $c_1$, $c_2$ and $c_\alpha$ are positive numbers. That such a function exists was proved in Theorem 2 in [13], Ch. VI. § 2. Lemma 2.9. The function $\varphi\in D^\infty_N(\Omega)$ can be represented in the form $\varphi(\sigma)=\psi_k(\sigma)\triangle^k(\sigma, N)$, where $k \in \mathbb{Z}_+$, $\psi_k(\sigma)\in D^\infty_N(\Omega)$, and the map $F_k(\varphi)=\psi_k$ is continuous in $D^\infty_N(\Omega)$. Proof. We use the representation of the function $\varphi\in D^\infty_N(\Omega)$ in the form $\varphi=(1-\mu_\varepsilon)\varphi+\mu_\varepsilon\varphi$, where the function $\mu_\epsilon(\sigma)$ satisfies (2.7).
For $(1-\mu_\varepsilon)\varphi$ the assertion of the lemma holds because for $k \in \mathbb{Z}_+$ the function $(1-\mu_\varepsilon(\sigma))\triangle^{-k}(\sigma, N)\in C^\infty(\Omega)$ is a multiplier in the space $D^\infty_N(\Omega)$. That the assertion holds for $\mu_\varepsilon(\sigma)\varphi(\sigma)$ is based, as in the proof of Theorem 2.1, on the representation
$$
\begin{equation*}
\mu_\varepsilon(\sigma) \varphi(\sigma)=\mu_\epsilon(\sigma) \biggl(\varphi(\sigma)- \sum_{|\gamma|\leqslant j-1} \partial^\gamma \varphi(\eta)\frac{(\sigma-\eta)^\gamma}{\gamma!}\biggr) = \mu_\varepsilon(\sigma) R_j(\sigma,\eta),
\end{equation*}
\notag
$$
where $\eta=\eta(\sigma)\in N$, $d(\sigma,N)=|\sigma-\eta(\sigma)|$ and $R_j(\sigma,\eta)$ is the remainder in Taylor’s formula.
Consider the function $\psi_{k\epsilon}(\sigma)=\mu_\epsilon(\sigma)\varphi(\sigma)\triangle^{-k}(\sigma,N)$, $k\in\mathbb{Z}_+$ for $\sigma\in\Omega\setminus N$. It is obvious that $\psi_k(\sigma)\in C^\infty(\Omega\setminus N)$. Using that $R_j(\sigma,\eta)$ is a Lagrange remainder we obtain
$$
\begin{equation*}
|\psi_{k\epsilon}(\sigma)| \leqslant \triangle^{-k}(\sigma,N) |\mu_\epsilon(\sigma)| \biggl|\sum_{|\alpha|=j} \varphi^\alpha(\sigma+t(\eta-\sigma))\frac{(\sigma-\eta)^\alpha}{\alpha!}\biggr|, \qquad 0\leqslant t\leqslant 1.
\end{equation*}
\notag
$$
From inequalities (2.18) for $|\alpha|=j$ it follows that
$$
\begin{equation*}
|(\sigma-\eta(\sigma))^\alpha | \triangle^{-k}(\sigma,N) \leqslant d^j(\sigma,N) \triangle^{-k}(\sigma,N) \leqslant c_j d^{j-k}(\sigma,N).
\end{equation*}
\notag
$$
Hence
$$
\begin{equation*}
|\psi_{k\epsilon}(\sigma)| \leqslant c_j' |\mu_\epsilon(\sigma)| d^{j-k}(\sigma,N) \sum_{|\alpha|=j} |\varphi^\alpha(\sigma+t(\eta-\sigma))|.
\end{equation*}
\notag
$$
Since for $j>k$ the right-hand side of this inequality tends to zero when $\sigma_n\to\sigma_0\in N$ as $n\to\infty$, we can extend the function $\psi_{k\epsilon}(\sigma)$ to $N$ by setting $\psi_{k\epsilon}(\sigma)=0$ for $\sigma\in N$. Then $\psi_{k\epsilon}(\sigma)\in C(\Omega)$. In a similar way we show that the function $\psi_{k_\varepsilon}(\sigma)$ has derivatives of any order, which vanish on $N$. The proof uses Leibniz’s formula, (2.18), (2.19) and the inequality
$$
\begin{equation}
| \partial^\alpha\triangle^{-p}(\sigma, N)|\leqslant C_\alpha(K, p)[d(\sigma, N)]^{-p-|\alpha|}, \qquad \sigma \in K\setminus N,
\end{equation}
\tag{2.20}
$$
where $K\Subset\Omega$, $p \in \mathbb{N}$ and $\alpha\in \mathbb{Z}^n_+$.
This inequality holds for $|\alpha|=1$ because of (2.18) and (2.19):
$$
\begin{equation*}
|\partial_i\triangle^{-p}(\sigma, N) | =p|\triangle^{-p-1}(\sigma, N)\partial_i\triangle(\sigma, N)|\leqslant c_i[d(\sigma, N)]^{-p-1}.
\end{equation*}
\notag
$$
Assume that (2.20) holds for all $\alpha \in \mathbb{Z}^n_+$ with $|\alpha|\leqslant j$, and let $\alpha'=\alpha+\gamma_i$, $|\gamma_i|=1$. Then using Leibniz’s formula and relations (2.19) for $\sigma\in K\setminus N$ we obtain
$$
\begin{equation*}
\begin{aligned} \, &| {\partial^\alpha}'\triangle^{-p}(\sigma, N)|=p|\partial^\alpha(\triangle^{-p-1}(\sigma, N)\partial_i\triangle(\sigma, N)| \\ &\qquad \leqslant c'_\alpha(p)\sum_{\beta\leqslant\alpha}|\partial^{\alpha-\beta}\triangle^{-p-1}(\sigma, N)|\,|\partial^\beta\partial_i\triangle(\sigma, N)| \\ &\qquad\leqslant c'_\alpha(P, K)\sum_{\beta\leqslant\alpha}[d(\sigma,N)]^{-p-|\alpha|+|\beta|-1}[d(\sigma, N)]^{1-|\beta|-1} \\ &\qquad\leqslant c_\alpha(p, k)[d(\sigma, N)]^{-p-|\alpha'|}. \end{aligned}
\end{equation*}
\notag
$$
Therefore, $\psi_{k\varepsilon}\in D(\Omega)$ and $\partial^\alpha\psi_{k\varepsilon}(\sigma)|_N=0$ for $k \in \mathbb{Z}^n_+$, that is, $\psi_{k\varepsilon} \in D^\infty_N(\Omega)$. The proof is complete. § 3. Multipliers in function spacesOne aim of introducing the function spaces ${D^\infty_N}'(\Omega)$, ${D^\infty_{NF}}'(\Omega)$ and ${\mathcal{E}^\infty_N}'(\Omega)$ is to construct extensions of the operator of multiplication of distributions by a function whose derivatives are singular on a relatively closed subset $N$ of $\Omega$. In accordance with the general philosophy of distribution theory, this can be accomplished by extending the class of functions under consideration. Let $a(\sigma)\in C^\infty(\Omega\setminus N)$, and assume that for each compact set $K\Subset\Omega$ and any $\alpha\in \mathbb{Z}^n_+$ there exist numbers $C_\alpha(K)>0$ and $q_\alpha(K)\leqslant0$ such that
$$
\begin{equation}
|\partial^\alpha a(\sigma)|\leqslant C_\alpha(K)[d(\sigma, N)]^{q_\alpha(K)}, \qquad \sigma\in K\setminus N.
\end{equation}
\tag{3.1}
$$
Theorem 3.1. Multiplication by a function $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfying (3.1) is well defined and continuous in $D^\infty_N(\Omega)$. Proof. We show that elements of $D^\infty_N(\Omega)$ can be multiplied by any function $a(\sigma)$ satisfying (3.1). More precisely, for each $\varphi\in D^\infty_N(\Omega)$ the function $a(\sigma)\varphi(\sigma)$, ${\sigma\in\Omega\setminus N}$, can uniquely be extended to a function in $D^\infty_N(\Omega)$. In fact, it is sufficient to prove that for each multi-index $\alpha$, each point $\sigma_0\in N$ and each sequence $\sigma_n$ tending to $\sigma_0$ as $n\to +\infty$ we have Using Leibniz’s formula, the proof of this reduces to the proof of the equality
Repeating the arguments in the proof of Theorem 2.1 we write
$$
\begin{equation*}
\psi(\sigma_n)\partial^\alpha a(\sigma_n)=\partial^\alpha a(\sigma_n)\biggl(\psi(\sigma_n)-\sum_{|\gamma|\leqslant j-1}\partial^\gamma\psi(\eta_n)\frac{(\sigma_n-\eta_n)^\gamma}{\gamma!}\biggr),
\end{equation*}
\notag
$$
where $\eta_n(\sigma_n)\in N$, $|\sigma_n-\eta_n |=d(\sigma_n, N)$. This can be done for sufficiently large $n$.
Using inequalities (3.1) and an estimate for the remainder in Taylor’s formula, as in the proof of Theorem 2.1, for sufficiently large $n$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &|\psi(\sigma_n)\,\partial^\alpha a(\sigma_n) | \\ &\qquad \leqslant c'_\alpha[d(\sigma_n, N)]^{q_\alpha(K_0)}[d(\sigma_n, N)]^j\sum_{|\gamma|=j}|\partial^\gamma\psi(\sigma_n+t_n(\eta_n-\sigma_n))|, \end{aligned}
\end{equation}
\tag{3.4}
$$
where $0\leqslant t_n(\sigma_n)\leqslant 1$ and $K_0$ is a ball with centre $\sigma_0$.
Since we can choose $j$ to be arbitrarily large, the right-hand side of this inequality tends to zero if $\sigma_n\to\sigma_0\in N$ as $n\to +\infty$. Hence (3.3), and therefore also (3.2), holds. It follows from (3.2) that the extension of $a(\sigma)\varphi(\sigma)$ to $N$ by continuity belongs to the space $D^\infty_N(\Omega)$. We denote it by $a\varphi$ in what follows. Since $(1-\mu_\varepsilon(\sigma))a(\sigma)\in C^\infty(\Omega)$ if $\mu_\varepsilon(\sigma)$ satisfies (2.7), this function is a multiplier in $D^\infty_N(\Omega)$. This is because for each compact set $K\Subset \Omega$ we have
$$
\begin{equation*}
\begin{aligned} \, &\sup_{\sigma\in K}|\partial^\alpha[(1-\mu_\varepsilon(\sigma))a(\sigma)]|\leqslant C_\alpha\sup_{\sigma\in K}\sum_{\beta\leqslant\alpha}|\partial^{\alpha-\beta}(1-\mu_\varepsilon(\sigma))|\,| \partial^\beta a(\sigma)| \\ &\qquad \leqslant C_\alpha(K, \varepsilon)\sum_{\beta\leqslant\alpha}\sup_{\sigma\in K\setminus N^{\varepsilon/4}}[d(\sigma, N)]^{q_\beta(K)}\leqslant C'_\alpha(K, \varepsilon), \end{aligned}
\end{equation*}
\notag
$$
where $C_\alpha>0$, $C_\alpha(K,\varepsilon)>0$ and $C'_\alpha(K,\varepsilon)>0$ are numbers depending on $\alpha\in \mathbb{Z}^n_+$, the compact set $K$ and the fixed number $\varepsilon>0$.
It follows from the inequalities obtained that for each compact set $K\Subset\Omega$, each $p \in \mathbb{Z}_+$ and any function $\varphi\in D^\infty_N(\Omega)$ with $\operatorname{supp}\varphi\subset K$ we have
$$
\begin{equation}
\sum_{|\alpha|\leqslant p}\sup_{\sigma\in K}|\partial^\alpha[(1-\mu_\varepsilon(\sigma))a(\sigma)\varphi(\sigma)]|\leqslant C_p(K, \varepsilon)\sum_{|\alpha|\leqslant p}\sup_{\sigma\in K}|\partial^\alpha\varphi(\sigma)|.
\end{equation}
\tag{3.5}
$$
Thus, multiplication by $(1-\mu_\varepsilon(\sigma))a(\sigma)$ is continuous in $D^\infty_N(\Omega)$.
We show that multiplication by $\mu_\varepsilon(\sigma)a(\sigma)$ is also continuous in $D^\infty_N(\Omega)$. Using Leibniz’s formula and an inequality similar to (3.4), for any $\alpha\in \mathbb{Z}^n_+$, $j\in \mathbb{Z}_+$, any compact set $K\Subset \Omega$ and an arbitrary function $\varphi\in D^\infty_N(\Omega)$ with $\operatorname{supp}\varphi\subset K$, for $\sigma\in K\setminus N$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag &|\partial^\alpha(\mu_\varepsilon(\sigma)a\varphi(\sigma)) |\leqslant C_\alpha\sum_{\beta\leqslant\alpha}|\partial^\beta a(\sigma)|\,|\partial^{\alpha-\beta}(\mu_\varepsilon(\sigma)\varphi(\sigma))| \\ &\qquad\leqslant C_\alpha(K)\sum_{\beta\leqslant\alpha}[d(\sigma, N)]^{q_\beta(K)}[d(\sigma, N)]^j\sum_{|\gamma|=j}|\partial^{\alpha-\beta+\gamma}(\mu_\varepsilon\varphi) (\sigma+t(\eta(\sigma)-\sigma))|. \end{aligned}
\end{equation}
\tag{3.6}
$$
The right-hand side of (3.6) is well defined for sufficiently small $\varepsilon$, because $\mu_\varepsilon\varphi\in D^\infty_N(\Omega)\subset D(\mathbb{R}^n)$, $K\cap N^\varepsilon\Subset \Omega$ if $\varepsilon$ is sufficiently small, and for any $\sigma\in K\cap N^\varepsilon$ there exists $\eta\in N$ such that $d(\sigma, N)=|\eta-\sigma|$ by Lemmas 2.4 and 2.5. In addition, points in the line segment $\sigma+t(\eta(\sigma)-\sigma)$, $0\leqslant t\leqslant 1$, either belong to $K$, or the function $\mu_\varepsilon\varphi$ and its derivatives vanish at these points because $\operatorname{supp}\varphi\subset K$. We denote $[\sup_{|\beta|\leqslant p}|q_\beta(K)|]+ 1$, where $[a]$ is the integer part of $a$, by $q_p(K)$. Then we have the inequality
$$
\begin{equation*}
\sup_{\sigma\in K\cap N^\varepsilon}[d(\sigma, N)]^{q_\beta(K)}[d(\sigma, N)]^{q_{|\alpha|}(K)}\leqslant C_1, \qquad \beta\leqslant\alpha.
\end{equation*}
\notag
$$
From it and (3.6) we obtain
$$
\begin{equation*}
\sup_{\sigma\in K}|\partial^\alpha(\mu_\varepsilon(\sigma)a\varphi(\sigma))|\leqslant C'_\alpha(K)\sum_{\beta\leqslant\alpha}\sum_{|\gamma|=q_{|\alpha|}(K)}\sup_{\sigma\in K\cap N^\varepsilon}|\partial^{\alpha-\beta+\gamma}(\mu_\varepsilon\varphi)(\sigma)|.
\end{equation*}
\notag
$$
Using Leibniz’s formula on the right-hand side and relying on (2.7) we obtain
$$
\begin{equation*}
\sup_{\sigma\in K}|\partial^\alpha(\mu_\varepsilon(\sigma)a\varphi(\sigma))|\leqslant C'_p(K,\varepsilon)\sum_{|\alpha|\leqslant p+q_p(K)}\sup_{\sigma\in K\cap N^\varepsilon}|\partial^\alpha\varphi(\sigma)|, \qquad |\alpha|\leqslant p.
\end{equation*}
\notag
$$
From this and inequality (3.5), for each compact set $K\Subset\Omega$ and any $p\in \mathbb{Z}_+$, for an arbitrary function $\varphi\in D^\infty_N(\Omega)$ with $\operatorname{supp}\varphi\subset K$ we deduce the inequality
$$
\begin{equation}
\sum_{|\alpha|\leqslant p}\sup_{\sigma\in K}|\partial^\alpha a\varphi(\sigma)|\leqslant \widetilde{C}_p(K, \varepsilon)\sum_{|\alpha|\leqslant p+q_p(K)}\sup_{\sigma\in K}|\partial^\alpha\varphi(\sigma)|,
\end{equation}
\tag{3.7}
$$
where $\widetilde{C}_p(K,\varepsilon)>0$ is a number depending on $K$ and $\varepsilon>0$.
Hence multiplication by the function $a(\sigma)\in C^\infty(\Omega\setminus N)$, which satisfies the assumptions of Theorem 3.1, is continuous in $D^\infty_N(\Omega)$. The proof is complete. A similar result holds for the space $\mathcal{E}^\infty_N(\Omega)$. Theorem 3.2. Multiplication by a function $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfying (3.1) is well defined and continuous in the space $\mathcal{E}^\infty_N(\Omega)$. Proof. Similarly to the proof of Theorem 3.1 we can justify the extension of $a(\sigma)\varphi(\sigma)\in C^\infty(\Omega\setminus N)$, $\varphi\in C^\infty(\Omega)$, to a continuous function in $\mathcal{E}^\infty_N(\Omega)$.
In just the same way we prove that $(1-\mu_\varepsilon(\sigma))a(\sigma)$, where $\mu_\varepsilon(\sigma)$ is a function satisfying (2.7), is a multiplier in $\mathcal{E}^\infty_N(\Omega)$, and for each compact set $K\Subset\Omega$, any $p\in \mathbb{Z}_+$ and an arbitrary function $\varphi\in \mathcal{E}^\infty_N(\Omega)$ we have
$$
\begin{equation}
\sum_{|\alpha|\leqslant p}\sup_{\sigma\in K}|\partial^\alpha[(1-\mu_\varepsilon(\sigma))a(\sigma)\varphi(\sigma)]|\leqslant C_p(K, \varepsilon)\sum_{|\alpha|\leqslant p}\sup_{\sigma\in K}|\partial^\alpha\varphi(\sigma)|.
\end{equation}
\tag{3.8}
$$
We show that multiplication by $\mu_\varepsilon(\sigma)a(\sigma)$ is a continuous map of $\mathcal{E}^\infty_N(\Omega)$ to itself. Using Leibniz’s formula, inequalities (3.1) and an estimate of the form (3.4), for each $\alpha\in \mathbb{Z}^n_+$, $|\alpha|\leqslant p$, an arbitrary $\varphi\in \mathcal{E}^\infty_N(\Omega)$ and a compact set $K\Subset\Omega$, for sufficiently small $\varepsilon$ we obtain the inequalities
$$
\begin{equation*}
\begin{aligned} \, &|\partial^\alpha(\mu_\varepsilon(\sigma)a\varphi(\sigma))|\leqslant C'_\alpha\sum_{\beta\leqslant\alpha}|\partial^\beta a(\sigma)|\,|\partial^{\alpha-\beta}(\mu_\varepsilon\varphi)(\sigma)| \\ &\qquad \leqslant C_\alpha\sum_{\beta\leqslant\alpha}[d(\sigma,N)]^{q_\beta(K)}[d(\sigma, N)]^j\sum_{|\gamma|=j}|\partial^{\alpha-\beta+\gamma}(\mu_\varepsilon\varphi) (\sigma+t(\eta(\sigma)-\sigma))| \end{aligned}
\end{equation*}
\notag
$$
for $\sigma\in K\setminus N$.
This estimate is based on the facts that $\mu_\varepsilon\varphi \in \mathcal{E}^\infty_N(\Omega)$, $\operatorname{supp}\mu_\varepsilon\varphi \subset N^\varepsilon$ and for $\sigma\in K\cap N^\varepsilon$ the points $\sigma+t(\eta(\sigma)-\sigma)$ belong to the compact set $K_\varepsilon=(K\cap N^\varepsilon)_\varepsilon\Subset\Omega$ for sufficiently small $\varepsilon$ by Lemma 2.5. If $j=q_{|\alpha|}(K)=[\sup_{|\beta|\leqslant|\alpha|}|q_\beta(K)|]+1$, then it follows from the above inequality that
$$
\begin{equation*}
\begin{aligned} \, &\sup_{\sigma\in K}|\partial^\alpha(\mu_\varepsilon(\sigma)a\varphi(\sigma))| \\ &\qquad\leqslant C'_\alpha(K,\varepsilon)\sum_{\beta\leqslant\alpha}\sum_{|\gamma|=q_{|\alpha|}(K)}\sup_{\sigma\in K\cap N^\varepsilon}|\partial^{\alpha-\beta+\gamma}(\mu_\varepsilon\varphi)(\sigma+t(\eta(\sigma)-\sigma))|. \end{aligned}
\end{equation*}
\notag
$$
Using Leibniz’s formula and (2.7) on the right-hand side, since for $\sigma\in K\cap N^\varepsilon$ the points $(\sigma+t(\eta(\sigma)-\sigma))$ belong to the set $(K\cap N^\varepsilon)_\varepsilon$, which is a compact subset of $\Omega$ by Lemma 2.5, we obtain
$$
\begin{equation*}
\sup_{\sigma\in K}|\partial^\alpha(\mu_\varepsilon(\sigma)a\varphi(\sigma))|\leqslant C_\alpha(K, \varepsilon)\sum_{|\beta|\leqslant|\alpha|+q_{|\alpha|}(K)}\sup_{\sigma\in (K\cap N^\varepsilon)_\varepsilon}|\partial^\beta\varphi(\sigma)|.
\end{equation*}
\notag
$$
It follows from this and (3.8) that for each compact set $K\Subset\Omega$ and any $p\in \mathbb{Z}_+$, for an arbitrary function $\varphi\in \mathcal{E}^\infty_N(\Omega)$ we have
$$
\begin{equation}
\sup_{|\alpha|\leqslant p,\, \sigma\in K}|\partial^\alpha(a\varphi)(\sigma)|\leqslant C_p(K,\varepsilon)\sum_{|\alpha|\leqslant p+q_p(K)}\sup_{\sigma\in K'_\varepsilon}|\partial^\beta\varphi(\sigma)|,
\end{equation}
\tag{3.9}
$$
where $ K'_\varepsilon=K\cup(K\cap N^\varepsilon)_\varepsilon\Subset\Omega$ and $C_p(K,\varepsilon)>0$ is a quantity depending on $K$, $p$ and the choice of $\varepsilon>0$.
Hence multiplication by the function $a(\sigma)\in C^\infty(\Omega\setminus N)$, which satisfies (3.1), is continuous in $\mathcal{E}^\infty_N(\Omega)$. The proof is complete. The following result is a consequence of Theorem 3.1. Theorem 3.3. A function $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfying (3.1) is a multiplier in the space ${D^\infty_N}'(\Omega)$. Proof. By Theorem 3.1, for any $\varphi\in D^\infty_N(\Omega)$ the function $a\varphi\in D^\infty_N(\Omega)$ is well defined.
Let $f\in {D^\infty_N}'(\Omega)$. Then a linear form $af$ is defined on $D^\infty_N(\Omega)$ by
$$
\begin{equation*}
(af,\varphi)\equiv (f,a\varphi), \qquad \varphi\in D^\infty_N(\Omega).
\end{equation*}
\notag
$$
From (2.13) and (3.7), for each compact set $K\Subset\Omega$ and any $\varphi\in D^\infty_N(\Omega)$ such that $\operatorname{supp}\varphi\subset K$ we obtain
$$
\begin{equation}
\begin{aligned} \, \notag |(af,\varphi| &\leqslant C(K)\sum_{|\alpha|\leqslant j(K)}\sup_{\sigma\in K}|\partial^\alpha a\varphi(\sigma)| \\ &\leqslant C'(K)\sum_{|\beta|\leqslant j(K)+q_j(K)}\sup_{\sigma\in K}|\partial^\beta\varphi(\sigma)|. \end{aligned}
\end{equation}
\tag{3.10}
$$
Therefore, the form $af$ is continuous on $D^\infty_N(\Omega)$, that is, $af\in {D^\infty_N}'(\Omega)$.
If a sequence $f_n\in {D^\infty_N}'(\Omega)$ converges to $f\in {D^\infty_N}'(\Omega)$, then
$$
\begin{equation*}
\lim_{n\to+\infty}(af_n,\varphi)=\lim_{n\to+\infty}(f_n,a\varphi)=(f, a\varphi)=(af,\varphi) \quad\text{for } \varphi\in D^\infty_N(\Omega).
\end{equation*}
\notag
$$
Thus, multiplication by the function $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfying (3.1) is a continuous map of ${D^\infty_N}'(\Omega)$ into itself. The proof is complete. It follows from Theorem 3.2 and inequalities (2.14) and (3.9) that for each distribution $f\in {\mathcal{E}^\infty_N}'^p(\Omega)$ such that $\operatorname{supp}f\subset K$ we have
$$
\begin{equation}
|(af,\varphi)|\leqslant C_p(K, \varepsilon)\sum_{|\alpha|\leqslant p+q_p(K)}\sup_{\sigma\in K'_\varepsilon}|\partial^\alpha\varphi(\sigma)| \quad\text{for } \varphi\in C^\infty(\Omega),
\end{equation}
\tag{3.11}
$$
where the positive quantity $C_p(K,\varepsilon)>0$ depends on $K$ and $\varepsilon>0$, and $K'_\varepsilon$ is the compact set from the proof of Theorem 3.2. Therefore, the following result holds. Theorem 3.4. A function $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfying inequalities (3.1), is a multiplier in the space ${\mathcal{E}^\infty_N}'(\Omega)$. When we consider multiplication of elements of ${D^\infty_{NF}}'(\Omega)$ and a function $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfying (3.1), it is natural to assume that the following condition holds for each $p \in \mathbb{Z}_+$:
$$
\begin{equation}
q_p=\Bigl[\sup_{|\alpha|\leqslant p,\,K\Subset\Omega}|q_\alpha(K)|\Bigr]+ 1 < +\infty.
\end{equation}
\tag{3.12}
$$
Theorem 3.5. If $a(\sigma)$ is a function in $C^\infty(\Omega\setminus N)$ and inequalities (3.1) hold for each compact set $K \Subset \Omega$, where condition (3.12) is satisfied, then for each distribution $f\in {D^\infty_N}'^p(\Omega)$ the distribution $af$ belongs to ${D^\infty_N}'^{p+q_p}(\Omega)$ and Theorem 3.5 follows from the definition of ${D^\infty_N}'^p(\Omega)$ and inequalities (3.10) and (3.12). An important example of $a(\sigma)\in C^\infty(\Omega\setminus N)$ satisfying (3.1) is the function $a(\sigma)=1/{P(\sigma)}$, where $P(\sigma)$ is a polynomial and $N$ is the set of its real zeros. It follows from the well-known estimate for the modulus of a polynomial of real variables
$$
\begin{equation}
\frac{1}{|P(\sigma)|}\leqslant C(1+|\sigma|)^\rho\, d^\varkappa(\sigma,N), \qquad \sigma\in \mathbb{R}^n\setminus N
\end{equation}
\tag{3.14}
$$
(see [4], the appendix, Theorem A.3), where $C>0$, $\rho\geqslant 0$ and $\varkappa <0$ are some numbers, that we have inequalities of the form (3.1),
$$
\begin{equation}
\biggl|\partial^\alpha\frac{1}{P(\sigma)}\biggr|\leqslant C_\alpha(K)[d(\sigma, N)]^{\varkappa_\alpha}, \qquad \sigma\in K\setminus N,
\end{equation}
\tag{3.15}
$$
where $C_\alpha(K)>0$ is a quantity depending on $K$ and $\alpha\,{\in}\, \mathbb{Z}^n_+$ and ${(|\alpha|\,{+}\,1)\varkappa\,{\leqslant}\,\varkappa_\alpha\,{<}\,0}$. Inequalities (3.15) are immediately verified for $|\alpha|=1$:
$$
\begin{equation*}
\biggl|\partial_i\frac{1}{P(\sigma)}\biggr|=\frac{1}{|P^2(\sigma)|}|\partial_i P(\sigma)|\leqslant C_i(K)\,d^{2\varkappa}(\sigma,N), \qquad \sigma\in K\setminus N.
\end{equation*}
\notag
$$
Assume that (3.15) holds for all $\alpha\in \mathbb{Z}^n_+$ such that $|\alpha|=p$, and let $\alpha'=\alpha+\gamma_i$, $|\gamma_i|=1$. For $\sigma\in K\setminus N$ set
$$
\begin{equation*}
\begin{aligned} \, \biggl|{\partial^\alpha}'\frac{1}{P(\sigma)}\biggr| &=\biggl|\partial^\alpha\biggl(\partial_i\frac{1}{P(\sigma)}\biggr)\biggr| =\biggl|\partial^\alpha\biggl(\frac{1}{P^2(\sigma)}\,\partial_i P(\sigma) \biggr)\biggr| \\ &\leqslant C_\alpha\sum_{\beta\leqslant\alpha}\biggl|\partial^\beta\frac{1}{P(\sigma)}\biggr|\, \biggl|\partial^{\alpha-\beta}\,\frac{\partial_iP(\sigma)}{P(\sigma)}\biggr| \\ &\leqslant C'_\alpha\sum_{\beta\leqslant\alpha}\biggl|\partial^\beta\frac{1}{P(\sigma)}\biggr| \sum_{\gamma\leqslant\beta}\biggl|\partial^{\alpha-\beta-\gamma}\frac{1}{P(\sigma)}\biggr|\, |\partial^\gamma\partial_iP(\sigma)| \\ &\leqslant C''_\alpha(K)\sum_{\gamma\leqslant\beta\leqslant\alpha} [d(\sigma,N)]^{\varkappa_\beta+\varkappa_{\alpha-\beta-\gamma}}. \end{aligned}
\end{equation*}
\notag
$$
Since $\varkappa_\beta+\varkappa_{\alpha-\beta-\gamma}\geqslant(|\beta|+1+|\alpha|-|\beta|-|\gamma|+1)\varkappa\geqslant(|\alpha|-|\gamma|+2)\varkappa\geqslant(|\alpha'+1)\varkappa$, it follows that
$$
\begin{equation*}
\biggl|\partial^{\alpha'}\frac{1}{P(\sigma)}\biggr| \leqslant C_{\alpha'}(K)[d(\sigma, N)]^{\varkappa_{\alpha'}} \quad\text{for } \sigma\in K\setminus N, \quad (|\alpha'|+1)\varkappa\leqslant\varkappa_{\alpha'}.
\end{equation*}
\notag
$$
Hence (3.15) is true. Thus, we have proved the following result. Theorem 3.6. A function $a(\sigma)={1}/{P(\sigma)}$, where $P(\sigma)$ is a polynomial with set of real zeros $N$, is a multiplier in the spaces $D^\infty_{N\cap\Omega}(\Omega)$, ${D^\infty_{N\cap \Omega}}'(\Omega)$, $D^\infty_{N\cap\Omega F}(\Omega)$, $\mathcal{E}^\infty_{N\cap\Omega}(\Omega)$ and ${\mathcal{E}^\infty_{N\cap\Omega}}'(\Omega)$ for each domain $\Omega\subset \mathbb{R}^n$. A further important example of a multiplier in the spaces from Theorem 3.6 is the regularized distance $\triangle(\sigma, N)$. This follows from inequalities (2.18) and (2.19) and Theorem 3.1–3.5. Other multipliers in these spaces are functions $b(\sigma)\triangle^k(\sigma, N)$ such that $b(\sigma)\in C^\infty(\Omega)$, $k\in \mathbb{N}$, because using Leibniz’s formula and the inequality
$$
\begin{equation}
|\partial^\alpha\triangle^p(\sigma, N)|\leqslant C_\alpha(K, p)[d(\sigma, N)]^{p-|\alpha|}, \qquad \sigma\in K\setminus N,
\end{equation}
\tag{3.16}
$$
which holds for any compact set $K\Subset\Omega$ and $p\in \mathbb{N}$, we obtain inequalities of the form (3.1). In this case condition (3.12) is satisfied. Inequality (3.16) can be established similarly to (2.20). The functions satisfying (3.1) exhaust the set of multipliers in the space $D^\infty_N(\Omega)$. Theorem 3.7. A function $a(\sigma)$ is a multiplier in the space $D^\infty_N(\Omega)$ if and only if $a(\sigma)\in C^\infty(\Omega\setminus N)$ and (3.1) holds. Proof. A function satisfying (3.1) is a multiplier in $D^\infty_N(\Omega)$ by Theorem 3.1.
We establish the converse result. Let $a(\sigma)$ be a multiplier in $D^\infty_N(\Omega)$, that is, assume that each element of $D^\infty_N(\Omega)$ can be multiplied by $a(\sigma)$ and this map is continuous in $D^\infty_N(\Omega)$. It is obvious that such a function belongs to $C^\infty(\Omega\setminus N)$. Since multiplication by $a(\sigma)$ is continuous in $D^\infty_N(\Omega)$, for each compact set $K\Subset\Omega$ and any $r\in \mathbb{Z}_+$ there exist numbers $C_r(K)$ and $m_r(K)$ such that for any function $\varphi\in D^\infty_N(\Omega)$ with $\operatorname{supp}\varphi\subset K$ we have
$$
\begin{equation}
|\partial^\alpha(a\varphi)(\sigma)|\leqslant C_r(K)\sum_{|\beta|\leqslant m_r(K)}\sup_{\sigma\in K}|\partial^\beta\varphi(\sigma)|, \qquad |\alpha|\leqslant r, \qquad \sigma\in K.
\end{equation}
\tag{3.17}
$$
It follows from these inequalities that multiplication by $a(\sigma)$ can be extended to a continuous map from $C^{m_r(K)}_{0N}(\mathring{K})$ to $C^r_{0N}(\mathring{K})$, provided that $\mathring{K}$, the interior of $K$, is nonempty. To do this, for any function $\varphi\in C^{m_r(K)}_{0N}(\Omega)$ we consider a sequence $\varphi_n\in D^\infty_N(\Omega)$ converging to $\varphi$ in $C^{m_r(K)}_{0N}(\Omega)$. Such a sequence exists by Lemma 2.7. It follows from (3.17) that the functions $a\varphi_n$ form a Cauchy sequence in $C^r_{0N}(\Omega)$. Hence this sequence has a limit as $n\to+\infty$, which we denote by $a\varphi$. It is easy to show that it does not depend on the choice of the sequence $\varphi_n$ and satisfies (3.17). Hence the required extension of the operator of multiplication by $a(\sigma)$ is well defined. Assume that $\mathring{K}\cap N\neq \varnothing$, and let $\psi(\sigma)\in C^{m_r(K)}_0(\mathring{K})$. Consider the function $\varphi_0(\sigma)=\psi(\sigma)\triangle^p(\sigma, N)$. If $p=m_r(K)+1$, then $\partial^\alpha\varphi_0(\sigma)|_N=0$ for $|\alpha|\leqslant m_r(K)$ and $\varphi_0$ belongs to $C^{m_r(K)}_{0N}(\mathring{K})$: this follows from (3.16). We choose $\psi(\sigma)$ so that on a prescribed compact set $K'\Subset \mathring{K}$, $K'\cap N\neq \varnothing$, we have Using the construction of $\varphi_0(\sigma)$ and inequality (3.17), for $\sigma\in K'\setminus N$ we obtain
$$
\begin{equation*}
\begin{aligned} \, |a(\sigma)[\triangle(\sigma, N)]^{m_r(K)+1}| &\leqslant \biggl|\frac{1}{b}a(\sigma)[\triangle(\sigma, N)]^{m_r(K)+1}\psi(\sigma)\biggr| \\ &\leqslant\frac{c_r(K)}{b}\sum_{|\beta|\leqslant m_r(K)}\sup_{\sigma\in K'}|\partial^\alpha\varphi_0(\sigma)|. \end{aligned}
\end{equation*}
\notag
$$
For the compact set $K'$ selected the right-hand side of this inequality is a fixed quantity. Hence
$$
\begin{equation*}
|a(\sigma)|\leqslant \widetilde{C}_0(K')[\triangle(\sigma, N)]^{-m_r(K)-1}\leqslant c'_0(K')[d(\sigma, N)]^{-m_r(K)-1}, \qquad \sigma\in K'\setminus N.
\end{equation*}
\notag
$$
Since $K'^\varepsilon\Subset\Omega$ for sufficiently small $\varepsilon$, the above reasoning means that the inequality obtained holds for each compact set $K'\Subset\Omega$. In a similar way we prove estimates of the form (3.1) for derivatives of $a(\sigma)$, because they are multipliers too. To show this, for any $\varphi\in D(\Omega\setminus N)$ we look at the equality
$$
\begin{equation}
\partial_i a(\sigma)\varphi(\sigma)=\partial_i(a\varphi)(\sigma)-a\,\partial_i\varphi(\sigma)\in D(\Omega\setminus N).
\end{equation}
\tag{3.18}
$$
Since the space $D(\Omega\setminus N)$ is dense in $D^\infty_N(\Omega)$ by Theorem 2.1, for each function $\varphi\in D^\infty_N(\Omega)$ there exists a sequence of functions $\varphi_n\in D(\Omega\setminus N)$ converging to $\varphi$ in $D^\infty_N(\Omega)$. Since $a(\sigma)$ is a multiplier in $D^\infty_N(\Omega)$, the limit
$$
\begin{equation*}
\lim_{n\to+\infty}[\partial_i(a\varphi_n)(\sigma)-a\,\partial_i\varphi_n(\sigma)]
\end{equation*}
\notag
$$
exists in $D^\infty_N(\Omega)$. Then it follows from (3.18) that the limit $\lim_{n\to+\infty}\partial_i a(\sigma)\varphi_n(\sigma)$ also exists in $D^\infty_N(\Omega)$. Hence the operator of multiplication by $\partial_i a(\sigma)$ extends to the whole of $D^\infty_N(\Omega)$ and (3.18) holds for $\varphi\in D^\infty_N(\Omega)$. This means that the operator in question is continuous.
Using induction and following the above scheme we can show that the derivative of $a(\sigma)$ of any order is a multiplier in $D^\infty_N(\Omega)$. Hence inequalities (3.1) hold for all $\alpha\in \mathbb{Z}^n_+$. The proof is complete. Theorem 3.8. A function $a(\sigma)$ is a multiplier in the space $\mathcal{E}^\infty_N(\Omega)$ if and only if $a(\sigma)\in C^\infty(\Omega\setminus N)$ and (3.1) is satisfied. Theorem 3.8 follows from Theorems 3.2 and 3.7 because a multiplier in $\mathcal{E}^\infty_N(\Omega)$ is at the same time a multiplier in $D^\infty_N(\Omega)$. § 4. Constructing regularizations for distributionsConsider the set of distributions in $D'(\mathbb{R}^n\setminus N)$, where $N$ is a closed set, that extend to distributions in $D'(\mathbb{R}^n)$. ‘We denote this set by $D'(\mathbb{R}^n\setminus N, \mathbb{R}^n)$, and let $D'_F(\mathbb{R}^n\setminus N, \mathbb{R}^n)$ denote its subset of elements extending to distributions in $D'_F(\mathbb{R}^n)$. The difference of two extensions of a distribution $f \in D'(\mathbb{R}^n\setminus N)$ is a distribution $h \in D'(\mathbb{R}^n)$ with $\operatorname{supp}h\subset N$. Hence under the embedding of $D'(\mathbb{R}^n)$ in ${D^\infty_N}'(\mathbb{R}^n)$ the images of two extensions coincide. Therefore, for a description of distributions in $D'(\mathbb{R}^n\setminus N)$ that have such an extension it is natural to look at the space ${D^\infty_N}'(\mathbb{R}^n)$. As the embedding
$$
\begin{equation*}
\tau_\infty \colon D(\mathbb{R}^n\setminus N)\to D^\infty_N(\mathbb{R}^n)
\end{equation*}
\notag
$$
is continuous, we have the embeddings
$$
\begin{equation*}
\tau^*_\infty \colon {D^\infty_N}'(\mathbb{R}^n)\to D'(\mathbb{R}^n\setminus N)\quad\text{and} \quad \tau^*_\infty \colon {D^\infty_{NF}}'(\mathbb{R}^n)\to D'_F(\mathbb{R}^n\setminus N).
\end{equation*}
\notag
$$
The next theorem provides a description of the space $\tau^*_\infty({D^\infty_{NF}}'(\mathbb{R}^n))$ in ${D'_F(\mathbb{R}^n\setminus N)}$. Proof. That $\tau^*$ is injective follows from Theorem 2.1.
Let $f \in {D^\infty_{NF}}'(\mathbb{R}^n)$. Then there exists $j \in \mathbb{Z}_+$ such that $f \in {D^\infty_N}'^j(\mathbb{R}^n)$. The composite map $f \circ\tau_\infty$ also belongs to $D'^j(\mathbb{R}^n\setminus N)$. This follows from the inequality
$$
\begin{equation*}
|(f \circ \tau_\infty, \varphi)| =|(f, \varphi)|\leqslant C(K)\sum_{|\alpha|\leqslant j}\sup_{\sigma\in K}|\partial^\alpha\varphi(\sigma)|,
\end{equation*}
\notag
$$
which holds by assumption for each compact set $K\Subset \mathbb{R}^n\setminus N$ and all functions $\varphi\in D(\mathbb{R}^n\setminus N)\subset D^\infty_N(\mathbb{R}^n)$ with $\operatorname{supp} \varphi\subset K$. Hence $\tau^*_\infty({D^\infty_{NF}}'(\mathbb{R}^n))\subset D'_F(\mathbb{R}^n\setminus N)$.
Let $f\in D'_F(\mathbb{R}^n\setminus N, \mathbb{R}^n)$, and let $\widetilde{f}\in D'_F(\mathbb{R}^n)$ be an extension of $f$, so that $(f, \varphi)=(\widetilde{f}, \varphi)$ for $\varphi\in D(\mathbb{R}^n\setminus N)$. Inequalities (2.1) and (2.13) and Theorem 2.1 yield
$$
\begin{equation*}
i^*_\infty(\widetilde{f})\in {D^\infty_{NF}}'(\mathbb{R}^n)\quad\text{and} \quad \tau^*_\infty(i^*_\infty(\widetilde{f}))=f.
\end{equation*}
\notag
$$
Therefore, each element of $D'_F(\mathbb{R}^n\setminus N, \mathbb{R}^n)$ belongs to $\tau^*_\infty({D^\infty_{NF}}'(\mathbb{R}^n))$.
To prove that $\tau^*_\infty(f)\in D'_F(\mathbb{R}^n\setminus N, \mathbb{R}^n)$ for all $f\in {D^\infty_{NF}}'(\mathbb{R}^n)$, for each $j \in \mathbb{Z}_+$ we construct a continuous linear map $F_j$ from $D(\mathbb{R}^n)$ to $C^j_{0N}(\mathbb{R}^n)$ using one of Whiteney’s continuation theorems. It follows from Theorem 17.2 in [4] that for each $j\in \mathbb{Z}_+$ there exists a linear map from $D(\mathbb{R}^n)$ to $C^j(\mathbb{R}^n)$ that to $\varphi\in D(\mathbb{R}^n)$ assigns a function $\widetilde{\varphi}_j\in C^j(\mathbb{R}^n$ such that: 1) $\partial^\alpha\widetilde{\varphi}_j(\sigma)=\partial^\alpha\varphi(\sigma)$, $\sigma\in N$, $|\alpha|\leqslant j$; 2) the inequality
$$
\begin{equation}
\sup_{|\alpha|\leqslant j}|\partial^\alpha\widetilde{\varphi}_j(\sigma)|\leqslant C\sup_{|\alpha|\leqslant j,\,\eta\in N_\sigma} \biggl\{|\partial^\alpha\varphi(\sigma)|, \frac{|\partial^\alpha_\sigma R_j (\sigma, \eta)|}{|\sigma-\eta|^{j-|\alpha|}}\biggr\}
\end{equation}
\tag{4.1}
$$
holds, where $R_j(\sigma,\eta)=\varphi(\sigma)-\sum_{|\alpha|\leqslant j}\varphi^\alpha(\eta)(\sigma-\eta)^\alpha/{\alpha!}$ and $N_\sigma=\{\eta\in N\colon |\sigma-\eta|\leqslant 10 \sqrt{n}\}$.
From inequality (4.1), since $\operatorname{supp}\varphi$ is compact, it follows that $\operatorname{supp}\widetilde{\varphi}_j(\sigma)$ is too. Using Taylor’s formula with Lagrange remainder we can write (4.1) as
$$
\begin{equation}
\sup_{|\alpha|\leqslant j}|\partial^\alpha\widetilde{\varphi}_j(\sigma)|\leqslant C'\sup_{|\alpha|\leqslant j+1,\,\sigma\in\operatorname{supp}\varphi}|\partial^\alpha\varphi(\sigma)|.
\end{equation}
\tag{4.2}
$$
The required map is defined by
$$
\begin{equation*}
F_j(\varphi)\equiv \varphi^j_N(\sigma)=\varphi(\sigma)-\widetilde{\varphi}_j(\sigma).
\end{equation*}
\notag
$$
It follows from the construction of $\varphi^j_N$ that $\varphi^j_N(\sigma)\in C^j_{0N}(\mathbb{R}^n)$ and
$$
\begin{equation}
\sum_{|\alpha|\leqslant j}\sup_{\sigma\in\operatorname{supp}\varphi^j_N}|\partial^\alpha\varphi^j_N(\sigma)|\leqslant C_j\sum_{|\alpha|\leqslant j+1}\sup_{\sigma\in\operatorname{supp}\varphi}|\partial^\alpha\varphi(\sigma)|,
\end{equation}
\tag{4.3}
$$
as a consequence of (4.2).
If $f \in {D^\infty_N}'^j(\mathbb{R}^n)$, then it follows from Lemma (2.8) that for each function ${f\in D(\mathbb{R}^n)}$ the quantity $(f,\varphi^j_N)$ is well defined. Hence we obtain a linear form $[f]_j$ on $D(\mathbb{R}^n)$:
$$
\begin{equation*}
([f]_j, \varphi)=(f, \varphi^j_N), \qquad \varphi\in D(\mathbb{R}^n).
\end{equation*}
\notag
$$
It follows from this equality and (4.3) and (2.13) that for each compact set $K$ and any function $\varphi\in D(\mathbb{R}^n)$ such that $\operatorname{supp}\varphi\subset K$ we have the inequalities
$$
\begin{equation}
\begin{aligned} \, \notag |([f]_j, \varphi)| &=|(f, \varphi^j_N)|\leqslant C'(K)\sum_{|\alpha|\leqslant j}\sup_{\sigma\in K}|\partial^\alpha\varphi^j_N(\sigma)| \\ &\leqslant C(K)\sum_{|\alpha|\leqslant j+1}\sup_{\sigma\in K}|\partial^\alpha\varphi(\sigma)|. \end{aligned}
\end{equation}
\tag{4.4}
$$
Therefore, $[f]_j\in D'^{j+1}(\mathbb{R}^n)$ because the value of the form is independent of $K$.
For an arbitrary function $\varphi\in D(\mathbb{R}^n\setminus N)$ we have $\varphi^j_N = \varphi$. Therefore, $([f]_j, \varphi)=(f,\varphi)$ for $\varphi\in D(\mathbb{R}^n\setminus N)$, that is, $[f]_j$ is an extension of the distribution $\tau^*_\infty(f)\in D'_F(\mathbb{R}^n\setminus N)$. Thus we have proved that $\tau^*_\infty({D^\infty_{NF}}'(\mathbb{R}^n))=D'_F(\mathbb{R}^n\setminus N, \mathbb{R}^n)$. The proof is complete. We call the distribution $[f]_j\in D'_F(\mathbb{R}^n)$ the regularization of the distribution $\tau^*_\infty(f)\in D_F(\mathbb{R}^n\setminus N)$. A similar result holds for the space $\mathcal{E}'(\mathbb{R}^n\setminus N)$. Theorem 4.2. The injective linear map $\tau^*_\infty$ takes ${\mathcal{E}^\infty_N}'(\mathbb{R}^n)$ to the subspace of $D'_F(\mathbb{R}^n\setminus N)$ consisting of the distributions extending to elements of $\mathcal{E}'(\mathbb{R}^n)$. Proof. Since ${\mathcal{E}^\infty_N}'(\mathbb{R}^n)$ is a subspace of ${D^\infty_{NF}}'(\mathbb{R}^n)$, it follows from Theorem 4.1 that
We show that elements of the subspace $\tau^*_\infty({\mathcal{E}^\infty_N}'(\mathbb{R}^n))$ have extensions belonging to $\mathcal{E}'(\mathbb{R}^n)$. If $f\in {\mathcal{E}^\infty_N}'(\mathbb{R}^n)$, then there exist a compact set $K$ and $j\in \mathbb{Z}_+$ such that $f\in {\mathcal{E}^\infty_N}'^j(K)$, and for each function $\varphi\in \mathcal{E}^\infty_N(\mathbb{R}^n)$ we have
$$
\begin{equation}
|(f,\varphi)|\leqslant C(K)\sum_{|\alpha|\leqslant j}\sup_{\sigma\in K}|\partial^\alpha\varphi(\sigma)|.
\end{equation}
\tag{4.5}
$$
Using this inequality we can extend the distribution $f$ to a map from $C^j_N(\mathbb{R}^n)$ to $\mathbb{R}$ preserving inequality (4.5). For an arbitrary function $\varphi\in C^j_N(\mathbb{R}^n)$ consider the function $\psi_\varepsilon=\chi_\varepsilon\varphi$, where $\chi_\varepsilon(\sigma)\in C^\infty_0(\mathbb{R}^n)$ and $\chi_\varepsilon(\sigma)=1$ for $\sigma\in K^\varepsilon$. It follows from Lemma 2.8 that there exists a sequence $\varphi_n(\sigma)\in D(\Omega\setminus N)$ converging to $\psi_\varepsilon$ in $C^j_N(\mathbb{R}^n)$ as $n\to +\infty$. Hence $\lim_{n\to +\infty}\sup_{\sigma\in K^\varepsilon}|\partial^\alpha\varphi_n(\sigma)-\partial^\alpha\varphi(\sigma)|=0$ for $|\alpha|\leqslant j$. It follows from (4.5) that $(f,\varphi_n)$ is a Cauchy sequence, and its limit is independent of the choice of the sequence $\varphi_n$ converging to $\psi_\varepsilon$. Hence to each function $\psi_\varepsilon$ there corresponds the number $(f,\psi_\varepsilon)$. Since $\operatorname{supp}f\subset K$, this number is independent of $\varepsilon$. Thus, on $C^j_N(\mathbb{R}^n)$ we obtain a linear form satisfying inequality (4.5) for each $\varphi\in C^j_N(\mathbb{R}^n)$. Similarly to the proof of Theorem 4.1, for $j\in \mathbb{Z}_+$ we construct a linear map $H_j$ from $\mathcal{E}(\mathbb{R}^n)$ to $C^j_N(\mathbb{R}^n)$. It follows from Theorem 17.2 in [4] that for each $j\in \mathbb{Z}_+$ there exists a linear map from $\mathcal{E}(\mathbb{R}^n)$ to the space $C^j(\mathbb{R}^n)$ that to a function $\varphi\in \mathcal{E}(\mathbb{R}^n)$ assigns a function $\widetilde{\varphi}_j\in C^j(\mathbb{R}^n)$ with properties indicated in the proof of Theorem 4.1. The required map $H_j$ is defined by
$$
\begin{equation*}
H_j(\varphi)\equiv \varphi^j_N(\sigma)=\varphi(\sigma)-\widetilde{\varphi}_j(\sigma).
\end{equation*}
\notag
$$
It follows from the construction of $\varphi^j_N(\sigma)$ that $\varphi^j_N\in C^j_N(\mathbb{R}^n)$ and
$$
\begin{equation}
\sum_{|\alpha|\leqslant j}\sup_{\sigma\in K}|\partial^\alpha\varphi^j_N(\sigma)|\leqslant C'_j(K)\sum_{|\alpha|\leqslant j+1}\sup_{\sigma\in K'}|\partial^\alpha\varphi(\sigma)|,
\end{equation}
\tag{4.6}
$$
where $K'$ is a compact set constructed from $K$.
Inequality (4.6) follows from (4.1). Using in it Taylor’s formula with Lagrange remainder we obtain
$$
\begin{equation*}
\sum_{|\alpha|\leqslant j}\sup_{\sigma\in K}|\partial^\alpha\widetilde{\varphi}_j(\sigma)|\leqslant C''_j(K)\sum_{|\alpha|\leqslant j+1}\sup_{\sigma\in K'}|\partial^\alpha\varphi(\sigma)|,
\end{equation*}
\notag
$$
where $K'=\{\xi\in \mathbb{R}^n\colon \xi=\sigma+t(\eta-\sigma),\, \sigma\in K, \, 0\leqslant t\leqslant 1, \, \eta\in N, \, (\sigma-\eta)\leqslant 10\sqrt{n}\}$.
Using this inequality in estimate for $|\partial^\alpha\varphi^j_N(\sigma)|$ we arrive at (4.6). Thus we can define a linear form $[f]_j$ on $\mathcal{E}(\mathbb{R}^n)$:
$$
\begin{equation*}
([f]_j,\varphi)=(f,\varphi^j_N), \qquad \varphi\in \mathcal{E}(\mathbb{R}^n).
\end{equation*}
\notag
$$
It follows from this equality, (4.5) and (4.6) that for each function $\varphi\in \mathcal{E}(\mathbb{R}^n)$ we have the inequalities
$$
\begin{equation*}
|([f]_j, \varphi)|=|(f,\varphi^j_N)|\leqslant C_j(K)\sum_{|\alpha|\leqslant j}\sup_{\sigma\in K}|\partial^\alpha\varphi^j_N(\sigma)|\leqslant C_j(K)\sum_{|\alpha|\leqslant j+1}\sup_{\sigma\in K'}|\partial^\alpha\varphi(\sigma)|,
\end{equation*}
\notag
$$
where $K'$ is a compact set constructed from $K$.
Therefore, $[f]_j\in \mathcal{E}'^{j+1}(K')$ and the proof is complete. We use the results obtained to construct a regularization for the distribution $a(\sigma)f$, where $f\in D'(\mathbb{R}^n)$ and $a(\sigma)\in C^\infty(\mathbb{R}^n\setminus N)$. Theorem 4.3. If $a(\sigma)\in C^\infty(\mathbb{R}^n\setminus N)$ is a function satisfying (3.1), where condition (3.12) holds, then there exists a regularization $[af]$ of the distribution $a(\sigma)f\in D'(\mathbb{R}^n\setminus N)$ for $f\in D'_F(\mathbb{R}^n)$ that belongs to $D'_F(\mathbb{R}^n)$. Proof. Since $D'_F(\mathbb{R}^n)\!\subset\! {D^\infty_{NF}}'(\mathbb{R}^n)$, it follows from Theorem 3.5 that $af\!\in\! {D^\infty_{NF}}'(\mathbb{R}^n)$. Therefore, $\tau^*_\infty(af)=a(\sigma)f \in D'(\mathbb{R}^n\setminus N)$ extends to a distribution in the space $D'_F(\mathbb{R}^n)$ by Theorem 4.1.
If $f \in D'^j(\mathbb{R}^n)$, $j \in \mathbb{Z}_+$, then this extension can for any $\varphi\in D(\mathbb{R}^n)$ be defined by where $\varphi^{j+q_j}_N\in C^{j+q_j}_{0N}(\mathbb{R}^n)$ is the function constructed from $\varphi$ in the proof of Theorem 4.1.From (4.4) and (3.7), for each compact set $K$ and any function $\varphi\in D(\mathbb{R}^n)$ such that $\operatorname{supp}\varphi\subset K$ we obtain Therefore, $[af]_j\in D'^{j+q_j+1}(\mathbb{R}^n)$. The proof is complete.Since two regularizations of a distribution in $D'(\mathbb{R}^n\setminus N)$ differ by a distribution with support in $N$, Theorem 4.3 ensures the following result. Corollary 4.1. If a function in $C^\infty(\mathbb{R}^n\setminus N)$ satisfies inequality (3.1), where condition (3.12) holds, then for each $f\in D'^j(\mathbb{R}^n)$ the distribution $a(\sigma)f\in D'(\mathbb{R}^n\setminus N)$ has a regularization $[af]_j\in D'^{j+q_j+1}(\mathbb{R}^n)$, and the set of its regularizations has the form A result similar to Theorem 4.3 holds in the space $\mathcal{E}'(\mathbb{R}^n)$. Theorem 4.4. If a function $a(\sigma)\in C^\infty(\mathbb{R}^n\setminus N)$ satisfies (3.1) for all $\alpha\in \mathbb{Z}^n_+$, $|\alpha|\leqslant j$, then for each distribution $f \in \mathcal{E}'^j(K)$, where $K$ is a compact set such that $K\cap N\neq \varnothing$, the distribution $a(\sigma)f \in D'(\mathbb{R}^n\setminus N)$ has a regularization $[af]_j$ in the space $\mathcal{E}'^{j+q_j+1}(K')$, where $K'$ is a compact set depending on $K$, and where $C_j(K)>0$ is a number depending on $K$. Proof. Since $f\in \mathcal{E}'^j(K)$, for each function $\varphi\in \mathcal{E}(\mathbb{R}^n)$ we have
$$
\begin{equation}
|(f,\varphi)|\leqslant\|f\|_{\mathcal{E}'^j(K)}\sup_{|\alpha|\leqslant j,\,\sigma\in K}|\partial^\alpha\varphi(\sigma)|.
\end{equation}
\tag{4.8}
$$
It follows from (3.11) that $af\in {\mathcal{E}^\infty_N}'^{j+q_j}(K')$, where $K'$ is a certain compact set constructed from $K$. Hence by Theorem 4.2, $\tau^*_\infty(af)\in \mathcal{E}'(\mathbb{R}^n\setminus N)$ has an extension to a distribution in $\mathcal{E}'(\mathbb{R}^n)$. This extension $[af]_j$ is defined by
$$
\begin{equation}
([af]_j,\varphi)=(f, a\varphi^{j+q_j}_N), \qquad \varphi\in \mathcal{E}(\mathbb{R}^n),
\end{equation}
\tag{4.9}
$$
where $\varphi^{j+q_j}_N\in C^{j+q_j}_N(\mathbb{R}^n)$ is the function from the proof of Theorem 4.2.
In (4.9) we use inequalities (4.7), (3.9) and (4.2); then we obtain
$$
\begin{equation*}
\begin{aligned} \, |([af]_j,\varphi)| &\leqslant \|f\|_{\mathcal{E}'^j(K)}\sup_{|\alpha|\leqslant j,\, \sigma\in K}|\partial^\alpha(a\varphi^{j+q_j}_N)(\sigma)| \\ &\leqslant C'_j(K)\|f\|_{\mathcal{E}'^j(K)}\sum_{|\alpha|\leqslant j+q_j}\sup_{\sigma\in K''}|\partial^\alpha\varphi^{j+q_j}_N(\sigma)| \\ & \leqslant C''_j(K)\|f\|_{\mathcal{E}'^j(K)}\sum_{|\alpha|\leqslant j+q_j+1}\sup_{\sigma\in K'}|\partial^\alpha\varphi(\sigma)|. \end{aligned}
\end{equation*}
\notag
$$
Hence $[af]_j\in \mathcal{E}'^{j+q_j+1}(K')$ and (4.7) holds: The proof is complete. Applying Theorem 4.4 to $a(\sigma)={1}/{P(\sigma)}$, where $P(\sigma)$ is a polynomial, we obtain the following. Corollary 4.2. For each distribution $f \in \mathcal{E}'^p(K)$, where $K$ is a compact set ${K\cap N\neq \varnothing}$, the distribution $({1}/{P(\sigma)})f \in D'(\mathbb{R}^n\setminus N)$ has a regularization $[({1}/{P(\sigma)})f]_p$ in the space $\mathcal{E}'^{\varkappa_p}(K')$, where $K'$ is a compact set depending on $K$ and $\varkappa_p=[p(1+ |\varkappa|]+1$, where $\varkappa$ is the exponent in (3.14), and the inequality holds, where $C_p(K)$ is a number depending on $K$. This follows directly from Theorem 4.4 and inequalities (3.14) and (3.15). § 5. Using regularizations of distributions in the division problemIn many problems in analysis we need to multiply a distribution by a function whose derivatives have singularities. The problem of division of a distribution by a polynomial is one of them. Consider the equation where $g\in D'(\mathbb{R}^n)$ and $P(\sigma)$ is a polynomial. Let $N$ denote the set of its real zeros.To construct a solution of (5.1) we use the ‘localization’ property of this equation: we can glue together solutions defined in neighbourhood of single points using a partition of unity. In a neighbourhood of a point at which $P(\sigma) \!\neq\! 0$ the solution of (5.1) is $(1/{P(\sigma)})f$. We construct a solution of (5.1) in a neighbourhood of an arbitrary nonsingular point $\sigma_0$ of $N$. Let $\sigma_0\in N$ be a point with a neighbourhood $U(\sigma_0)$ such that
$$
\begin{equation}
U(\sigma_0)\cap N=\biggl\{\sigma\in U(\sigma_0)\colon \partial^\alpha P(\sigma)=0,\, |\alpha|\leqslant k-1, \, \sum_{|\alpha|=k}|\partial^\alpha P(\sigma_0)|\neq 0 \biggr\}.
\end{equation}
\tag{5.2}
$$
Theorem 5.1. If $\sigma_0\in N$ has a neighbourhood $U(\sigma_0)$ satisfying (5.2) and the set $U(\sigma_0)\cap N$ is a smooth manifold, then this point has a neighbourhood $U'(\sigma_0)$ such that the equation Proof. Let $U$ be a neighbourhood of the point $\sigma_0 \in N$, whose selection will be specified in what follows.
We seek a solution of (5.3) in the form where $[({1}/{P(\sigma)})f]$ is a regularization of the distribution $({1}/{P(\sigma)})f\in D'(U \setminus N)$.That this regularization exists was shown in Corollary 4.2. By construction $[({1}/{P(\sigma)}) f ] \in \mathcal{E}'^{\varkappa_p}(K')$, where $K'$ is a compact set constructed from $K$. Inequality (4.10) means that $[({1}/{P(\sigma)})f]$ depends continuously on $f$. It order that $u$ in (5.5) be a solution of (5.3), it is necessary and sufficient that the distribution $v$ solve the equation
$$
\begin{equation}
P(\sigma) v=g \equiv f - P(\sigma) \biggl[ \frac{1}{P(\sigma)} f \biggr].
\end{equation}
\tag{5.6}
$$
It follows from the construction of a regularization that
$$
\begin{equation*}
P(\sigma) \biggl[ \frac{1}{P(\sigma)} f\biggr] =P(\sigma) \frac{1}{P(\sigma)}f=f, \qquad \sigma \in U \setminus N.
\end{equation*}
\notag
$$
Hence $\operatorname{supp}g \subset K'\cap N$. Equality (5.6) implies that $g \in \mathcal{E}'^{\varkappa_p}(K'\cap N)$. The problem of the solvability of (5.3) is reduced to the solvability of the equation
$$
\begin{equation}
P(\sigma)v=h \quad\text{for } h \in \mathcal{E}'^s(K'\cap N), \qquad K' \Subset U.
\end{equation}
\tag{5.7}
$$
It follows from the hypotheses that there exists an analytic diffeomorphism $\psi$ of a neighbourhood $U'$ of $\sigma_0$ onto a domain $V \subset R^n$ such that
$$
\begin{equation*}
\psi(U'\cap N)=V\cap \bigl\{ (\eta', \eta'') \in R^{n_1+n_2}\colon \eta''=0\bigr\}\equiv W.
\end{equation*}
\notag
$$
We assume without loss of generality that $U'=U$ and $\psi(\sigma_0)=0$.
After ‘straightening’ $N$ in $U$, equation (5.7) takes the form
$$
\begin{equation}
b(\eta)\widetilde{v}=\widetilde{h} \quad\text{for } \widetilde{h} \in \mathcal{E}'^s(\widetilde{K}), \qquad \widetilde{K} \Subset W,
\end{equation}
\tag{5.8}
$$
where $b(\eta)=P(\psi^{-1}(\eta))$, $b(\eta',0)=0$, $(\eta',0) \in W$ and $\widetilde{K}=\psi(K'\cap N)$.
Since $b(\eta)$ is an analytic function, it can be represented by a series in a neighbourhood of the point $\eta''=0$:
$$
\begin{equation*}
b(\eta)=\sum ^{\infty}_{|\alpha|=0}b_\alpha(\eta') \frac{\eta''^\alpha}{\alpha!}, \qquad \alpha\in Z^{n_2}_+,
\end{equation*}
\notag
$$
where the $b_\alpha(\eta')$ are analytic functions.
By the hypotheses of the lemma $b_\alpha(\eta')=\partial^\alpha_{\eta''}b(\eta', 0)$, for $(\eta', 0)\in W$ and any $\alpha$ such that $|\alpha|\leqslant k-1$ we have $b_\alpha(\eta')=0$ and there exists $\gamma_0\in Z^n_+$, $|\gamma_0|=k$, such that $b_{\gamma_0}(0)\neq 0$. We assume that $b_{\gamma_0}(\eta')\neq 0$ for all $(\eta',0)\in W$. This can be ensured by the choice of $U$, by taking its smaller if necessary. It follows from the description of distributions with compact support concentrated on a linear submanifold (see [12], Theorem 2.3.5) that
$$
\begin{equation*}
\widetilde{h}=\sum_{|\alpha|\leqslant q} h_\alpha\otimes \delta^\alpha_{\eta''}, \quad\text{where } h_\alpha \in \mathcal{E}'^{s-|\alpha|}(\widetilde{\widetilde{K}}), \qquad q\leqslant s,\quad\text{and} \quad \widetilde{\widetilde{K}} \times \{\eta''=0\}=\widetilde{K}.
\end{equation*}
\notag
$$
We seek the solution of (5.8) in the same form
$$
\begin{equation*}
\widetilde{v}=\sum_{|\alpha|\leqslant r} v_\alpha\otimes \delta^\alpha_{\eta''}, \qquad v_\alpha \in \mathcal{E}'^{s-|\alpha|}(\widetilde{\widetilde{K}}), \qquad r\geqslant q.
\end{equation*}
\notag
$$
For each function $\varphi \in C^\infty(V)$ we must have
$$
\begin{equation*}
\sum_{|\alpha|\leqslant r}(v_\alpha,(-1)^{|\alpha|}\, \partial^\alpha_{\eta''}(b\varphi)(\eta',0) )=\sum_{|\beta|\leqslant q}(h_\beta,(-1)^{|\beta|}\, \partial^\beta_{\eta''} \varphi(\eta',0) ).
\end{equation*}
\notag
$$
Using Leibniz’s formula, we can write this equality as
$$
\begin{equation*}
\sum_{|\alpha|\leqslant r}(\omega_\alpha,(-1)^{|\alpha|}\, \partial^\alpha_{\eta''}\varphi(\eta',0) )=\sum_{|\alpha|\leqslant q}(h_\alpha,(-1)^{|\alpha|}\, \partial^\alpha_{\eta''} \varphi(\eta',0) ),
\end{equation*}
\notag
$$
where
$$
\begin{equation*}
\omega_\alpha=\sum_{\beta\leqslant r}(-1)^{|\beta|-|\alpha|}\varepsilon_{\beta-\alpha}\, \partial^{\beta-\alpha}_{\eta''}b(\eta',0)v_\beta
\end{equation*}
\notag
$$
and
$$
\begin{equation*}
\varepsilon_{\beta-\alpha}=\begin{cases} \dfrac{\beta!}{\alpha!\,(\beta-\alpha)!}\quad &\text{for } \beta-\alpha \in Z^{n_2}_+, \\ 0 &\text{for }\beta-\alpha \not\in Z^{n_2}_+. \end{cases}
\end{equation*}
\notag
$$
Since $\varphi$ is arbitrary, this is equivalent to the system of equalities
$$
\begin{equation*}
\sum_{|\beta| \leqslant r}(-1)^{|\beta|-|\alpha|}\varepsilon_{\beta-\alpha}\,\partial^{\beta-\alpha}_{\eta''} b(\eta',0)v_\beta=\widehat{h}_\alpha, \qquad |\alpha|\leqslant r,
\end{equation*}
\notag
$$
where $\widehat{h}_\alpha=0$ for $|\alpha|>q$ and $\widehat{h}_\alpha=h$ for $|\alpha|\leqslant q$.
We write these equalities by setting $\beta-\alpha=\gamma$ and denoting the derivative $(-1)^{|\gamma|}\, \partial^\gamma_{\eta''} b (\eta',0)\equiv (-1)^{|\gamma|} b_\gamma (\eta')$ by $b'_\gamma(\eta')$:
$$
\begin{equation}
\sum_{|\gamma|\leqslant r-|\alpha|}\binom{\gamma+\alpha}{\alpha} b'_\gamma(\eta')v_{\gamma+\alpha}=\widehat{h}_\alpha, \qquad |\alpha|\leqslant r.
\end{equation}
\tag{5.9}
$$
That (5.8) has a solution of the form indicated is equivalent to the solvability of the system of linear equations (5.9). To describe the scheme of solution of (5.9) we define the following order relation on the set of multi-indices:
$$
\begin{equation*}
\alpha>\beta \quad \text{if } |\alpha|>|\beta|, \qquad \alpha_i=\beta_i \quad\text{for } 1\leqslant i \leqslant m, \quad\text{and}\quad \alpha_m>\beta_m.
\end{equation*}
\notag
$$
We show that (5.9) has a solution of the required form for $r=q+k$. The coefficients in equations in (5.9) that correspond to multi-indices $\alpha$ satisfying $q<|\alpha|\leqslant r$ are identically equal to zero because $|\gamma|\leqslant r-|\alpha|<k$. Therefore, the equations in (5.9) corresponding to these multi-indices are identities. Next consider the equations in (5.9) corresponding to multi-indices of order $q$:
$$
\begin{equation}
\sum_{|\gamma|=k}\binom{\gamma+\alpha}{\alpha} b'_\pi(\eta')v_{\gamma+\alpha}=h_\alpha, \qquad |\alpha|=q.
\end{equation}
\tag{5.10}
$$
We arrange the equations in (5.10) and the terms in these equation in the ‘decreasing’ order of multi-indices. By assumption $b_\gamma(\eta')\equiv 0$ for all $\gamma$ such that $|\gamma|< k$. We can assume without loss of generality that $\gamma_0=(k,0,\dots, 0)$ and $b_{\gamma_0}(\eta')\neq 0$.
Consider the scheme of solution of (5.10). 1) In the last equation we set all unknowns with indices $\gamma<\gamma_0+(0,\dots, 0, q)=(k,0,\dots,0,q)$ equal to zero and determine $v_{(k,0,\dots,0,q)}$ from the equation
$$
\begin{equation*}
v_{(k,0,\dots,0,q)}=\frac{1}{\binom{(k,0,\dots,0,q)}{(0,\dots,0,q)}} \,\frac{}{b'^{(\gamma')}_{(k,0,\dots,0)}}h_{(0,\dots,0,q)}.
\end{equation*}
\notag
$$
2) Into the next to the last equation we substitute the values of the unknowns found at the previous step, set the other unknowns with indices $\gamma<(k,0,\dots,0, 1,q-1)$ equal to zero and determine $v_{(k,0,\dots,0,1,q-1)}$ from the equation
$$
\begin{equation*}
v_{(k,0,\dots,0,1,q-1)} =\frac{1}{\binom{(k,0,\dots,0,1,q-1)}{(0,\dots,0,1,q-1)}}\, \frac{}{b'^{(\gamma')}_{(k,0,\dots,0)}}\widehat{h}_{(0,\dots,0,1,q-1)},
\end{equation*}
\notag
$$
where $\widehat{h}_{(0,\dots,0,1,q-1)}$ is different from $h_{(0,\dots,0,1,q-1)}$ by terms of the form $b'_\gamma(\eta')v_{\gamma+\alpha}$, which we know already.
3) Going over to the next equation is similar to going over from the last equation to the next to the last one. That this scheme can actually be used for the solution of (5.10) follows from the fact that an unknown $v_{\gamma_0+\alpha_0}$ defined from the equation corresponding to a multi-index $\alpha_0$ by means of division by $b'_{\gamma_o}(\eta')$ could not be determined before in this way because there is a unique index with the prescribed properties of the summands. The unknowns $v_{\gamma'+\alpha'}$ set equal to zero at the previous step cannot be found at the current step by means of division by functions $b'_\gamma(\sigma)$. For let $\gamma_0+\alpha_0=\gamma'+\alpha'$. If $v_{\gamma'+\alpha'}\equiv 0$ and $\gamma'<\gamma_0$, then $\alpha'>\alpha_0$, so that the equation corresponding to the multi-index $\alpha'$ was not involved in the previous step. Using the above scheme we find the values of part of the unknowns. Setting the values of the rest to zero we obtain a solution of (5.10). Now consider the equations in (5.9) corresponding to multi-indices of order $q-1$:
$$
\begin{equation*}
\sum_{k\leqslant|\gamma|\leqslant k+1}\binom{\alpha+\gamma}{\alpha}b'_\gamma(\eta')v_{\gamma+\alpha}=h_\alpha, \qquad |\alpha|=q-1.
\end{equation*}
\notag
$$
Substituting here the values of unknowns found from the equations corresponding to multi-indices of order $q+k$ we obtain a system similar to (5.10):
$$
\begin{equation}
\sum_{|\gamma|=k}\binom{\alpha+\gamma}{\alpha}b'_\gamma(\eta')v_{\gamma+\alpha}=\widehat{h}_\alpha, \qquad |\alpha|=q-1,
\end{equation}
\tag{5.11}
$$
where $\widehat{h}_\alpha=h_\alpha-\sum_{|\gamma|=k+1}\binom{\alpha+\gamma}{\alpha}b'_\gamma(\eta')v_{\gamma+\alpha}$.
Using the above scheme we find a solution of (5.11). Continuing this procedure for the systems of equations corresponding to multi-indices of orders $q-2, q-3, \dots$, we obtain a solution of (5.9). In finding a solution of (5.9) we have performed linear operations with terms of equations, multiplied by analytic functions and also divided by the analytic function $b'_{\gamma_0}(\eta')\neq 0$, $(\eta',0)\in W$. Hence the unknowns to be determined in (5.9) belong to the spaces $\mathcal{E}'^p(\widetilde{\widetilde{K}})$, where $p$ depends on $s$ and the order of the equation used to find the unknown in question. By construction $h_\alpha\in \mathcal{E}'^{s-|\alpha|}(\widetilde{\widetilde{K}})$. It follows from the above scheme that the distributions $v_\alpha$ are linear combinations of the elements $h_\beta$ for $|\beta|\geqslant |\alpha|-k$, with coefficients that are analytic functions. Therefore, $v_\alpha\in \mathcal{E}'^{s-|\alpha|+k}(\widetilde{\widetilde{K}})$ and we have the inequality
$$
\begin{equation}
\|v_\alpha\|_{\mathcal{E}'^{s-|\alpha|+k}(\widetilde{\widetilde{K}})}\leqslant C_\alpha(\widetilde{\widetilde{K}})\sum^q_{|\beta|\geqslant |\alpha|-k}\|h_\beta\|_{\mathcal{E}'^{s-|\beta|}(\widetilde{\widetilde{K}})}.
\end{equation}
\tag{5.12}
$$
If
$$
\begin{equation*}
\varphi(\eta')\in C^\infty(\mathbb{R}^{n_1}), \quad \psi_\alpha(\eta'')\in C^\infty(\mathbb{R}^{n_2})\quad\text{and} \quad \partial^\beta_{\eta''}\psi_\alpha(0)=\delta_{\alpha\beta}=\begin{cases} 1, &\alpha=\beta, \\ 0, &\alpha\neq\beta, \end{cases}
\end{equation*}
\notag
$$
then
$$
\begin{equation*}
\begin{aligned} \, |(\upsilon_\alpha,\varphi)| &=\biggl|\sum^r_{|\beta|=0}(\upsilon_\beta,\varphi)\,\partial^\beta_{\eta''}\psi_\alpha(0)\biggr| =|(\widetilde{\upsilon},\varphi\psi_\alpha)| \\ &\leqslant\|\widetilde{\upsilon}\|_{\mathcal{E}'^{s+k}(\widetilde{K})}\sup_{\substack {\eta\in \widetilde{K} \\ |\nu|\leqslant s+k}}|\partial^\nu_\eta(\varphi\psi_\alpha)(\eta)| \\ &\leqslant C'_\alpha(\widetilde{K})\|\widetilde{\upsilon}\|_{\mathcal{E}'^{s+k}(\widetilde{K})} \sup_{\substack{\eta\in \widetilde{K} \\ |\nu|\leqslant s+k}}\sum_{\gamma<v}|\partial^{\nu-\gamma}_\eta\varphi(\eta')|\, |\partial^\gamma_\eta\psi_\alpha(\eta'')| \\ &\leqslant C''_\alpha(\widetilde{K})\|\widetilde{\upsilon}\|_{\mathcal{E}'^{s+k}(\widetilde{K})} \sup_{\substack{ \eta\in \widetilde{K} \\ |\beta|\leqslant s+k-|\alpha|}}\sum|\partial^\beta\varphi(\eta')|, \end{aligned}
\end{equation*}
\notag
$$
because by construction
$$
\begin{equation*}
|\partial^\gamma_\eta\psi_\alpha(a)|=\partial^{\gamma_1}_{\eta'} \partial^{\gamma_2}_{\eta''}\psi_\alpha(0)= \begin{cases} 0 & \text{if }\gamma_1\neq 0\text{ or }\gamma_2\neq\alpha, \\ 1 & \text{if } \gamma_1=0 \text{ and } \gamma_2=\alpha. \end{cases}
\end{equation*}
\notag
$$
Hence
$$
\begin{equation}
\|\upsilon_\alpha\|_{\mathcal{E}'^{s+k-|\alpha|}(\widetilde{\widetilde{K}})}\leqslant C_\alpha(\widetilde{K})\|\widetilde{\upsilon}\|_{\mathcal{E}'^{s+k}(\widetilde{K})}.
\end{equation}
\tag{5.13}
$$
In a similar way we can show that
$$
\begin{equation}
\|h_\beta\|_{\mathcal{E}'^{s-|\beta|}(\widetilde{\widetilde{K}})}\leqslant \widetilde{C}_\beta(\widetilde{K})\|\widetilde{h}\|_{\mathcal{E}'^s(\widetilde{K})}.
\end{equation}
\tag{5.14}
$$
It follows from (5.12), (5.14) and the structure of $\widetilde{\upsilon}$ that
$$
\begin{equation}
\|\widetilde{\upsilon}\|_{\mathcal{E}'^{s+k}(\widetilde{K})}\leqslant C_1\sum^r_{|\alpha|=0}\|\upsilon_\alpha\|_{\mathcal{E}'^{s+k-|\alpha|}(\widetilde{\widetilde{K}})}\leqslant C_2\|\widetilde{h}\|_{\mathcal{E}'^s(\widetilde{K})}.
\end{equation}
\tag{5.15}
$$
Inequalities (5.15) show that the solution $\widetilde{\upsilon}$ of (5.8) depends continuously on $\widetilde{h}$.
The solvability of (5.7) is equivalent to that of (5.8). It follows from the proof of Theorem 6.1.2 in [7] that the diffeomorphism $\psi\colon U\to V$ we have used induces the isomorphism
$$
\begin{equation*}
\psi*\colon \mathcal{E}'^s(\widetilde{K})\to\mathcal{E}'^s(K).
\end{equation*}
\notag
$$
Therefore, inequalities of the form (5.15) hold for the solution of (5.7):
$$
\begin{equation}
\|v\|_{{\mathcal{E}'^{s+k}(K)}}\leqslant C \|h\|_{\mathcal{E}'^s(K)}.
\end{equation}
\tag{5.16}
$$
Using this result in the solution of (5.6) we obtain a solution of (5.3) of the form (5.5). Inequalities (4.10) and (5.16) imply that (5.4) holds. Theorem 5.1 is proved. Using Theorem 5.1 on the local solvability of (5.1) in a neighbourhood of a smooth point of the manifold of real zeros $N$ of $P(\sigma)$ we can construct a global solution of this equation using a partition of unity in the case when $N$ has no singular points. Theorem 5.2. If $N$ is a smooth manifold and all points in $N$ satisfy (5.2), then for each distribution $f\in D'^p(\mathbb{R}^n)$ equation (5.1) has a solution in the space $D'^{\varkappa_p}(\mathbb{R}^n)$, where $\varkappa_p=[p(1+|\varkappa|)]+1$ and $\varkappa$ is the exponent in (3.14). Proof. By Theorem 5.1 each point $\sigma \in N\subset \mathbb{R}^n$ has a neighbourhood in which the ‘localized’ equation (5.1) is solvable. The system of these neighbourhoods and the set $\mathbb{R}^n\setminus N$ make up an open cover of $\mathbb{R}^n$. Repeating the reasoning in [14], § 1.2, we select a locally finite subcover and construct a partition of unity $\{(\varphi_i,U_i), i \in N\}$ subordinated to this cover so that the compact sets $K_i=\operatorname{supp}\varphi_i$ satisfy $K'_i=K_i$.
In each open set $U_i$ consider the equation It follows from Lemma 2.3 that $\varphi_if \in \mathcal{E}'^p(K_i)$.If $U_i\cap N\neq \varnothing$, then by Theorem 5.1 equation (5.17) has a solution $u_i \in \mathcal{E}'^{\varkappa_p}(K'_i)$ satisfying
$$
\begin{equation}
\|u_i\|_{\mathcal{E}'^{\varkappa_p}(K_i)}\leqslant C_i(K_i)\|\varphi_if\|_{\mathcal{E}'^p(K_i)}.
\end{equation}
\tag{5.18}
$$
If $U_i\cap N=\varnothing$, then $u_i=({1}/{P(\sigma)})\varphi_if \in \mathcal{E}'^p(K_i)$ and inequalities (3.13) yield the estimate
$$
\begin{equation}
\|u_i\|_{\mathcal{E}'^p(K_i)}\leqslant C'_i(K_i)\|\varphi_if\|_{\mathcal{E}'^p(K)}.
\end{equation}
\tag{5.19}
$$
Consider the distribution For each compact set $K$ only a finite number of terms in (5.20) are distinct from zero on $K$, and therefore the linear form $u$ is well defined and for each function $\varphi \in D(\mathbb{R}^n)$, $\operatorname{supp}\varphi\subset K$, we have where the number $k$ depends on $K$.From (5.21) and inequalities (5.18) and (5.19) we obtain
$$
\begin{equation*}
\begin{aligned} \, |(u,\varphi)| &\leqslant \sum^{k}_{j=1}|(u_{i_j},\varphi)|\leqslant C_1\sum^{k}_{j=1}\|u_{i_j}\|_{\mathcal{E}'^{\varkappa_p}(K_{i_j})}\sup_{\sigma \in K'_{ij}|\alpha|\leqslant \varkappa_p}|\partial^\alpha\varphi(\sigma)| \\ &\leqslant C_2\sum^{k}_{j=1}\|\varphi_{i_j}f\|_{\mathcal{E}'^p(K_{i_j})}\sup_{\sigma \in K|\alpha|\leqq \varkappa_p}|\partial^\alpha\varphi(\sigma)|. \end{aligned}
\end{equation*}
\notag
$$
Hence $u$ is a distribution in the space $D'^{\varkappa_p}(\mathbb{R}^n)$. The distribution $u$ is a solution of (5.1): The proof is complete.
Citation in format AMSBIB
\Bibitem{Pav23} |
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