June 04, 2026
We prove the existence of a bounded Sobolev extension operator \(E:W^{p,1}\left( B,P \right) \rightarrow W^{p,1}\left( \ell^{2} ,P \right)\) using a completely new method, where \(B\subset \ell^{2}\) is the unit ball and \(P\) is any non-trivial centered Gaussian measure on \(\ell^{2}\). This solves an open problem posed in [1]–[3].
Of great significance to modern analysis, Sobolev spaces in finite dimensions (e.g., [4], [5]) are particularly crucial to partial differential equation theory and its applications in mathematical physics. Additionally, they are indispensable to approximation theory, control theory, differential geometry and beyond.
Let \(\Omega\subset \mathbb{R}^n\) be a domain. Let \(1\le p \le \infty\) and \(m\) be an integer. Denote by \(W^{p,m}(\Omega)\) the Sobolev space on \(\Omega\). It consists of locally integrable functions whose distributional partial derivatives of all orders up to \(m\) are \(L^p-\) integrable. This is a Banach space with norm \[\|f\|_{W^{p,m}(\Omega)}= \sum_{|\alpha|\le m} \|D^\alpha f\|_{L^p(\Omega)}.\] An operator \(E\) is called the bounded extension operator if there exists \(C>0\) such that \[\| Ef\|_{W^{p,m}(\mathbb{R}^n)} \le C\| f \|_{W^{p,m}(\Omega)}, \qquad Ef|_\Omega (x) = f(x), \quad \forall f \in W^{p,m}(\Omega).\] Then we say \(\Omega\) is a \(W^{p,m}\)-extension domain.
Since the existence of bounded extension operator for \(\Omega\) guarantees that \(W^{p,m}(\Omega)\) inherits many properties of \(W^{ p,m}(\mathbb{R}^n)\), lots of efforts have been made to give the criteria of the \(W^{p,m}\)-extension domains. In [6], Calderón proved that Lipschitz domains are \(W^{p,m}\)-extension domains for \(1<p<\infty, m\ge1\). In [7], Stein showed that Lipschitz domains are \(W^{p,m}\)-extension domains for \(p=1, \infty\). In [8], Jones introduced the \((\varepsilon, \delta)\)-domains and proved that they are \(W^{p,m}\)-extension domain for \(1\le p \le \infty, m\ge1\). To show that \(\Omega\) is a \(W^{p,m}\)-extension domain, a natural approach is explicitly constructing a bounded extension operator. Indeed, Calderón in [6] defined the extension operator locally and using the partition of unity, constructed the bounded extension operator. Similarly, Stein’s extension operator relies on Whitney’s decomposition, and he also employed a partition of unity to assemble the global extension operator from locally defined extensions([7]). Jones in [8] also adopted Whitney’s decomposition, and by incorporating polynomial fitting on the cubes, he constructed a bounded extension operator with the aid of a partition of unity.
Sobolev spaces in infinite dimensions also play an important role in various areas of mathematics including Malliavin analysis and infinite-dimensional real and complex analysis (e.g., [9]–[14]). However, compared to the finite dimensions, much less is known about the extension problems in infinite dimension. According to [2], this problem has been discussed by V.I. Bogachev first with G. Da Prato and P. Malliavin in the 1990s, and later also with A. Lunardi, but only a trivial positive result was known about extension from half-spaces by reflection.
This problem remains open for several decades partially due to the fact that many basic tools – such as convolutions, mollifiers, and standard covering arguments– are not directly avaible in the infinite-dimensional setting. In this paper, we prove the existence of a bounded Sobolev extension operator \(E:W^{p,1}\left( B,P \right) \rightarrow W^{p,1}\left( \ell^{2} ,P \right)\) using a completely new method, where \(B\subset \ell^{2}\) is the unit ball and \(P\) is any non-trivial centered Gaussian measure on \(\ell^{2}\). This solves an open problem posed in [1]–[3].
In this paper, we use \(\mathbb{N}\) to denote the set of all positive integers, and \(\mathbb{N}_{0}\) to denote the set of all non-negative integers. we adopt the following notation.
We use \(\mathrm{int}(A)\) to denote the interior points of subset \(A\) in a topological space, and \(\partial A\) to denote the boundary points of subset \(A\) in a topological space. Let \[\delta_{ij} \triangleq \begin{cases}1,&i=j\\ 0,&i\neq j\end{cases}.\]
For any nonempty set \(S\), let \[\begin{align} \ell^2(S)\triangleq \left\{\boldsymbol{x}=(x_i)_{i\in S}\in \mathbb{R}^{S}:\sum_{i\in S}|x_i|^2<\infty\right\}. \end{align}\] There is a natural norm on \(\ell^2(S)\) defined by \[\begin{align} \left\lVert \boldsymbol{x}\right\rVert _{\ell^2(S)}\triangleq \left(\sum_{i\in S}|x_i|^2\right)^{\frac{1}{2}} ,\qquad\forall\,\boldsymbol{x}=(x_i)_{i\in S}\in\ell^2(S), \end{align}\] For a topological space \(X\), we denote by \(\mathscr{B}(X)\) the Borel \(\sigma\)-algebra on \(X\). We denote by \(B\left( \boldsymbol{x} ,r \right)\) the open ball in the \(\ell^{2}\) space centered at \(\boldsymbol{x}\) with radius \(r>0\). We use \(B\) to denote the open unit ball in \(\ell^{2}\), and \(B_{m}\) to denote the open unit ball in \(\mathbb{R}^{m}\).
For each \(k\in\mathbb{N}\), let \(C_b^{\infty}(\mathbb{R}^k)\) denote the set of all \(C^{\infty}\) real-valued functions on \(\mathbb{R}^k\) such that the function and all its partial derivatives of all orders are bounded.Note that any \(f\in C_b^{\infty}(\mathbb{R}^k)\) can be regarded as a cylinder function on \(\ell^2\) that depends only on the first \(k\) variables. We define \[\mathscr {C}_b^{\infty}\triangleq \bigcup_{k=1}^{\infty}C_b^{\infty}(\mathbb{R}^k).\] We fix \(\left\{ a_{i} \right\}_{i=1}^{\infty} \subset \left( 0,+\infty \right)\) satisfying \(\sum\limits_{i=1}^{\infty} a_{i}^{2}<\infty\). Similar to the reasoning in Section 2.2 of [15], one obtains a probability measure \(P\) on \(\ell^2\). Following (8) in [15], for each \(k\in\mathbb{N}\), we set \(\mathcal{N}^k\triangleq\prod\limits_{i=1}^{k}\mathcal{N}_{a_i}\), where \[\mathcal{N}_a(B)\triangleq \frac{1}{\sqrt{2\pi a^2}}\int_Be^{-\frac{x^2}{2a^2}}\mathrm{d}x,\quad\,\forall\, B\in\mathscr{B}(\mathbb{R}).\] Since \(\ell^2\) can be identified with \(\mathbb{R}^k\times \ell^{2}(\mathbb{N}\setminus\{1,2,\ldots,k\})\), we have the decomposition \(P=\mathcal{N}^k\times P^{\widehat{1,2,\ldots,k}}\). Here \(P^{\widehat{1,\ldots,k}}\) denotes the product measure obtained by omitting the \(1,2,\ldots,k\)-th components; i.e., it is the restriction of the product measure \(\prod\limits_{j\in\mathbb{N}\setminus\{1,\ldots,k\}}\mathcal{N}_{a_j}\) to the space \[\left(\ell^{2}(\mathbb{N}\setminus\{1,\ldots,k\}),\mathscr{B}\big(\ell^{2}(\mathbb{N}\setminus\{1,\ldots,k\})\big)\right).\]
In this paper, we always assume that \[\delta_{i} f\triangleq \frac{\partial}{\partial x_{i}} f-\frac{x_{i}}{a_{i}^{2}} f,\partial_{i} f\triangleq \frac{\partial f}{\partial x_{i}} ,i=1,2,\cdots\] Further we fix \(\lambda \in \left( 0,1 \right) ,N\in \mathbb{N}\) such that \[\begin{align} \sum_{i>N} a_{i}^{2}<\frac{1}{5} ,\lambda \sum_{i=1}^{N} a_{i}^{2}+\sum_{i>N} a_{i}^{2}<\frac{1}{2}.\label{1f3} \end{align}\tag{1}\] and \[\alpha_{i}\triangleq \lambda a_{i}^{2},1\leqslant i\leqslant N,\alpha_{i}\triangleq a_{i}^{2},i>N.\] For \(\mathbf{x} =\left( x_{1},x_{2},\cdots ,x_{n} \right) \in \mathbb{R}^{n}\), let \(\mathrm{d}\mathbf{x}\) denote the Lebesgue measure on \(\mathbb{R}^n\) and \(\mathrm{d} S\) denote the standard surface measure on \(\mathbb{R}^{n}\). Define the region \[D_{n}\triangleq \left\{ \boldsymbol{x} \in \mathbb{R}^{n} :\sum\limits_{i=1}^{n} a_{i}^{2}x_{i}^{2}<1 \right\}.\]
Proposition 1. \[g:\mathbb{R} \times \partial D_{n}\rightarrow \mathbb{R}^{n} \setminus \left\{ \boldsymbol{0} \right\} ,\left( t,\xi \right) \mapsto g\left( t,\xi \right) =\left( e^{\alpha_{1}t}\xi_{1} ,e^{\alpha_{2}t}\xi_{2} ,\cdots ,e^{\alpha_{n}t}\xi_{n} \right),\] is a homeomorphism and pushes forward the measure \[\frac{e^{\sum\limits_{i=1}^{n} \left( \alpha_{i}t-\frac{1}{2} e^{2\alpha_{i}t}\xi_{i}^{2} \right)}\sum\limits_{i=1}^{n} \alpha_{i}a_{i}^{2}\xi_{i}^{2}}{\sqrt{\sum\limits_{i=1}^{n} a_{i}^{4}\xi_{i}^{2}}} \mathrm{d} t\otimes \mathrm{d} S,\] to the standard Gaussian measure \(e^{-\frac{1}{2} \sum\limits_{i=1}^{n} x_{i}^{2}}\mathrm{d} \boldsymbol{x}\) on \(\mathbb{R}^{n}\setminus \left\{ \boldsymbol{0} \right\}\). Moreover, we have \[\begin{align} g\left( t,\xi \right) \in \mathbb{R}^{n} \setminus \overline{D_{n}} \Leftrightarrow t>0,g\left( t,\xi \right) \in D_{n}\Leftrightarrow t<0.\label{11f1ff3w} \end{align}\qquad{(1)}\]
Proof. Step 1
For each \(\mathbf{x} =\left( x_{1},x_{2},\cdots ,x_{n} \right) \in \mathbb{R}^{n} \setminus \left\{ \boldsymbol{0} \right\}\), from \[\begin{gather}\frac{d}{dt} \left( \sum_{i=1}^{n} a_{i}^{2}e^{-2\alpha_{i} t}x_{i}^{2} \right) =-2\sum_{i=1}^{n} \alpha_{i} a_{i}^{2}e^{-2\alpha_{i} t}x_{i}^{2}<0,\\ \lim_{t\rightarrow +\infty} \sum_{i=1}^{n} a_{i}^{2}e^{-2\alpha_{i} t}x_{i}^{2}=0,\lim_{t\rightarrow -\infty} \sum_{i=1}^{n} a_{i}^{2}e^{-2\alpha_{i} t}x_{i}^{2}=+\infty ,\end{gather}\] we know that there exists a unique \(t=t\left( \mathbf{x} \right) \in \mathbb{R}\) such that \[\begin{align} \sum_{i=1}^{n} a_{i}^{2}e^{-2\alpha_{i} t\left( \mathbf{x} \right)}x_{i}^{2}=1.\label{1f11f} \end{align}\tag{2}\] In this case, \[\xi_{i} =\xi_{i} \left( \mathbf{x} \right) =e^{-\alpha_{i} t\left( \mathbf{x} \right)}x_{i},i=1,2,\cdots ,n,\] are uniquely determined. Hence \(g:\mathbb{R} \times \partial D_{n}\rightarrow \mathbb{R}^{n} \setminus \left\{ \boldsymbol{0} \right\}\) is a homeomorphism.
Step 2
When \(\left( t,\xi \right) \in \mathbb{R} \times \partial D_{n}\), we have \[\begin{gather}\sum_{i=1}^{n} a_{i}^{2}e^{2\alpha_{i} t}\xi_{i}^{2} >1\Leftrightarrow t>0,\\ \sum_{i=1}^{n} a_{i}^{2}e^{2\alpha_{i} t}\xi_{i}^{2} <1\Leftrightarrow t<0,\end{gather}\] that is, ?? is also proved.
Step 3 By 2 and the implicit function theorem, for a non-negative measurable function \(f\) defined on \(\mathbb{R}^{n} \setminus \left\{ \boldsymbol{0} \right\}\), we have from the coarea formula \[\begin{align} &\int_{\mathbb{R}^{n} \setminus \left\{ \boldsymbol{0} \right\}} f\left( \mathbf{x} \right) e^{-\frac{1}{2} \sum\limits_{i=1}^{n} x_{i}^{2}}\mathrm{d} \mathbf{x} =\int_{-\infty}^{\infty} \mathrm{d} t\int_{t\left( \mathbf{x} \right) =t} \frac{f\left( \mathbf{x} \right) \cdot \left( \sum\limits_{j=1}^{n} \alpha_{j} a_{j}^{2}e^{-2\alpha_{j} t\left( \mathbf{x} \right)}x_{j}^{2} \right)}{\sqrt{\sum\limits_{i=1}^{n} a_{i}^{4}e^{-4\alpha_{i} t\left( \mathbf{x} \right)}x_{i}^{2}}} e^{-\frac{1}{2} \sum\limits_{i=1}^{n} x_{i}^{2}}\mathrm{d} S\left( \mathbf{x} \right)\\ &=\int_{-\infty}^{\infty} \mathrm{d} t\int_{\sum\limits_{i=1}^{n} a_{i}^{2}\xi_{i}^{2} =1} \frac{f\left( g\left( t,\xi \right) \right) \cdot \left( \sum\limits_{j=1}^{n} \alpha_{j} a_{j}^{2}\xi_{j}^{2} \right) e^{-\frac{1}{2} \sum\limits_{i=1}^{n} e^{2\alpha_{i} t}\xi_{i}^{2}}}{\sqrt{\sum\limits_{i=1}^{n} a_{i}^{4}e^{-4\alpha_{i} t}x_{i}^{2}}} \frac{e^{\sum\limits_{j=1}^{n} \alpha_{j} t}\cdot \sqrt{\sum\limits_{i=1}^{n} a_{i}^{4}e^{-4\alpha_{i} t}x_{i}^{2}} \left( \sum\limits_{j=1}^{n} \alpha_{j} a_{j}^{2}\xi_{j}^{2} \right)}{\sum\limits_{j=1}^{n} \alpha_{j} a_{j}^{2}\xi_{j}^{2} \cdot \sqrt{\sum\limits_{j=1}^{n} e^{2\alpha_{j} t}a_{j}^{4}e^{-2\alpha_{j} t}\xi_{j}^{2}}} \mathrm{d} S\left( \xi \right)\\ &=\int_{-\infty}^{\infty} \mathrm{d} t\int_{\sum\limits_{i=1}^{n} a_{i}^{2}\xi_{i}^{2} =1} f\left( g\left( t,\xi \right) \right) \frac{e^{\sum\limits_{j=1}^{n} \left( \alpha_{j} t-\frac{1}{2} e^{2\alpha_{j} t}\xi_{j}^{2} \right)}\left( \sum\limits_{j=1}^{n} \alpha_{j} a_{j}^{2}\xi_{j}^{2} \right)}{\sqrt{\sum\limits_{j=1}^{n} a_{j}^{4}\xi_{j}^{2}}} \mathrm{d} S\left( \xi \right),\end{align}\] thus \(g\) pushes forward the measure \[\frac{e^{\sum\limits_{i=1}^{n} \left( \alpha_{i}t-\frac{1}{2} e^{2\alpha_{i}t}\xi_{i}^{2} \right)}\sum\limits_{i=1}^{n} \alpha_{i}a_{i}^{2}\xi_{i}^{2}}{\sqrt{\sum\limits_{i=1}^{n} a_{i}^{4}\xi_{i}^{2}}} \mathrm{d} m\otimes \mathrm{d} S,\] to the standard Gaussian measure \(e^{-\frac{1}{2} \sum\limits_{i=1}^{n} x_{i}^{2}}\mathrm{d} \boldsymbol{x}\) on \(\mathbb{R}^{n}\setminus \left\{ \boldsymbol{0} \right\}\).
This completes the proof of Proposition 1. ◻
Take \(\theta \in C^{\infty}\left( \mathbb{R} \right)\) such that \[0\leqslant \theta \leqslant 1,\theta \left( s \right) =1,s\leqslant \frac{1}{4} ,\theta \left( s \right) =0,s\geqslant \frac{1}{2}.\] Define \[\vartheta \left( \boldsymbol{x} \right) \triangleq \theta \left( \sum\limits_{i=1}^{N} a_{i}^{2}x_{i}^{2} \right).\]
Definition 1. Let \(\Omega \subset \ell^{2}\) be a nonempty open set. Let \(1\le p <\infty\). \(W^{p,1}\left( \Omega ,P \right)\) is defined as the closure of \(\mathscr{C}^{\infty}_{b}\) with respect to the norm \[\left| \left| f \right| \right|_{p,1,\Omega} \triangleq \left( \int_{\Omega} \left| f \right|^{p} \mathrm{d} P\right)^{\frac{1}{p}} + \left( \int_{\Omega} \left( \sum\limits_{j=1}^{\infty} a_{j}^{2}\left| \partial_{j} f \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P\right)^{\frac{1}{p}}.\] Let \(\Omega \subset \mathbb{R}^{m}\) be a nonempty open set. \(W^{p,1}\left( \Omega ,\mathcal{N}^{m} \right)\) is defined as the closure of \(C_{b}^{\infty}\left( \mathbb{R}^{m} \right)\) with respect to the norm \[\left| \left| f \right| \right|_{p,1,\Omega} \triangleq \left( \int_{\Omega} \left| f \right|^{p} \mathrm{d} \mathcal{N}^{m} \right)^{\frac{1}{p}} + \left( \int_{\Omega} \left( \sum\limits_{j=1}^{m} a_{j}^{2}\left| \partial_{j} f \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} \mathcal{N}^{m} \right)^{\frac{1}{p}}.\]
Remark 2. Naturally, we have \(W^{p,1}\left( \mathbb{R}^{m} ,\mathcal{N}^{m} \right) \subset W^{p,1}\left( \ell^{2} ,P \right).\)
Lemma 1. There exist bounded linear operators \[E_{m}:W^{p,1}\left( B_{m},\mathcal{N}^{m} \right) \rightarrow W^{p,1}\left( \mathbb{R}^{m} ,\mathcal{N}^{m} \right),\] such that \[\left( E_{m}f \right) |_{B_{m}}=f,a.e, \forall f\in W^{p,1}\left( B_{m},\mathcal{N}^{m} \right),\] and \[\sup_{m>N} \left| \left| E_{m} \right| \right| <\infty.\]
Remark 3. In the proof of Lemma 1, for \(f\in C^{\infty}\left( \overline{B_{m}} \right)\), \(E_{m}f\) is independent of \(p\).
Proof. We only need to find \(C>0\) independent of \(m\) such that for every \(f\in C^{\infty}\left( \overline{B_{m}} \right)\) there exists an extension \(F\in W^{p,1}\left( \mathbb{R}^{m} ,\mathcal{N}^{m} \right)\) satisfying \[\left| \left| F \right| \right|_{p,1,\mathbb{R}^{m}} \leqslant C\left| \left| f \right| \right|_{p,1,B_{m}} .\] Consider the change of variables \[x_{1}=a_{1}y_{1},x_{2}=a_{2}y_{2},\cdots ,x_{m}=a_{m}y_{m}.\] It is sufficient to find \(C>0\) independent of \(m\) such that for every \(f\in C^{\infty}\left( \overline{D_{m}} \right)\), there exists \[F\in C^{\infty}\left( \mathbb{R}^{m} \setminus \overline{D_{m}} \right) \bigcap C\left( \mathbb{R}^{m} \setminus D_{m} \right),\] satisfying \[F|_{\partial D_{m}}=f|_{\partial D_{m}},\] and \[\begin{align} \begin{aligned} &\left( \int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left| F \right|^{p} e^{-\frac{1}{2} \left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}} +\left( \int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left( \sum\limits_{j=1}^{m} \left| \partial_{j} F \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}}\\ &\; \leqslant C\left( \int_{D_{m}} \left| f \right|^{p} e^{-\frac{1}{2} \left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}} +C\left( \int_{D_{m}} \left( \sum\limits_{j=1}^{m} \left| \partial_{j} f \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}} .\end{aligned}.\label{11ff3f} \end{align}\tag{3}\] Step 1 Set \[\widetilde{f} \left( \boldsymbol{x} \right) \triangleq \left( 1-\vartheta \left( \boldsymbol{x} \right) \right) f\left( \boldsymbol{x} \right) ,\boldsymbol{x} \in \overline{D_{m}}.\]
Define \[\begin{align}W_{2i-1}&\triangleq \left\{ \boldsymbol{x} \in \mathbb{R}^{m} :\frac{1}{2a_{i}\sqrt{N}} <x_{i}<\frac{2}{a_{i}} ,\sum\limits_{j\neq i,1\leqslant j\leqslant m} a_{j}^{2}x_{j}^{2}<1-\frac{1}{4N} \right\} ;\\ W_{2i}&\triangleq \left\{ \boldsymbol{x} \in \mathbb{R}^{m} :-\frac{2}{a_{i}} <x_{i}<-\frac{1}{2a_{i}\sqrt{N}} ,\sum\limits_{j\neq i,1\leqslant j\leqslant m} a_{j}^{2}x_{j}^{2}<1-\frac{1}{4N} \right\} ;\\ U_{2i-1}&\triangleq \left\{ \boldsymbol{x} \in \mathbb{R}^{m} :\frac{1}{3a_{i}\sqrt{N}} <x_{i}<\frac{3}{a_{i}} ,\sum\limits_{j\neq i,1\leqslant j\leqslant m} a_{j}^{2}x_{j}^{2}<1-\frac{1}{9N} \right\} ;\\ U_{2i}&\triangleq \left\{ \boldsymbol{x} \in \mathbb{R}^{m} :-\frac{3}{a_{i}} <x_{i}<-\frac{1}{3a_{i}\sqrt{N}} ,\sum\limits_{j\neq i,1\leqslant j\leqslant m} a_{j}^{2}x_{j}^{2}<1-\frac{1}{9N} \right\} ,\end{align} i=1,2,\cdots ,N.\] We have \[W_{i}\bigcap \partial D_{m}\neq \emptyset ,i=1,2,\cdots ,2N,\partial D_{m}\bigcap \left\{ \boldsymbol{x} \in \mathbb{R}^{m} :\sum\limits_{i=1}^{N} a_{i}^{2}x_{i}^{2}\geqslant \frac{1}{2} \right\} \subset \bigcup_{i=1}^{2N} W_{i}.\] For each \[\boldsymbol{x}_{0} \in \partial D_{m}\setminus \left( \bigcup\limits_{i=1}^{2N} W_{i} \right) ,\] there exists an open bounded neighborhood \(U_{\boldsymbol{x}_{0}}\) of \(\boldsymbol{x}_{0}\) such that \[U_{\boldsymbol{x}_{0}}\bigcap \left\{ \boldsymbol{x} \in \mathbb{R}^{m} :\sum\limits_{i=1}^{N} a_{i}^{2}x_{i}^{2}\geqslant \frac{1}{2} \right\} \bigcap \overline{D_{m}} =\emptyset.\] It follows that \[\partial D_{m}\setminus \left( \bigcup\limits_{i=1}^{2N} W_{i} \right) \subset \bigcup\limits_{\boldsymbol{x}_{0} \in \partial D_{m}\setminus \left( \bigcup\limits_{i=1}^{2N} W_{i} \right)} U_{\boldsymbol{x}_{0}}.\] Since \(\partial D_{m}\setminus \left( \bigcup\limits_{i=1}^{2N} W_{i} \right)\) is compact, there exist finite sets \[U_{2N+j}\triangleq U_{\boldsymbol{x}_{i_{j}}},\boldsymbol{x}_{i_{j}} \in \partial D_{m}\setminus \left( \bigcup\limits_{i=1}^{2N} W_{i} \right) ,j=1,2,\cdots ,T,\] such that \[\partial D_{m}\setminus \left( \bigcup\limits_{i=1}^{2N} W_{i} \right) \subset \bigcup_{j=1}^{T} U_{2N+j}.\] By [16], there exist open sets \(W_{2N+j}\subset U_{2N+j},j=1,2,\cdots , T,\) such that \[\overline{W_{2N+j}} \subset U_{2N+j},j=1,2,\cdots ,T,\partial D_{m}\setminus \left( \bigcup\limits_{i=1}^{2N} W_{i} \right) \subset \bigcup_{j=1}^{T} W_{2N+j}.\] By the (b) of[17], we can take \(\eta_{j} \in C_{c}^{\infty}\left( U_{2N+j} \right) ,j=1,2,\cdots , T,\) such that \[\eta_{j} |_{W_{2N+j}}=1,j=1,2,\cdots ,T.\] Set \[\begin{align}\delta&\triangleq \min_{1\leqslant i\leqslant N} d\left( W_{2i-1},\mathbb{R}^{m} \setminus U_{2i-1} \right) =\min_{1\leqslant i\leqslant N} d\left( W_{2i},\mathbb{R}^{m} \setminus U_{2i} \right)\\ &=\min_{1\leqslant i\leqslant N} \left\{ \frac{1}{6a_{i}\sqrt{N}} ,\frac{\sqrt{1-\frac{1}{9N}} -\sqrt{1-\frac{1}{4N}}}{\max\limits_{j\neq i,1\leqslant j<\infty} a_{j}} \right\} >0.\end{align}\] Consider \[J_{m} \left( \boldsymbol{x} \right) \triangleq \begin{cases}\frac{1}{\int_{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}} <1} e^{-\frac{1}{1-\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}}\mathrm{d} \boldsymbol{x}} e^{-\frac{1}{1-\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}},&\left| \left\vert \boldsymbol{x} \right\vert \right|_{\mathbb{R}^{m}} <1,\\ 0,&\left| \left\vert \boldsymbol{x} \right\vert \right|_{\mathbb{R}^{m}} \geqslant 1,\end{cases}\] and \[\eta_{i} \left( \boldsymbol{x} \right) \triangleq \left( \frac{8}{\delta} \right)^{m} \int_{\mathbb{R}^{m}} J_{m}\left( \frac{8\left( \boldsymbol{x} -\boldsymbol{y} \right)}{\delta} \right) \theta \left( \frac{d\left( \boldsymbol{y} ,W_{i} \right)}{\delta} \right) \mathrm{d} \boldsymbol{y} ,i=1,2,\cdots ,2N,\] where \(d\) denotes the standard distance function. For each \(i=1,2,\cdots ,2N\), it is easy to see that \(0\leqslant \eta_{i} \leqslant 1\) and for \(\boldsymbol{x} \in W_{i},\) we have \[\begin{align} \eta_{i} \left( \boldsymbol{x} \right) =\int_{\left| \left| \boldsymbol{y} \right| \right|_{\mathbb{R}^{m}} <1} J_{m}\left( \boldsymbol{y} \right) \theta \left( \frac{d\left( \boldsymbol{x} -\frac{\delta}{8} \boldsymbol{y} ,W_{i} \right)}{\delta} \right) \mathrm{d} \boldsymbol{y} =\int_{\left| \left| \boldsymbol{y} \right| \right|_{\mathbb{R}^{m}} <1} J_{m}\left( \boldsymbol{y} \right) \mathrm{d} \boldsymbol{y} =1.\label{13f31} \end{align}\tag{4}\] For \(\boldsymbol{x} \in U_{i}\) satisfying \(d\left( \boldsymbol{x} ,W_{i} \right) \geqslant \frac{6}{8} \delta\), it follows that \[d\left( \boldsymbol{x} -\frac{\delta}{8} \boldsymbol{y} ,W_{i} \right) \geqslant \frac{\delta}{2}, \quad \forall \left| \left| \boldsymbol{y} \right| \right|_{\mathbb{R}^{m}} <1.\] By 4 we have \(\eta_{i} \left( \boldsymbol{x} \right) =0\) and hence \(\eta_{i} \in C_{b}^{\infty}\left( U_{i} \right)\). Since for every \(\boldsymbol{x},\boldsymbol{z}\in \mathbb{R}^{m}\), \[\begin{align} &\left| \eta_{i} \left( \boldsymbol{x} \right) -\eta_{i} \left( \boldsymbol{z} \right) \right| \leqslant \int_{\mathbb{R}^{m}} J_{m}\left( \boldsymbol{y} \right) \left| \theta \left( \frac{d\left( \boldsymbol{x} -\frac{\delta}{8} \boldsymbol{y} ,W_{i} \right)}{\delta} \right) -\theta \left( \frac{d\left( \boldsymbol{z} -\frac{\delta}{8} \boldsymbol{y} ,W_{i} \right)}{\delta} \right) \right| \mathrm{d} \boldsymbol{y}\\ &\; \leqslant \sup_{\mathbb{R}} \left| \theta^{\prime} \right| \int_{\mathbb{R}^{m}} J_{m}\left( \boldsymbol{y} \right) \left| \frac{d\left( \boldsymbol{x} -\frac{\delta}{8} \boldsymbol{y} ,W_{i} \right) -d\left( \boldsymbol{z} -\frac{\delta}{8} \boldsymbol{y} ,W_{i} \right)}{\delta} \right| \mathrm{d} \boldsymbol{y}\\ &\; \leqslant \sup_{\mathbb{R}} \left| \theta^{\prime} \right| \frac{\left| \left| \boldsymbol{x} -\boldsymbol{z} \right| \right|_{\mathbb{R}^{m}}}{\delta} \int_{\mathbb{R}^{m}} J_{m}\left( \boldsymbol{y} \right) \mathrm{d} \boldsymbol{y} =\sup_{\mathbb{R}} \left| \theta^{\prime} \right| \frac{\left| \left| \boldsymbol{x} -\boldsymbol{z} \right| \right|_{\mathbb{R}^{m}}}{\delta} ,\end{align}\] thus we obtain the estimate \[\begin{align} \sqrt{\sum\limits_{i=1}^{m} \left| \partial_{j} \eta_{i} \right|^{2}} \leqslant \frac{\sup\limits_{\mathbb{R}} \left| \theta^{\prime} \right|}{\delta}.\label{1fg3gg1} \end{align}\tag{5}\] Consider \[\rho_{1} \triangleq \eta_{1} ,\rho_{i} \triangleq \eta_{i} \prod_{j=1}^{i-1} \left( 1-\eta_{j} \right) ,i=2,3,\cdots,2N+T.\] Since \(\partial D_{m}\subset \bigcup\limits_{j=1}^{2N+T} W_{j}\), for each \(\boldsymbol{x} \in \partial D_{m}\), there exists \(j\in \left\{ 1,2,\cdots ,2N+T \right\}\) such that \(\boldsymbol{x} \in W_{j}\). Then we have \[\eta_{j} \left( \boldsymbol{x} \right) =1\Rightarrow \sum\limits_{j=1}^{2N+T} \rho_{j} \left( \boldsymbol{x} \right) =1-\prod_{j=1}^{2N+T} \left( 1-\eta_{j} \left( \boldsymbol{x} \right) \right) =1.\] By 5 and the basic inequality \[\left| \prod_{i=1}^{n} a_{i}-\prod_{i=1}^{n} b_{i} \right| \leqslant \sum\limits_{i=1}^{n} \left| a_{i}-b_{i} \right| ,\forall a_{i},b_{i}\in \left[ 0,1 \right],\] we obtain for \(i=1,2,\cdots ,2N,\boldsymbol{x} ,\boldsymbol{y} \in \mathbb{R}^{m}\), \[\left| \rho_{i} \left( \boldsymbol{x} \right) -\rho_{i} \left( \boldsymbol{y} \right) \right| \leqslant \sum\limits_{j=1}^{i} \left| \eta_{i} \left( \boldsymbol{x} \right) -\eta_{i} \left( \boldsymbol{y} \right) \right| \leqslant \frac{2N}{\delta} \sup_{\mathbb{R}} \left| \theta^{\prime} \right| \cdot \left| \left| \boldsymbol{x} -\boldsymbol{y} \right| \right|_{\mathbb{R}^{m}}.\] In summary, we obtain a partition of unity \(\left\{ \rho_{i} \right\}_{i=1}^{2N+T} \subset C_{c}^{\infty}\left( \mathbb{R}^{m} \right)\) subordinate to \(\left\{ U_{i} \right\}_{i=1}^{2N+T}\) and it holds that
\(0\leqslant \rho_{i} \leqslant 1,i=1,2,\cdots ,2N+T\);
for each \(i=1,2,\cdots,2N+T\), \(\mathrm{supp} \rho_{i}\) is a compact subset contained in \(U_{i}\);
\(\sum\limits_{i=1}^{2N+T} \rho_{i} \left( \boldsymbol{x} \right) =1,\forall \boldsymbol{x} \in \partial D_{m}\);
\(\sqrt{\sum\limits_{j=1}^{m} \left| \partial_{j} \rho_{i} \left( \boldsymbol{x} \right) \right|^{2}} \leqslant \frac{2N}{\delta} \sup\limits_{\mathbb{R}} \left| \theta^{\prime} \right| ,i=1,2,\cdots ,2N\).
First, the mapping \[\left( x_{1},x_{2},\cdots ,x_{m} \right) \mapsto \left( \frac{2}{a_{1}} \sqrt{1-\sum\limits_{j=2}^{m} a_{j}^{2}x_{j}^{2}} -x_{1},x_{2},\cdots ,x_{m} \right),\] maps \(U_{1}\bigcap D_{m}\) onto an open subset \(V_{1}\) of \(\mathbb{R}^{m}\setminus \overline{D_{m}}\).
Set \[\widetilde{f_{1}} \left( \boldsymbol{x} \right) \triangleq \begin{cases}\left( \rho_{1} \widetilde{f} \right) \left( -x_{1}+\frac{2}{a_{1}} \sqrt{1-\sum\limits_{j=2}^{m} a_{j}^{2}x_{j}^{2}} ,x_{2},\cdots ,x_{m} \right) ,&\boldsymbol{x} \in V_{1}\\ 0,&\boldsymbol{x} \in \mathbb{R}^{m} \setminus \left( \overline{D_{m}} \bigcup V_{1} \right)\end{cases} .\] Clearly, we have \[\widetilde{f_{1}} \left( \boldsymbol{x} \right) =\left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{x} \right) ,\forall \boldsymbol{x} \in \partial D_{m},\] and hence \(\widetilde{f_{1}}\) is an extension of \(\rho_{1} \widetilde{f}\). Now for \(\boldsymbol{x}\in V_{1}\), set \[\boldsymbol{y} \triangleq \left( -x_{1}+\frac{2}{a_{1}} \sqrt{1-\sum\limits_{j=2}^{m} a_{j}^{2}x_{j}^{2}} ,x_{2},\cdots ,x_{m} \right).\] By direct computation we have \[\begin{align}\sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f_{1}} \left( \boldsymbol{x} \right) \right|^{2}&=\left| \partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} +\sum\limits_{j=2}^{m} \left\vert -\frac{2}{a_{1}} \frac{a_{j}^{2}x_{j}\partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right)}{\sqrt{1-\sum\limits_{k=2}^{m} a_{k}^{2}x_{k}^{2}}} +\partial_{j} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right\vert^{2}\\ &\leqslant \left| \partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} +\frac{8}{a_{1}^{2}} \sum\limits_{j=2}^{m} \frac{a_{j}^{4}x_{j}^{2}\left| \partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2}}{1-\sum\limits_{k=2}^{m} a_{k}^{2}x_{k}^{2}} +2\sum\limits_{j=2}^{m} \left| \partial_{j} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2}\\ &\leqslant 2\sum\limits_{j=1}^{m} \left| \partial_{j} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} +\frac{32N}{a_{1}^{2}} \sum\limits_{j=2}^{m} a_{j}^{4}x_{j}^{2}\left| \partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2}\\ &\leqslant 2\sum\limits_{j=1}^{m} \left| \partial_{j} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} +\frac{32N}{a_{1}^{2}} \left| \partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} \sqrt{\left( \sum\limits_{j=2}^{m} a_{j}^{4}x_{j}^{4} \right) \left( \sum\limits_{j=2}^{m} a_{j}^{4} \right)}\\ &\leqslant 2\sum\limits_{j=1}^{m} \left\vert \partial_{j} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right\vert^{2} +\frac{32N}{a_{1}^{2}} \left| \partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} \sqrt{\left( \sum\limits_{j=2}^{\infty} a_{j}^{4} \right) \left( \sum\limits_{j=2}^{m} a_{j}^{2}x_{j}^{2} \right)^{2}}\\ &\leqslant 2\sum\limits_{j=1}^{m} \left| \partial_{j} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} +\frac{32N}{a_{1}^{2}} \left| \partial_{1} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2} \left( \sum\limits_{j=2}^{\infty} a_{j}^{4} \right)^{\frac{1}{2}}\\ &\leqslant \left( 2+\frac{32N}{a_{1}^{2}} \left( \sum\limits_{j=2}^{\infty} a_{j}^{4} \right)^{\frac{1}{2}} \right) \sum\limits_{j=1}^{m} \left| \partial_{j} \left( \rho_{1} \widetilde{f} \right) \left( \boldsymbol{y} \right) \right|^{2}\\ &\leqslant \left( 4+\frac{64N}{a_{1}^{2}} \left( \sum\limits_{j=2}^{\infty} a_{j}^{4} \right)^{\frac{1}{2}} \right) \left( \sum\limits_{j=1}^{m} \left| \partial_{j} \rho_{1} \left( \boldsymbol{y} \right) \cdot \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} +\sum\limits_{j=1}^{m} \left| \rho_{1} \left( \boldsymbol{y} \right) \cdot \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)\\ &\leqslant \left( 4+\frac{64N}{a_{1}^{2}} \left( \sum\limits_{j=2}^{\infty} a_{j}^{4} \right)^{\frac{1}{2}} \right) \left( 1+\frac{4N^{2}}{\delta^{2}} \sup\limits_{\mathbb{R}} \left| \theta^{\prime} \right|^{2} \right) \left( \left\vert \widetilde{f}\left( \boldsymbol{y} \right) \right\vert^{2} +\sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right) .\end{align}\] Set \[D_{1}\triangleq \left( 4+\frac{64N}{a_{1}^{2}} \left( \sum\limits_{j=2}^{\infty} a_{j}^{4} \right)^{\frac{1}{2}} \right) \left( 1+\frac{4N^{2}}{\delta^{2}} \sup\limits_{\mathbb{R}} \left| \theta^{\prime} \right|^{2} \right).\] Then we have \[\begin{align} &\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left\vert \widetilde{f_{1}} \left( \boldsymbol{x} \right) \right\vert^{p} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x} =\int_{V_{1}} \left| \rho_{1} \left( \boldsymbol{y} \right) \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x}\\ &\; \leqslant \int_{V_{1}} \left| \rho_{1} \left( \boldsymbol{y} \right) \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x} =\int_{D_{m}\bigcap U_{1}} \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left( \frac{2}{a_{1}} \sqrt{1-\sum\limits_{j=2}^{m} a_{j}^{2}y_{j}^{2}} \right)^{2}}{2} +\frac{y_{1}^{2}}{2}}e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\\ &\; \leqslant e^{\frac{9}{2a_{1}^{2}}}\int_{D_{m}} \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y} ,\end{align}\] and \[\begin{align} &\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left( \sum\limits_{j=1}^{m} \left\vert \partial_{j} \widetilde{f_{1}} \left( \boldsymbol{x} \right) \right\vert^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x} \leqslant D_{1}^{\frac{p}{2}}\int_{V_{1}} \left( \left\vert \widetilde{f}\left( \boldsymbol{y} \right) \right\vert^{2} +\sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x}\\ &\; =D_{1}^{\frac{p}{2}}\int_{V_{1}} \left( \left\vert \widetilde{f}\left( \boldsymbol{y} \right) \right\vert^{2} +\sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{\left( \frac{2}{a_{1}} \sqrt{1-\sum\limits_{j=2}^{m} a_{j}^{2}y_{j}^{2}} \right)^{2}}{2} +\frac{y_{1}^{2}}{2}}e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\\ &\; \leqslant D_{1}^{\frac{p}{2}}e^{\frac{9}{2a_{1}^{2}}}\int_{V_{1}} \left( \left\vert \widetilde{f}\left( \boldsymbol{y} \right) \right\vert^{2} +\sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\\ &\; \leqslant D_{1}^{\frac{p}{2}}e^{\frac{9}{2a_{1}^{2}}}2^{\frac{p}{2}}\int_{D_{m}} \left[ \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} +\left( \sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} \right] e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y} .\end{align}\] Consequently, \[\begin{align} &\left(\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left\vert \widetilde{f_{1}} \left( \boldsymbol{x} \right) \right\vert^{p} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x}\right)^{\frac{1}{p}} + \left(\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left( \sum\limits_{j=1}^{m} \left\vert \partial_{j} \widetilde{f_{1}} \left( \boldsymbol{x} \right) \right\vert^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}} \\ &\; \leqslant e^{\frac{9}{2pa_{1}^{2}}} \left(\int_{D_{m}} \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}} +\sqrt{2D_{1}} e^{\frac{9}{2pa_{1}^{2}}}\left(\int_{D_{m}} \left[ \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} +\left( \sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} \right] e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}}\\ &\; \leqslant \left( \sqrt{2D_{1}} +1 \right) e^{\frac{9}{2pa_{1}^{2}}}\left(\int_{D_{m}} \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}} +\sqrt{2D_{1}} e^{\frac{9}{2pa_{1}^{2}}}\left(\int_{D_{m}} \left( \sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}}\\ &\; \leqslant \left( \sqrt{2D_{1}} +1 \right) e^{\frac{9}{2pa_{1}^{2}}}\left( \left(\int_{D_{m}} \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}} +\left(\int_{D_{m}} \left( \sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}} \right) .\end{align}\] Set \(\widetilde{D_{1}} \triangleq \left( \sqrt{2D_{1}} +1 \right) e^{\frac{9}{2pa_{1}^{2}}}\). Similarly, considering \(i=1,2,\cdots,2N\), there exists extension \(\widetilde{f_{i}}\) of \(\rho_{i} f\) and \(\widetilde{D_{i}} >0\) such that \[\begin{align} &\left(\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left\vert \widetilde{f_{i}} \left( \boldsymbol{x} \right) \right\vert^{p} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x}\right)^{\frac{1}{p}} + \left(\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left( \sum\limits_{j=1}^{m} \left\vert \partial_{j} \widetilde{f_{i}} \left( \boldsymbol{x} \right) \right\vert^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left| \boldsymbol{x} \right| \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{x}\right)^{\frac{1}{p}}\\ &\; \leqslant \widetilde{D_{i}} \left( \left(\int_{D_{m}} \left| \widetilde{f}\left( \boldsymbol{y} \right) \right|^{p} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}} + \left(\int_{D_{m}} \left( \sum\limits_{j=1}^{m} \left| \partial_{j} \widetilde{f}\left( \boldsymbol{y} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{\left| \left\vert \boldsymbol{y} \right\vert \right|_{\mathbb{R}^{m}}^{2}}{2}}\mathrm{d} \boldsymbol{y}\right)^{\frac{1}{p}} \right) .\end{align}\] When \(i=2N+1,\cdots, 2N+T\), we have \(\rho_{i} \widetilde{f}=0\). Therefore set \[F_{1}\triangleq \sum\limits_{i=1}^{2N} \widetilde{f_{i}}.\] It follows that \[F_{1}\in C^{\infty}\left( \mathbb{R}^{m} \setminus \overline{D_{m}} \right) \bigcap C\left( \mathbb{R}^{m} \setminus D_{m} \right) ,F_{1}|_{\partial D_{m}}=\widetilde{f} |_{\partial D_{m}},\] and \[\begin{align} \left| \left| F_{1} \right| \right|_{p,1,\mathbb{R}^{m} \setminus \overline{D_{m}}} \leqslant \sum\limits_{i=1}^{2N} \left| \left| \widetilde{f_{i}} \right| \right|_{p,1,\mathbb{R}^{m} \setminus \overline{D_{m}}} \leqslant \left( \sum\limits_{i=1}^{2N} \widetilde{D_{i}} \right) \cdot \left| \left| \widetilde{f} \right| \right|_{p,1,D_{m}}.\label{6} \end{align}\tag{6}\]
Step 2 Consider \[F_{2}\left( g\left( t,\xi \right) \right) \triangleq \vartheta \left( \xi \right) f\left( g\left( -t,\xi \right) \right) ,t>0,\xi \in \partial D_{m}.\]We have \[F_{2}\in C^{\infty}\left( \mathbb{R}^{m} \setminus \overline{D_{m}} \right) \bigcap C\left( \mathbb{R}^{m} \setminus D_{m} \right) ,F_{2}|_{\partial D_{m}}=\left( f-\widetilde{f} \right) |_{\partial D_{m}}.\] When \[\sum\limits_{i=1}^{N} a_{i}^{2}\xi_{i}^{2} \leqslant \frac{1}{2} ,\sum\limits_{i=1}^{m} a_{i}^{2}\xi_{i}^{2} =1,\] we have \[\begin{align} \begin{aligned}e^{\sum\limits_{i=1}^{m} \left( \alpha_{i} t-\frac{1}{2} e^{2\alpha_{i} t}\xi_{i}^{2} \right)}&=e^{\int_{0}^{t} \sum\limits_{i=1}^{m} \left( 2\alpha_{i} -\alpha_{i} \left( e^{2\alpha_{i} s}+e^{-2\alpha_{i} s} \right) \xi_{i}^{2} \right) \mathrm{d} s}e^{\sum\limits_{i=1}^{m} \left( -\alpha_{i} t-\frac{1}{2} e^{-2\alpha_{i} t}\xi_{i}^{2} \right)}\\ &\leqslant e^{\int_{0}^{t} \sum\limits_{i=1}^{m} \left( 2\alpha_{i} -2\alpha_{i} \xi_{i}^{2} \right) \mathrm{d} s}e^{\sum\limits_{i=1}^{m} \left( -\alpha_{i} t-\frac{1}{2} e^{-2\alpha_{i} t}\xi_{i}^{2} \right)}\\ &\leqslant e^{2\sum\limits_{i=1}^{m} \alpha_{i} \cdot t-2\sum\limits_{i=N+1}^{m} a_{i}^{2}\xi_{i}^{2} \cdot t}e^{\sum\limits_{i=1}^{m} \left( -\alpha_{i} t-\frac{1}{2} e^{-2\alpha_{i} t}\xi_{i}^{2} \right)}\\ &\leqslant e^{2\cdot \frac{1}{2} \cdot t-2\left( 1-\frac{1}{2} \right) \cdot t}e^{\sum\limits_{i=1}^{m} \left( -\alpha_{i} t-\frac{1}{2} e^{-2\alpha_{i} t}\xi_{i}^{2} \right)}\\ &=e^{\sum\limits_{i=1}^{m} \left( -\alpha_{i} t-\frac{1}{2} e^{-2\alpha_{i} t}\xi_{i}^{2} \right)}.\end{aligned}\label{1f1ge} \end{align}\tag{7}\]
On the one hand, by 7 and the fact that \(\vartheta \left( \xi \right) =0\) for \[\sum_{i=1}^{N} a_{i}^{2}\xi_{i}^{2} \geqslant \frac{1}{2} ,\sum_{i=1}^{m} a_{i}^{2}\xi_{i}^{2} =1,\] we have \[\begin{align} \begin{aligned}\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left| f\left( \boldsymbol{x} \right) \right|^{p} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x} \;&=\int_{0}^{\infty} \mathrm{d} t\int_{\partial D_{m}} \left| f\left( g\left( t,\xi \right) \right) \right|^{p} \frac{e^{\sum\limits_{i=1}^{m} \left( \alpha_{i} t-\frac{1}{2} e^{2\alpha_{i} t}\xi_{i}^{2} \right)}\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}{\sqrt{\sum\limits_{i=1}^{m} a_{i}^{4}\xi_{i}^{2}}} \mathrm{d} S\left( \xi \right)\\ &\leqslant \int_{0}^{\infty} \mathrm{d} t\int_{\partial D_{m}} \left| \vartheta \left( \xi \right) f\left( g\left( -t,\xi \right) \right) \right|^{p} \frac{e^{\sum\limits_{i=1}^{m} \left( -\alpha_{i} t-\frac{1}{2} e^{-2\alpha_{i} t}\xi_{i}^{2} \right)}\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}{\sqrt{\sum\limits_{i=1}^{m} a_{i}^{4}\xi_{i}^{2}}} \mathrm{d} S\left( \xi \right)\\ &=\int_{-\infty}^{0} \mathrm{d} t\int_{\partial D_{m}} \left| \vartheta \left( \xi \right) f\left( g\left( t,\xi \right) \right) \right|^{p} \frac{e^{\sum\limits_{i=1}^{n} \left( \alpha_{i} t-\frac{1}{2} e^{2\alpha_{i} t}\xi_{i}^{2} \right)}\sum\limits_{i=1}^{n} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}{\sqrt{\sum\limits_{i=1}^{m} a_{i}^{4}\xi_{i}^{2}}} \mathrm{d} S\left( \xi \right)\\ &\leqslant \int_{-\infty}^{0} \mathrm{d} t\int_{\partial D_{m}} \left| f\left( g\left( t,\xi \right) \right) \right|^{p} \frac{e^{\sum\limits_{i=1}^{m} \left( \alpha_{i} t-\frac{1}{2} e^{2\alpha_{i} t}\xi_{i}^{2} \right)}\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}{\sqrt{\sum\limits_{i=1}^{m} a_{i}^{4}\xi_{i}^{2}}} \mathrm{d} S\left( \xi \right)\\ &=\int_{D_{m}} \left| f\left( \boldsymbol{x} \right) \right|^{p} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x}.\end{aligned}\label{1f131} \end{align}\tag{8}\] According to Proposition 1, set \[y_{j}\triangleq e^{-2\alpha_{j}t}x_{j},j=1,2,\cdots,m,\boldsymbol{y} =\boldsymbol{y} \left( \boldsymbol{x} \right) =\left( y_{1},y_{2},\cdots ,y_{m} \right),\] then by direct computation we have \[\begin{align} &\frac{\partial t}{\partial x_{k}} =\frac{a_{k}^{2}e^{-\alpha_{k} t}\xi_{k}}{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}} ;\\ &\frac{\partial \xi_{j}}{\partial x_{k}} =e^{-\alpha_{j} t}\delta_{jk} -e^{-\alpha_{k} t}\frac{\alpha_{j} a_{k}^{2}\xi_{j} \xi_{k}}{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}} ;\\ &\frac{\partial y_{j}}{\partial x_{k}} =e^{-2\alpha_{j} t}\delta_{jk} -2e^{-\left( \alpha_{j} +\alpha_{k} \right) t}\frac{\alpha_{j} a_{k}^{2}\xi_{j} \xi_{k}}{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}} .\end{align} j,k=1,2,\cdots ,m.\] Set \[E\triangleq \begin{pmatrix}e^{-\alpha_{1} t}&&&\\ &e^{-\alpha_{2} t}&&\\ &&\ddots&\\ &&&e^{-\alpha_{m} t}\end{pmatrix} , P\triangleq \begin{pmatrix}\alpha_{1} \xi_{1}\\ \alpha_{2} \xi_{2}\\ \vdots\\ \alpha_{m} \xi_{m}\end{pmatrix} , Q\triangleq \begin{pmatrix}\frac{a_{1}^{2}e^{-\alpha_{1} t}\xi_{1}}{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}\\ \frac{a_{2}^{2}e^{-\alpha_{2} t}\xi_{2}}{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}\\ \vdots\\ \frac{a_{m}^{2}e^{-\alpha_{m} t}\xi_{m}}{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}\end{pmatrix} ,\widetilde{P} \triangleq \begin{pmatrix}\alpha_{1} e^{-\alpha_{1} t}\xi_{1}\\ \alpha_{2} e^{-\alpha_{2} t}\xi_{2}\\ \vdots\\ \alpha_{m} e^{-\alpha_{m} t}\xi_{m}\end{pmatrix} .\] Then \(E-PQ^{T}\) is the Jacobian matrix of the mapping \(\boldsymbol{x} \mapsto \xi\) and it satisfies \[\begin{align} \begin{aligned}\left| \left| E-PQ^{T} \right| \right|&\leqslant 1+\left| \left| Q \right| \right| \cdot \left| \left| P \right| \right| =1+\frac{\sqrt{\sum\limits_{i=1}^{m} \alpha_{i}^{2} \xi_{i}^{2}} \cdot \sqrt{\sum\limits_{i=1}^{m} a_{i}^{4}\xi_{i}^{2}}}{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}}\\ &\leqslant 1+\frac{\sqrt{\sum\limits_{i=1}^{m} \alpha_{i} a_{i}^{2}\xi_{i}^{2}} \cdot \sqrt{\sum\limits_{i=1}^{m} \frac{\alpha_{i}}{\lambda} a_{i}^{2}\xi_{i}^{2}}}{\sum\limits_{i=1}^{n} \alpha_{i} a_{i}^{2}\xi_{i}^{2}} =1+\frac{1}{\sqrt{\lambda}} .\end{aligned}\label{1} \end{align}\tag{9}\] Similarly, \(E^{2}-2Q\widetilde{P}^{T}\) is the Jacobian matrix of the mapping \(\boldsymbol{x} \mapsto \boldsymbol{y}\) and it satisfies \[\begin{align} \left| \left| E^{2}-2Q\widetilde{P}^{T} \right| \right| \leqslant 1+2\left| \left| Q \right| \right| \cdot \left| \left| \widetilde{P}^{T} \right| \right| \leqslant 1+\frac{2}{\sqrt{\lambda}}.\label{2} \end{align}\tag{10}\] Since \[\sum\limits_{i=1}^{m} \left| \frac{\partial \vartheta}{\partial \xi_{i}} \left( \xi \right) \right|^{2} \leqslant 4\sup_{\mathbb{R}} \left| \theta^{\prime} \right|^{2} \sum\limits_{i=1}^{N} a_{i}^{4}\xi_{i}^{2} \leqslant 4\sup_{i\geqslant 1} a_{i}^{2}\cdot \sup_{\mathbb{R}} \left| \theta^{\prime} \right|^{2} ,\forall \xi \in \partial D_{m},\] by 9 and the chain rule we obtain \[\begin{align} \sum\limits_{i=1}^{m} \left| \frac{\partial \vartheta \left( \xi \left( \boldsymbol{x} \right) \right)}{\partial x_{i}} \right|^{2} \leqslant 4\left( 1+\frac{1}{\sqrt{\lambda}} \right)^{2} \cdot \sup_{i\geqslant 1} a_{i}^{2}\cdot \sup_{\mathbb{R}} \left| \theta^{\prime} \right|^{2} ,\forall \boldsymbol{x} \in \mathbb{R}^{m} \setminus \left\{ \boldsymbol{0} \right\}.\label{3} \end{align}\tag{11}\] Combining 10 together with the chain rule yields \[\begin{align} \sum\limits_{i=1}^{m} \left| \frac{\partial f\left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right)}{\partial x_{i}} \right|^{2} \leqslant \left( 1+\frac{2}{\sqrt{\lambda}} \right)^{2} \sum\limits_{i=1}^{m} \left| \frac{\partial f}{\partial y_{i}} \left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2} ,\forall \boldsymbol{x} \in \mathbb{R}^{m} \setminus \overline{D_{m}}.\label{4} \end{align}\tag{12}\] Set \[D\triangleq \max \left\{ \left( 1+\frac{1}{\sqrt{\lambda}} \right)^{2} \cdot \sup_{i\geqslant 1} a_{i}^{2}\cdot \sup_{\mathbb{R}} \left| \theta^{\prime} \right|^{2} ,2\left( 1+\frac{2}{\sqrt{\lambda}} \right)^{2} \right\} >0.\] For \(\boldsymbol{x} \in \mathbb{R}^{m} \setminus \overline{D_{m}}\), using 11 and 12 , we get \[\begin{align} &\sum\limits_{k=1}^{m} \left| \partial_{k} F_{2}\left( \boldsymbol{x} \right) \right|^{2} =\sum\limits_{k=1}^{m} \left\vert \frac{\partial \vartheta \left( \xi \left( \boldsymbol{x} \right) \right)}{\partial x_{k}} \cdot f\left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) +\vartheta \left( \xi \left( \boldsymbol{x} \right) \right) \frac{\partial f}{\partial x_{k}} \left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right\vert^{2}\\ &\; \leqslant 2\left| f\left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2} \sum\limits_{k=1}^{m} \left\vert \frac{\partial \vartheta \left( \xi \left( \boldsymbol{x} \right) \right)}{\partial x_{k}} \right\vert^{2} +2\sum\limits_{k=1}^{m} \left\vert \frac{\partial f\left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right)}{\partial x_{k}} \right\vert^{2}\\ &\; \leqslant 8\left( 1+\frac{1}{\sqrt{\lambda}} \right)^{2} \cdot \sup_{i\geqslant 1} a_{i}^{2}\cdot \sup_{\mathbb{R}} \left| \theta^{\prime} \right|^{2} \cdot \left| f\left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2} +2\left( 1+\frac{2}{\sqrt{\lambda}} \right)^{2} \sum\limits_{i=1}^{m} \left| \frac{\partial f}{\partial y_{i}} \left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2}\\ &\; \leqslant D\left( \left| f\left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2} +\sum\limits_{i=1}^{m} \left| \frac{\partial f}{\partial y_{i}} \left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2} \right) .\end{align}\] Making use of 7 and the above inequality, following the same reasoning as in the proof of 8 , one arrives at \[\begin{align} &\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left( \sum\limits_{k=1}^{m} \left| \partial_{k} F_{2}\left( \boldsymbol{x} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x}\\ &\; \leqslant D^{\frac{p}{2}}\int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left( \left| f\left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2} +\sum\limits_{i=1}^{m} \left| \frac{\partial f}{\partial y_{i}} \left( \boldsymbol{y} \left( \boldsymbol{x} \right) \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x}\\ &\; \leqslant D^{\frac{p}{2}}\int_{D_{m}} \left( \left| f\left( \boldsymbol{x} \right) \right|^{2} +\sum\limits_{i=1}^{m} \left| \partial_{i} f\left( \boldsymbol{x} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x}\\ &\; \leqslant \left( 2D \right)^{\frac{p}{2}} \left( \int_{D_{m}} \left| f\left( \boldsymbol{x} \right) \right|^{p} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x} +\int_{D_{m}} \left( \sum\limits_{i=1}^{m} \left| \partial_{i} f\left( \boldsymbol{x} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x} \right),\end{align}\] that is, \[\begin{align} &\left( \int_{\mathbb{R}^{m} \setminus \overline{D_{m}}} \left( \sum\limits_{k=1}^{m} \left| \partial_{k} F_{2}\left( \boldsymbol{x} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}}\\ &\; \leqslant \sqrt{2D}\left[ \left( \int_{D_{m}} \left| f\left( \boldsymbol{x} \right) \right|^{p} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}} +\left( \int_{D_{m}} \left( \sum\limits_{i=1}^{m} \left| \partial_{i} f\left( \boldsymbol{x} \right) \right|^{2} \right)^{\frac{p}{2}} e^{-\frac{1}{2} \sum\limits_{i=1}^{m} x_{i}^{2}}\mathrm{d} \boldsymbol{x} \right)^{\frac{1}{p}} \right].\end{align}\] Now we have \[\begin{align} \left| \left| F_{2} \right| \right|_{p,1,\mathbb{R}^{m} \setminus \overline{D_{m}}} \leqslant \left( \sqrt{2D} +1 \right) \cdot \left| \left| f \right| \right|_{p,1,D_{m}}.\label{5} \end{align}\tag{13}\] Step 3 We are now in a position to define the extension operator.
Set \(F\left( \boldsymbol{x} \right) \triangleq F_{1}\left( \boldsymbol{x} \right) +F_{2}\left( \boldsymbol{x} \right)\). It follows that \[F(\boldsymbol{x})=\left( 1-\vartheta \left( \boldsymbol{x} \right) \right) f\left( \boldsymbol{x} \right) + \vartheta \left( \boldsymbol{x} \right) f\left( \boldsymbol{x} \right) = f(\boldsymbol{x}), ~ \forall \boldsymbol{x} \in \partial D_{m}.\] Then from 6 and 13 we obtain that \(F\left( \boldsymbol{x} \right)\) satisfies 3 with \(C\triangleq \sqrt{2D} +1+\sum_{i=1}^{2N} \widetilde{D_{i}}\).
Set \[E_m f(\boldsymbol{x})= \begin{cases} f(\boldsymbol{x}), & \boldsymbol{x}\in D_m,\\ F(\boldsymbol{x}), & \boldsymbol{x}\in \mathbb{R}^m \backslash \overline{D_{m}}. \end{cases}\] As a result, from 3 , we obtain \[\left| \left| E_m f \right| \right|_{p,1,\mathbb{R}^{m}} \leqslant \left| \left| F \right| \right|_{p,1,\mathbb{R}^{m} \setminus \overline{D_{m}}} +\left| \left| f \right| \right|_{p,1,D_{m}} \leqslant \left( 1+C \right) \left| \left| f \right| \right|_{p,1,D_{m}}.\] Clearly, since \(D,\widetilde{D_{i}}\) are independent of \(m\), \(C+1\) is independent of \(m\), hence we complete the proof of Lemma 1. ◻
Lemma 2. Suppose \(\left\{ f_{n}^{\left( k \right)} \right\}_{k,n=1} \subset L^{p}\left(\ell^{2} ,P \right)\) satisfies \[\begin{align} \int_{\ell^{2}} \left( \sum_{k=1}^{\infty} a_{k}^{2}\left| f_{n}^{\left( k \right)} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P<\infty.\label{1f31f} \end{align}\qquad{(2)}\] If for \(k\in \mathbb{N}\), as \(n\rightarrow+\infty\), \(f_{n}^{\left( k \right)}\) converges weakly to \(f^{\left( k \right)}\) in \(L^{p}\left( \ell^{2} ,P \right)\), then \[\begin{align} \int_{\ell^{2}} \left( \sum_{k=1}^{\infty} a_{k}^{2}\left| f^{\left( k \right)} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P\leqslant \varliminf_{n\rightarrow \infty} \int_{\ell^{2}} \left( \sum_{k=1}^{\infty} a_{k}^{2}\left| f_{n}^{\left( k \right)} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P.\label{1f1f3f1f3} \end{align}\qquad{(3)}\]
Proof. Note that \[\begin{align} &\int_{\ell^{2}} \left( \sum_{k=1}^{\infty} a_{k}^{2}\left| f^{\left( k \right)} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P=\lim_{m\rightarrow \infty} \int_{\ell^{2}} \left( \sum_{k=1}^{m} a_{k}^{2}\left| f^{\left( k \right)} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P\\ &\; \leqslant \varlimsup_{m\rightarrow \infty} \varliminf_{n\rightarrow \infty} \int_{\ell^{2}} \left( \sum_{k=1}^{m} a_{k}^{2}\left| f_{n}^{\left( k \right)} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P\leqslant \varliminf_{n\rightarrow \infty} \int_{\ell^{2}} \left( \sum_{k=1}^{\infty} a_{k}^{2}\left| f_{n}^{\left( k \right)} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P, \end{align}\] where the first equality follows from Levi’s theorem, and the second inequality can be obtained by considering the Banach space naturally defined by the Cartesian product of \(m\) copies of \(L^{p}\left(\ell^{2}, P\right)\) and then using the weak lower semicontinuity of the norm.
This completes the proof of Lemma 2. ◻
Theorem 1. Let \(1\le p <\infty\). There exist bounded linear operators \[E:W^{p,1}\left( B,P \right) \rightarrow W^{p,1}\left( \ell^{2} ,P \right),\] such that \[Ef |_{B}=f, ~ a.e, ~ \forall f\in W^{p,1}\left( B,P\right).\]
Remark 4. Clearly, Theorem 1 can be extended to \(B\left( \boldsymbol{0} ,r \right) ,r>0\), since one can consider constructing the measure \(P_{r}\) using \(ra_{1},ra_{2},\cdots\). Then consider the change of variables \(\boldsymbol{x} =r\boldsymbol{y}\).
Proof. For given \(n\in\mathbb{N}, f\in C_{b}^{\infty}\left( \mathbb{R}^{n} \right)\), by Lemma 1, we know that for any positive integer \(m>\max \left\{ n,N \right\}\), there exist \(F_{m}\in \bigcap_{q=1}^{\infty} W^{q,1}\left( \mathbb{R}^{m} ,\mathcal{N}^{m} \right)\) such that \[F_{m}|_{B_{m}}=f|_{B_{m}},a.e,\] and for every \(q\geqslant 1\), we have \[\begin{align} &\left( \int_{\mathbb{R}^{m}} \left\vert F_{m} \right\vert^{q} \mathrm{d} \mathcal{N}^{m} \right)^{\frac{1}{q}} +\left( \int_{\mathbb{R}^{m}} \left( \sum_{i=1}^{m} a_{i}^{2}\left| \partial_{i} F_{m} \right|^{2} \right)^{\frac{q}{2}} \mathrm{d} \mathcal{N}^{m} \right)^{\frac{1}{q}}\\ &\; \; \; \leqslant C_{q}\left[ \left( \int_{B_{m}} \left| f \right|^{q} \mathrm{d} \mathcal{N}^{m} \right)^{\frac{1}{q}} +\left( \int_{B_{m}} \left( \sum_{i=1}^{n} a_{i}^{2}\left| \partial_{i} f \right|^{2} \right)^{\frac{q}{2}} \mathrm{d} \mathcal{N}^{m} \right)^{\frac{1}{q}} \right] .\end{align}\] This is exactly \[\begin{align} &\left( \int_{\ell^{2}} \left| F_{m} \right|^{q} \mathrm{d} P \right)^{\frac{1}{q}} +\left( \int_{\ell^{2}} \left( \sum_{i=1}^{\infty} a_{i}^{2}\left\vert \partial_{i} F_{m} \right\vert^{2} \right)^{\frac{q}{2}} \mathrm{d} P \right)^{\frac{1}{q}}\\ &\; \; \leqslant C_{q}\left[ \left( \int_{\ell^{2}} \left| f \right|^{q} \chi_{B_{m}\times \left( \ell^{2} \setminus \left\{ 1,2,\cdots ,m \right\} \right)} \mathrm{d} P \right)^{\frac{1}{q}} +\left( \int_{\ell^{2}} \left( \sum_{i=1}^{n} a_{i}^{2}\left| \partial_{i} f \right|^{2} \right)^{\frac{q}{2}} \chi_{B_{m}\times \left( \ell^{2} \setminus \left\{ 1,2,\cdots ,m \right\} \right)} \mathrm{d} P \right)^{\frac{1}{q}} \right]\\ &\; \; \leqslant C_{q}\left[ \left( \int_{\ell^{2}} \left| f \right|^{q} \mathrm{d} P \right)^{\frac{1}{q}} +\left( \int_{\ell^{2}} \left( \sum_{i=1}^{\infty} a_{i}^{2}\left| \partial_{i} f \right|^{2} \right)^{\frac{q}{2}} \mathrm{d} P \right)^{\frac{1}{q}} \right] <\infty .\end{align}\] By the dominated convergence theorem we have \[\begin{align} \varlimsup_{m\rightarrow +\infty} \left| \left| F_{m} \right| \right|_{q,1,\ell^{2}} \leqslant C_{q}\left| \left| f \right| \right|_{q,1,B} ,\forall q\geqslant1.\label{q1} \end{align}\tag{14}\] By [18], we know that \(\left\{ F_{m} \right\}_{m=1}^{\infty} ,\left\{ \partial_{1} F_{m} \right\}_{m=1}^{\infty} ,\left\{ \partial_{2} F_{m} \right\}_{m=1}^{\infty} ,\cdots\) are uniformly integrable and form a bounded sequence in \(L^{p}\left( \ell^{2} ,P \right)\). Hence by [19] and [19], we can use a diagonal argument to obtain a strictly increasing sequence of positive integers \(\left\{ m_{k} \right\}_{k=1}^{\infty}\) such that \(F_{m_{k}},\partial_{1} F_{m_{k}},\cdots\) converge weakly in \(L^{p}\left( \ell^{2} ,P \right)\) to \(F,F^{1},\cdots\).
By the weak lower semicontinuity of the norm and Lemma 2, we have \[\begin{align} \int_{\ell^{2}} \left| F \right|^{p} \mathrm{d} P\leqslant \varliminf_{k\rightarrow \infty} \int_{\ell^{2}} \left\vert F_{m_{k}} \right\vert^{p} \mathrm{d} P,\quad \int_{\ell^{2}} \left( \sum_{i=1}^{\infty} a_{i}^{2}\left| F^{i} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P\leqslant \varliminf_{k\rightarrow \infty} \int_{\ell^{2}} \left( \sum_{i=1}^{\infty} a_{i}^{2}\left| \partial_{i} F_{m_{k}} \right|^{2} \right)^{\frac{p}{2}} \mathrm{d} P.\label{q2} \end{align}\tag{15}\]
For \(j=1,2,\cdots ,\phi \in \mathscr{C}_{b}^{\infty}\), we have \[\begin{align} \int_{\ell^{2}} F\cdot \delta_{j} \phi \mathrm{d} P=\lim_{k\rightarrow \infty} \int_{\ell^{2}} F_{m_{k}}\cdot \delta_{j} \phi \mathrm{d} P=-\lim_{k\rightarrow \infty} \int_{\ell^{2}} \partial_{j} F_{m_{k}}\cdot \phi \mathrm{d} P=-\int_{\ell^{2}} F^{j}\cdot \phi \mathrm{d} P.\label{q3} \end{align}\tag{16}\]
From 14 , 15 , 16 we obtain \(F\in W^{p,1}\left( \ell^{2} ,P \right)\) and \[\left| \left| F \right| \right|_{p,1,\ell^{2}} \leqslant C_{p}\left| \left| f \right| \right|_{p,1,B}.\] From \[F_{m_{k}}|_{B_{m_{k}}}=f|_{B_{m_{k}}}, \text{a.e.}, B\subset B_{m_{k}}\times \ell^{2} \left( \mathbb{N} \setminus \left\{ 1,2,\cdots ,m_{k} \right\} \right), k=1,2,\cdots\] we obtain \(F_{m_{k}}|_{B}=f|_{B}, \text{a.e},k=1,2,\cdots\). Moreover, for any \(\phi \in C_{0,F^{\infty}}^{\infty}\left( B \right)\), we have \[\int_{B} F\phi \mathrm{d} P=\lim_{k\rightarrow \infty} \int_{B} F_{m_{k}}\phi \mathrm{d} P=\int_{B} f\phi \mathrm{d} P,\] then by [14] we get \(F|_{B}=f|_{B}, \text{a.e.}\).
Since \(W^{p,1}\left( B,P \right)\) is separable and the restrictions to \(B\) of all functions in \(\mathscr{C}_{b}^{\infty}\) form a dense subset of \(W^{p,1}\left( B,P \right)\), there exist linearly independent \(\left\{ f_{i} \right\}_{i=1}^{\infty} \subset \mathscr{C}_{b}^{\infty}|_{B}\) such that \(\mathrm{span} \left\{ f_{1}|_{B},f_{2}|_{B},\cdots \right\}\) is dense in \(W^{p,1}\left( B,P \right)\). For each \(f_{i}\) we obtain an extension \(F_{i}\in W^{p,1}\left( \ell^{2} ,P \right)\) by the above procedure. The diagonal argument ensures that the subsequence \(\left\{ m_{k} \right\}\) we extract can be chosen to be the same for all \(i\in \mathbb{N}\).
Define a linear operator \[E:\mathrm{span} \left\{ f_{1}|_{B},f_{2}|_{B},\cdots \right\} \rightarrow W^{p,1}\left( \ell^{2} ,P \right),\] by \[Ef_{i}=F_{i},\quad i=1,2,\cdots.\] Note that for \(f\in \mathrm{span} \left\{ f_{1}|_{B},f_{2}|_{B},\cdots \right\}\), \(Ef\) can be obtained by the same method, hence we also have \[\left| \left| Ef \right| \right|_{p,1,\ell^{2}} \leqslant C_{p}\left| \left| f \right| \right|_{p,1,B} .\] Thus we can naturally extend \(E\) to \[E:W^{p,1}\left( B,P \right) \rightarrow W^{p,1}\left( \ell^{2} ,P \right)\] such that \[\left| \left| Ef \right| \right|_{p,1,\ell^{2}} \leqslant C_{p}\left| \left| f \right| \right|_{p,1,B},\quad \forall f\in W^{p,1}\left( B,P \right) .\]
This completes the proof of Theorem 1. ◻
School of Mathematics, Sichuan University, Chengdu 610064, China. E-mail address: wangzhouzhe@stu.scu.edu.cn.↩︎
School of Mathematics, Sichuan University, Chengdu 610064, China. E-mail address: zhang_xu@scu.edu.cn.↩︎
School of Mathematics, Sichuan University, Chengdu 610064, China. E-mail address: zhaoshiliang@scu.edu.cn.↩︎