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In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that there are classical regularity results based on these spaces.

For example, for each $p\in [1,\infty)$, the weighted $L^p_\mu(\mathbb{R}^d)$ space on $\mathbb{R}^d$ is defined to be the set of Lebesgue measurable functions such that $f\omega_\mu(x)\in L^p(\mathbb{R}^n)$, i.e., $$\|f\|_{L^p_\mu}=\int_{\mathbb{R}^d}|f|^p\omega^p_\mu(x)\,dx< \infty,$$ where $\omega_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for $\mu>0$, and the weighted sobolev space $W^{1,p}_\mu(\mathbb{R}^d)$ is defined to be the space of functions such that $u\omega_\mu\in L^p(\mathbb{R}^n)$ and $\partial_{x_i} u\omega_\mu\in L^p(\mathbb{R}^n)$, where $\partial_{x_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}_\mu(\mathbb{R}^d)$ such that $\partial_{x_ix_j}u\omega_\mu\in L^p$ for all $i,j$.

I am interested in a reference on the embedding properties between spaces of different orders. For example, it is pointed out (without a proof) in the book that the injection $$ W^{2,p}_\mu\hookrightarrow W^{1,p}_\nu \tag{1}$$ with $\nu>\mu$ is compact. Could you provide a reference for a proof of this statement? Does it follow from the results for the classical sobolev space?

Note that Corollary 3.3 in Hooton's paper implies the injection $W^{2,p}_\mu\hookrightarrow W^{1,p}_\mu \tag{2}$ is not compact.

In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that there are classical regularity results based on these spaces.

For example, for each $p\in [1,\infty)$, the weighted $L^p_\mu(\mathbb{R}^d)$ space on $\mathbb{R}^d$ is defined to be the set of Lebesgue measurable functions such that $f\omega_\mu(x)\in L^p(\mathbb{R}^n)$, i.e., $$\|f\|_{L^p_\mu}=\int_{\mathbb{R}^d}|f|^p\omega^p_\mu(x)\,dx< \infty,$$ where $\omega_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for $\mu>0$, and the weighted sobolev space $W^{1,p}_\mu(\mathbb{R}^d)$ is defined to be the space of functions such that $u\omega_\mu\in L^p(\mathbb{R}^n)$ and $\partial_{x_i} u\omega_\mu\in L^p(\mathbb{R}^n)$, where $\partial_{x_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}_\mu(\mathbb{R}^d)$ such that $\partial_{x_ix_j}u\omega_\mu\in L^p$ for all $i,j$.

I am interested in a reference on the embedding properties between spaces of different orders. For example, it is pointed out (without a proof) in the book that the injection $$ W^{2,p}_\mu\hookrightarrow W^{1,p}_\nu \tag{1}$$ with $\nu>\mu$ is compact. Could you briefly outline or provide a reference for a proof of this statement? Does it follow from the results for the classical sobolev space?

Note that Corollary 3.3 in Hooton's paper implies the injection $W^{2,p}_\mu\hookrightarrow W^{1,p}_\mu \tag{2}$ is not compact.

In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that there are classical regularity results based on these spaces.

For example, for each $p\in [1,\infty)$, the weighted $L^p_\mu(\mathbb{R}^d)$ space on $\mathbb{R}^d$ is defined to be the set of Lebesgue measurable functions such that $f\omega_\mu(x)\in L^p(\mathbb{R}^n)$, i.e., $$\|f\|_{L^p_\mu}=\int_{\mathbb{R}^d}|f|^p\omega^p_\mu(x)\,dx< \infty,$$ where $\omega_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for some $\mu>0$, and the weighted sobolev space $W^{1,p}_\mu(\mathbb{R}^d)$ is defined to be the space of functions such that $u\omega_\mu\in L^p(\mathbb{R}^n)$ and $\partial_{x_i} u\omega_\mu\in L^p(\mathbb{R}^n)$, where $\partial_{x_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}_\mu(\mathbb{R}^d)$ such that $\partial_{x_ix_j}u\omega_\mu\in L^p$ for all $i,j$.

I am interested in a reference on the embedding properties between spaces of different orders. For example, it is pointed out in the book that the injection $$ W^{2,p}_\mu\hookrightarrow W^{1,p}_\nu \tag{1}$$ with $\nu>\mu$ is compact. Does it follow from the results for the classical sobolev space?

Note that Corollary 3.3 in Hooton's paper implies the injection $W^{2,p}_\mu\hookrightarrow W^{1,p}_\mu \tag{2}$ is not compact.

In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that there are classical regularity results based on these spaces.

For example, for each $p\in [1,\infty)$, the weighted $L^p_\mu(\mathbb{R}^d)$ space on $\mathbb{R}^d$ is defined to be the set of Lebesgue measurable functions such that $f\omega_\mu(x)\in L^p(\mathbb{R}^n)$, i.e., $$\|f\|_{L^p_\mu}=\int_{\mathbb{R}^d}|f|^p\omega^p_\mu(x)\,dx< \infty,$$ where $\omega_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for $\mu>0$, and the weighted sobolev space $W^{1,p}_\mu(\mathbb{R}^d)$ is defined to be the space of functions such that $u\omega_\mu\in L^p(\mathbb{R}^n)$ and $\partial_{x_i} u\omega_\mu\in L^p(\mathbb{R}^n)$, where $\partial_{x_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}_\mu(\mathbb{R}^d)$ such that $\partial_{x_ix_j}u\omega_\mu\in L^p$ for all $i,j$.

I am interested in a reference on the embedding properties between spaces of different orders. For example, it is pointed out (without a proof) in the book that the injection $$ W^{2,p}_\mu\hookrightarrow W^{1,p}_\nu \tag{1}$$ with $\nu>\mu$ is compact. Could you provide a reference for a proof of this statement? Does it follow from the results for the classical sobolev space?

Note that Corollary 3.3 in Hooton's paper implies the injection $W^{2,p}_\mu\hookrightarrow W^{1,p}_\mu \tag{2}$ is not compact.

In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that there are classical regularity results based on these spaces.

For example, for each $p\in [1,\infty)$, the weighted $L^p_\mu(\mathbb{R}^d)$ space on $\mathbb{R}^d$ is defined to be the set of Lebesgue measurable functions such that $f\omega_\mu(x)\in L^p(\mathbb{R}^n)$, i.e., $$\|f\|_{L^p_\mu}=\int_{\mathbb{R}^d}|f|^p\omega^p_\mu(x)\,dx< \infty,$$ where $\omega_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for some $\mu>0$, and the weighted sobolev space $W^{1,p}_\mu(\mathbb{R}^d)$ is defined to be the space of functions such that $u\omega_\mu\in L^p(\mathbb{R}^n)$ and $\partial_{x_i} u\omega_\mu\in L^p(\mathbb{R}^n)$, where $\partial_{x_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}_\mu(\mathbb{R}^d)$ such that $\partial_{x_ix_j}u\omega_\mu\in L^p$ for all $i,j$.

I am interested in a reference on the embedding properties between spaces of different orders. For example, it is pointed out in the book that the injection $$ W^{2,p}_\mu\hookrightarrow W^{1,p}_\nu \tag{1}$$ with $\nu>\mu$ is compact. Does it follow from the results for the classical sobolev space? Is the injection $W^{2,p}_\mu\hookrightarrow W^{1,p}_\mu \tag{2}$ compact?

In the book by Bensoussan and Lions, they introduce the weighted spaces with exponentially decaying weights to study elliptic equations with bounded coefficients on the whole space $\mathbb{R}^n$. They mentioned that there are classical regularity results based on these spaces.

For example, for each $p\in [1,\infty)$, the weighted $L^p_\mu(\mathbb{R}^d)$ space on $\mathbb{R}^d$ is defined to be the set of Lebesgue measurable functions such that $f\omega_\mu(x)\in L^p(\mathbb{R}^n)$, i.e., $$\|f\|_{L^p_\mu}=\int_{\mathbb{R}^d}|f|^p\omega^p_\mu(x)\,dx< \infty,$$ where $\omega_\mu(x)=\exp(-\mu\sqrt{1+|x|^2})$ for some $\mu>0$, and the weighted sobolev space $W^{1,p}_\mu(\mathbb{R}^d)$ is defined to be the space of functions such that $u\omega_\mu\in L^p(\mathbb{R}^n)$ and $\partial_{x_i} u\omega_\mu\in L^p(\mathbb{R}^n)$, where $\partial_{x_i}$ denotes the weak derivative in the distribution sense. Similarly we define the high-order sobolev space $W^{2,p}_\mu(\mathbb{R}^d)$ such that $\partial_{x_ix_j}u\omega_\mu\in L^p$ for all $i,j$.

I am interested in a reference on the embedding properties between spaces of different orders. For example, it is pointed out in the book that the injection $$ W^{2,p}_\mu\hookrightarrow W^{1,p}_\nu \tag{1}$$ with $\nu>\mu$ is compact. Does it follow from the results for the classical sobolev space?

Note that Corollary 3.3 in Hooton's paper implies the injection $W^{2,p}_\mu\hookrightarrow W^{1,p}_\mu \tag{2}$ is not compact.