I am looking for a reference for the space $\mathcal{D}^{1, 2}(\mathbb{R}^n), n\ge 1$. I am interested on the exact definition of this space.
Also, having a sequence of functions $(f_n)_n$ bounded in $\mathcal{D}^{1, 2}(\mathbb{R}^n)$, you can say that there exists $f\in \mathcal{D}^{1, 2}(\mathbb R)$ such that $f_n \to f$? If yes, in which spaces do you have strong and weak convergences (if any)? I would be interested, in the strong convergence in $L^\infty_{loc}(\mathbb{R}^n)$.
Anyone could help?
The space $\mathcal{D}^{1, 2}(\mathbb{R}^n)$ refers to the space of functions with weak first-order derivatives that are square integrable. More precisely, $\mathcal{D}^{1, 2}(\mathbb{R}^n)$ is defined as the set of all functions $u: \mathbb{R}^n \to \mathbb{R}$ such that $u$ and its weak derivatives $\partial_{x_i}u$ (partial derivatives with respect to each variable) are square integrable, i.e., $u, \partial_{x_i}u \in L^2(\mathbb{R}^n)$ for $i = 1, 2, \ldots, n$. Here, $L^2(\mathbb{R}^n)$ denotes the space of square integrable functions on $\mathbb{R}^n$. Regarding the convergence of a sequence $(f_n)_n$ bounded in $\mathcal{D}^{1, 2}(\mathbb{R}^n)$, it is not guaranteed that there exists a limit function $f \in \mathcal{D}^{1, 2}(\mathbb{R}^n)$. The space $\mathcal{D}^{1, 2}(\mathbb{R}^n)$ does not possess the property of compactness, which is necessary to guarantee the existence of a convergent subsequence. However, under certain additional assumptions, you can obtain convergence results. One such assumption is the equicontinuity of the sequence $(f_n)_n$, which means that the functions in the sequence share a common modulus of continuity. If the sequence $(f_n)_n$ is equicontinuous and bounded in $\mathcal{D}^{1, 2}(\mathbb{R}^n)$, then by the Arzelà–Ascoli theorem, you can extract a subsequence that converges uniformly on compact subsets of $\mathbb{R}^n$. The limit function $f$ obtained in this case will belong to the space $\mathcal{D}^{1, 2}(\mathbb{R}^n)$ as well. However, it's important to note that uniform convergence on compact sets does not imply convergence in $L^\infty_{loc}(\mathbb{R}^n)$, which is the local $L^\infty$ space. The space $L^\infty_{loc}(\mathbb{R}^n)$ consists of functions that are essentially bounded on every compact subset of $\mathbb{R}^n$. It is possible to have uniform convergence on compact subsets without convergence in the $L^\infty_{loc}(\mathbb{R}^n)$ sense.
The strong convergence in $L^\infty_{loc}(\mathbb{R}^n)$ is not guaranteed for a bounded sequence in $\mathcal{D}^{1, 2}(\mathbb{R}^n)$. The convergence properties depend on additional assumptions such as equicontinuity and compactness, as discussed above.