Physlib.QuantumMechanics.HilbertSpaces.SpaceD.SobolevSubmodule

For a natural number $d$ and a real number $s$, the Sobolev submodule $H^s$ is the $\mathbb{C}$-linear submodule of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ consisting of those $L^2$ functions $\psi$ whose associated tempered distribution $T_\psi$ satisfies the condition for belonging to the Sobolev space $W^{s, 2}$ (denoted as $H^s$), specifically that $(1 + |\xi|^2)^{s/2} \hat{\psi} \in L^2$ where $\hat{\psi}$ is the Fourier transform.

For any real number $s$ and any element $\psi$ of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$, $\psi$ belongs to the Sobolev submodule $H^s$ if and only if the tempered distribution $T_\psi$ associated with $\psi$ satisfies the Sobolev condition for the space $W^{s, 2}$.

For any real number $s$ and any Schwartz function $g \in \mathcal{S}(\text{Space } d, \mathbb{C})$, the image of $g$ under the continuous linear inclusion $\iota : \mathcal{S}(\text{Space } d, \mathbb{C}) \to L^2(\text{Space } d, \mathbb{C})$ is an element of the Sobolev submodule $H^s$.

For any dimension $d \in \mathbb{N}$ and any real number $s$, the Schwartz submodule $\mathcal{S}$ of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ is contained within the Sobolev submodule $H^s$, that is, $\mathcal{S} \subseteq H^s$.

For any dimension $d \in \mathbb{N}$ and any real number $s$, the Sobolev submodule $H^s$ is a dense subset of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$.

For any dimension $d \in \mathbb{N}$, the Sobolev submodules $H^s$ of the Hilbert space $L^2(\text{Space } d, \mathbb{C})$ are antitone with respect to the regularity index $s \in \mathbb{R}$. This means that for any $s, s' \in \mathbb{R}$, if $s' \leq s$, then $H^s \subseteq H^{s'}$.