Physics interpretation of Sobolev space What is the physics interpretation of Sobolev space?
$H^{s,p}:=\left\{u\in L^p(\mathbb{R}^n):\mathcal{F}^{-1}((1+|\cdot|^2)^{s/2}\mathcal{F}(u))\in L^p(\mathbb{R}^n)\right\}$, $s\geq 0,\, 1<p<\infty$ (Sobolev space)
For example, if $n=3,\,s=2,\,p=2$. Is there an interpretation?
Ask this, to find out if the equation $\mathcal{F}^{-1}((1+|\cdot|^2)^{s/2}\mathcal{F}(u))=g$ with $g\in L^2(\mathbb{R}^n)$ has any physical interpretation or application.
 A: The (non-homogeneous) Sobolev spaces $H^{p,s}$ are the subspaces of $L^p$ of functions that admit $s$ weak derivatives belonging to the same Lebesgue space, where a weak derivative is a derivative in the sense of distributions.
The foremost ones of physical significance are the ones $H^s=H^{2,s}$ where the base space is $L^2$. This is due to the fact that $L^2$ is the prototypical Hilbert space for nonrelativistic quantum mechanics.
As a concrete example, $H^2$ is the natural domain of definition, and self-adjointness, of the Laplace operator $-\Delta$, that is the kinetic energy of a quantum particle.
Another example may be given by nonlinear Schrödinger equations of the type
$$i\partial_t \psi = -\Delta \psi + \lvert \psi\rvert^{p-1} \psi \; .$$
Such equations are crucial to describe the effective behavior of condensed matter systems, such as the motion of atoms in a BEC. The existence and uniqueness of solutions to such equations is typically investigated and proved in Sobolev spaces $H^s$.
A: By stretching the question just a little bit, there is in fact a very physical interpretation.
For a Euclidean QFT with say a scalar field $\phi$ in $d$ spacetime dimensions, the latter is typically (when Nelson-Symanzik positivity holds in addition to Osterwalder-Schrader positivity) realized as a random Schwartz distribution in $\mathscr{S}'(\mathbb{R}^d)$. One may then ask for which (weighted inhomogeneous) Besov spaces $B_{p,q}^{s}$ is $\phi$ almost surely in that space. The typical answer is: for all $s<-\Delta_{\phi}$ where $\Delta_{\phi}$ is an important physical quantity, namely the scaling dimension of the field $\phi$. This is just the distributional generalization of the Kolmogorov-Chentsov Theorem.
