Magnetic Sobolev Spaces in Analysis

• Magnetic Sobolev spaces are functional spaces that extend classical Sobolev spaces by incorporating magnetic potentials through the covariant derivative.
• They feature both local and fractional formulations, with equivalent norm descriptions that ensure gauge invariance and accommodate nonlocal interactions.
• These spaces underpin advanced analysis in PDEs, spectral theory, and quantum mechanics, with applications in variational methods and discrete magnetic models.

A magnetic Sobolev space is a functional space designed to encode the presence of an external magnetic field in variational and partial differential equations. It systematically extends the notion of classical (fractional or local) Sobolev spaces to accommodate gauge-variant vector fields, replacing the usual gradient by the covariant (magnetic) derivative. This framework appears in mathematical physics, spectral theory, nonlinear analysis, and PDEs with magnetic effects, and has both local (differential) and nonlocal (integral) implementations.

1. Magnetic Sobolev and Fractional Magnetic Sobolev Spaces

Let $\Omega \subset \mathbb{R}^n$ be an open set, and let $A \colon \Omega \to \mathbb{R}^n$ be a vector potential of class $C^1$ (or Lipschitz, as appropriate). The covariant (magnetic) gradient is

$\nabla_A u := \nabla u - i\,A(x)\,u,$

where $u \colon \Omega \to \mathbb{C}$. The first-order magnetic Sobolev space is

$$W^{1,p}_A(\Omega) := \{\,u \in L^p(\Omega;\mathbb{C})\ :\ \nabla_A u \in L^p(\Omega;\mathbb{C}^n)\,\}$$

with the norm

$$\|u\|_{W^{1,p}_A(\Omega)} := \big(\|u\|_{L^p}^p + \|\nabla_A u\|_{L^p}^p\big)^{1/p}.$$

For fractional orders $s\in(0,1)$ the magnetic Gagliardo seminorm generalizes this structure: $$[u]_{W^{s,p}_A(\Omega)} = \left( \iint_{\Omega\times\Omega} \frac{ |u(x) - \mathrm{e}^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^p }{ |x-y|^{n+sp} } dx\,dy \right)^{1/p}.$$ The corresponding fractional space is

$$W^{s,p}_A(\Omega) := \{u\in L^p(\Omega; \mathbb{C}): [u]_{W^{s,p}_A(\Omega)} < \infty\}$$

endowed with the norm

$$\|u\|_{W^{s,p}_A(\Omega)} = \big(\|u\|_{L^p(\Omega)}^p + [u]_{W^{s,p}_A(\Omega)}^p\big)^{1/p}.$$

This definition is directly parallel to classical Gagliardo–Sobolev theory, but the phase factor encodes parallel transport in the gauge $A$ (Bal et al., 2023, Nguyen et al., 2017).

2. Gauge Invariance and Functional Structure

A central property of magnetic Sobolev spaces is gauge invariance. For any smooth scalar $\phi \colon \Omega \to \mathbb{R}$, the transformations

$$A \mapsto A + \nabla\phi, \qquad u \mapsto e^{i\phi}u$$

leave $|\nabla_A u|$ and hence the $W^{1,p}_A$-norm invariant. This ensures that the functional analysis of these spaces depends only on the magnetic field $B = dA$ rather than the specific potential $A$. Norm equivalence, density of $C^\infty_c(\Omega)$, and compactness properties generally parallel those of standard Sobolev spaces, with additional control through the diamagnetic inequality: $$|\nabla|u||(x)| \leq |\nabla_A u(x)| \quad \text{a.e. } x,$$ which also extends to the fractional/nonlocal setting via

$$||u(x)| - |u(y)|| \leq |u(x) - \mathrm{e}^{i(x-y)\cdot A((x+y)/2)}u(y)|$$

(Devillanova et al., 2019, Bal et al., 2023, Bonder et al., 2018).

3. Nonlocal Characterizations, BBM-Type Formulas, and Interpolation

Magnetic Sobolev spaces admit exact nonlocal characterizations extending the Bourgain–Brezis–Mironescu (BBM) paradigm. For $A$ Lipschitz, $1<p<\infty$, consider the nonlocal functional

$$B_n[u] := \iint_{\mathbb{R}^n\times\mathbb{R}^n} \frac{|u(x) - \mathrm{e}^{i(x-y)\cdot A((x+y)/2)}u(y)|^p}{|x-y|^p} \rho_n(|x-y|)\, dx\,dy,$$

where $\rho_n$ is a sequence of mollifiers supported on small distances. Then for $u \in W^{1,p}_A(\mathbb{R}^n)$,

$$\lim_{n\to\infty} B_n[u] = p Q_{n,p} \int_{\mathbb{R}^n} |\nabla_A u(x)|^p dx,$$

where $Q_{n,p}$ is a geometric constant (Nguyen et al., 2017). The nonlocal Gagliardo-type formula with the magnetic phase provides an equivalent norm for $W^{1,p}_A$ as the scale parameter vanishes or as $s \to 1$. This extends to anisotropic settings and to Orlicz–Sobolev variations, where the Orlicz function $G$ replaces the power law (Nguyen et al., 2017, Bonder et al., 2018).

Furthermore, for $0fractional Sobolev spaces admit a real-interpolation structure: $$[L^p(\mathbb{R}^d), W^{1,p}_A(\mathbb{R}^d)]_{s,p} = W^{s,p}_A(\mathbb{R}^d)$$ with equivalence of norms (Nguyen et al., 2019).

4. Boundary Traces, Extension, and Variational Characterizations

Trace theory for magnetic Sobolev spaces is subtle due to the interaction of the gauge and the boundary geometry. For the upper half-space $\mathbb{R}^{d+1}_+$ and $A$ of regularity $C^1$ with bounded field $dA$, the trace of $W^{1,p}_A(\mathbb{R}^{d+1}_+)$ on $t=0$ is precisely $W^{1-1/p,p}_{A^\parallel}(\mathbb{R}^d)$, where $A^\parallel(x)$ is the boundary projection: $$A^\parallel(x) = (A_1,\dots,A_d)(x,0)\in\mathbb{R}^d.$$ The trace operator is continuous and surjective, and there exists a continuous right-inverse (extension operator) constructed via convolution with gauge-twisted Poisson kernels. These results extend to smooth bounded domains via charts and partition of unity, with constants dependent only on the geometry and $\|dA\|_\infty$ (Nguyen et al., 2019).

For the nonlocal (fractional) spaces, traces and extension operators can also be formulated in terms of the corresponding Gagliardo–type seminorms, and extension to the full space is achieved by suitable reflection, zero-extension, or cutoff procedures that respect the magnetic structure.

5. Inequalities, Compactness, and Spectral Theory

Magnetic Sobolev spaces enjoy compact embedding and Poincaré–Wirtinger inequalities analogous to the classical case. For fractional spaces, the magnetic Poincaré inequality on a bounded Lipschitz domain $\Omega$ takes the form

$$\|u\|_{L^q(\Omega)} \le S\left([u]_{W^{s,p}_A(\Omega)} - E_{s,A}^{p,q}\|u\|_{L^q(\Omega)}\right), \quad \forall u \text{ with } d_{s,A}^q(u)\ge \delta \|u\|_{L^q}$$

where $E_{s,A}^{p,q}$ is the infimum of the seminorm to norm ratio and $d_{s,A}^q(u)$ quantifies the distance from the ground state manifold. In punctured domains, no straightforward analog of the additivity of local magnetic seminorms holds, and improved inequalities require additional nonlocal terms on the complement, blending different exponents for the energies (Bal et al., 2023).

Spectral theory for the magnetic fractional Laplacian—defined on $W^{s,2}_A(\Omega)$ via

$$Q_A(u) = \iint_{\Omega\times\Omega} \frac{|u(x) - e^{i(x-y)\cdot A(\frac{x+y}{2})}u(y)|^2}{|x-y|^{n+2s}} dx\,dy$$

—exhibits a discrete spectrum with eigenvalues tending to infinity, obtained via variational Rayleigh–Ritz procedures leveraging the compactness of embeddings (Bal et al., 2023).

6. Extensions: Orlicz–Sobolev, Anisotropic, and Discrete Magnetic Spaces

Magnetic Sobolev spaces admit robust generalizations:

• Magnetic Orlicz–Sobolev spaces use a convex Young function $G$, extending the power-law structure to nonstandard growth laws. Nonlocal modulars and BBM-type formulas characterize limits as $s\uparrow1$ (localization to the magnetic Orlicz–Dirichlet energy) and as $s\to0$ (collapse to the Orlicz norm), with $\Gamma$-convergence of variational problems, and analogues of diamagnetic inequalities (Bonder et al., 2018, Maione et al., 2020).
• Anisotropic variants model preferred directions or convex gauge norms $|x|_K$, with the seminorm

$$[u]_{W^{1,p}_{A,K}} = \left( \int_{\mathbb{R}^N} |\nabla_A u(x)|_{Z^*_K}^p dx \right)^{1/p}$$

and nonlocal BBM limits formulated with anisotropic kernels (Nguyen et al., 2017).

• Discrete magnetic Sobolev spaces arise on graphs with edge signatures of modulus one, yielding discrete covariant gradients and Laplacians defined via edge weights and unitary phases. The resulting framework connects isoperimetric inequalities, Sobolev-type embedding, Cheeger constants, and spectral heat kernel bounds, mirroring continuum phenomena (Chávez-Domínguez, 2020).

7. Applications and Research Directions

Magnetic Sobolev spaces are essential in the analysis of Schrödinger operators and semilinear PDEs with magnetic potentials, spectral theory in the semiclassical limit, nonlinear variational methods, and the theory of nonlocal operators. They provide the function space setting for problems involving fractional and higher-order magnetic Laplacians, play a role in high-precision estimates of minimizer localization, and offer a unifying language for anisotropic diffusion, magnetic field effects in quantum systems, and magnetic perimeters in surface-energy minimization (Devillanova et al., 2019, Fournais et al., 2014, Nguyen et al., 2017).

These spaces admit further exploration in the context of Orlicz growth, limit theorems for nonlocal-to-local convergence, $\Gamma$-convergence of variational problems, extension to manifolds and metric measure spaces, and quantitative analysis of discrete/graph-theoretic magnetic models. The deep interplay between magnetic gauge, boundary phenomena, and nonlocality remains an active area of mathematical analysis.