Fourier-Symmetric Sobolev Space

• Fourier-Symmetric Sobolev space is a framework that measures function regularity via quadratic, scale-invariant averages symmetric under the Fourier transform.
• It generalizes classical Sobolev spaces by replacing derivative-based analysis with geometric averaging over metric balls, ensuring robust performance in diverse settings.
• This approach enables advanced applications in PDE, harmonic analysis, and approximation theory by providing intrinsic criteria for smoothness and decay.

The Fourier-Symmetric Sobolev space is a modern functional framework designed to encode regularity and decay properties of functions by means of quantities that are explicitly symmetric under the Fourier transform or, more generally, through symmetric averaging over metric balls, spheres, or other geometric constructs. This paradigm extends classical Sobolev space theory—where regularity is measured either by derivatives in the physical domain or by decay of Fourier coefficients—by introducing intrinsic, quadratic, and multiscale expressions that depend solely on the underlying metric and measure. Such spaces admit generalizations to metric measure settings, possess strong symmetry properties, and provide robust tools for harmonic analysis, PDE, and approximation theory.

1. Quadratic Multiscale Characterization

Classical Sobolev spaces $W^{\alpha,p}(\mathbb{R}^n)$ are defined via derivatives (including fractional ones using Fourier multipliers like $|\xi|^\alpha$) or weak differentiability. The Fourier-symmetric characterization, as developed in "A new characterization of Sobolev spaces on $\mathbb{R}^n$" (Alabern et al., 2010), replaces this approach with a single quadratic, multiscale condition. For $f \in L^p(\mathbb{R}^n)$, the defining square function is

$$S(f)(x)^2 = \int_0^{\infty} \left|\frac{f_B(x,t) - f(x)}{t}\right|^2 \frac{dt}{t}$$

where $f_B(x,t)$ denotes the average of $f$ over the ball $B(x,t)$ centered at $x$ with radius $t$. For higher or fractional order ($\alpha > 0$), one systematically subtracts the relevant Taylor polynomial up to integer $N = \lfloor \alpha/2 \rfloor$:

$$S_{\alpha}(f, g_1, \ldots, g_N)(x)^2 = \int_0^{\infty} \left| \frac{1}{|B(x, t)|} \int_{B(x, t)} \frac{f(y) - f(x) - g_1(x)|y - x|^2 - \ldots - g_N(x) |y - x|^{2N}}{t^\alpha} dy \right|^2 \frac{dt}{t}$$

This square function is fundamentally scale-invariant, quadratic, and captures smoothness without any reference to explicit derivatives.

Key Equivalence

The central equivalence is

$$\left\| S_\alpha(f, \cdots) \right\|_p \simeq \left\| (-\Delta)^{\alpha/2} f \right\|_p$$

This equivalence provides a metric- and measure-dependent method to detect membership in $W^{\alpha,p}$.

2. Symmetry and Cancellation Properties

Euclidean symmetry is vital: radial averaging and rotational invariance guarantee that odd-order Taylor terms integrate to zero over spheres or balls. This corresponds in Fourier language to the invariance of spherical functions under the transform and ensures that the multiscale expressions become insensitive to lower-degree polynomials. Differentiating inside metric averages never introduces signatures of unwanted lower order terms due to this symmetry. Such properties are deeply tied to the notion of "Fourier symmetry," where both the physical and frequency domains are treated equivalently.

3. Extension to Metric Measure Spaces

The quadratic characterization utilizes only the metric $d(x,y)$ and the underlying measure (Lebesgue or more general). Thus, one can transplant the definition to a doubling metric measure space $(X,d,\mu)$:

$$S_\alpha(f, \cdots)(x) = \left[ \int_0^D \left| \frac{1}{\mu(B(x,t))} \int_{B(x,t)} \frac{f(y) - f(x) - \text{higher order corrections}}{t^\alpha} d\mu(y) \right|^2 \frac{dt}{t} \right]^{1/2}$$

Membership in the Sobolev space is then characterized by the $L^p(\mu)$ norm of $S_\alpha(f, \cdots)$. This approach does not rely on a linear structure and supports analysis in fractal spaces, manifolds with rough geometry, or any metric space admitting a doubling measure.

4. Applications: PDE, Harmonic Analysis, and Potential Theory

The Fourier-symmetric characterization streamlines a variety of problems:

• Singular integral operators: Quadratic square functions are instrumental in establishing $L^p$ bounds.
• PDE theory: Regularity, integrability, and stability of solutions can be analyzed without explicit differentiability.
• Embeddings and inequalities: The theory yields new proofs of Poincaré and Sobolev embedding theorems relying purely on metric properties, facilitating extensions to non-Euclidean settings.
• Harmonic analysis: Bridge Fourier multiplier theory—where fractional Laplacians are realized in the frequency domain—with geometric averaging processes in the physical domain.

5. Examples of Square Functions for Various Orders

Order $\alpha$	Correction Term(s)	Square Function $S_\alpha$
$\alpha=1$	none	$\int_0^\infty |(f_B(x,t) - f(x))/t|^2 \frac{dt}{t}$
$0<\alpha<2$	none	$$\int_0^\infty |(1/|B(x,t)|) \int_{B(x,t)} (f(y)-f(x))/t^\alpha dy|^2 \frac{dt}{t}$$
$\alpha=2$	$-(1/2n)(\Delta f)_B(x,t)|y-x|^2$	$$\int_0^\infty |(1/|B(x,t)|) \int_{B(x,t)} [f(y)-f(x)-...]/t^2 dy|^2 \frac{dt}{t}$$

The "correction terms" are determined such that the expression vanishes for polynomials of degree lower than the order, which ensures scale invariance and avoids detection of artificial smoothness.

6. Broader Impact and Transplantation

The Fourier-symmetric Sobolev space theory transcends classical differentiability-based approaches, enabling:

• Analysis on fractals and spaces without smooth structure: Regularity notions defined via scale-invariant averaging.
• Robust numerical approaches: Quadratic criteria avoiding explicit derivative computation allow for stable discretization in numerical schemes.
• Connection with concentration compactness and loss of compactness: As analyzed in (Bahouri et al., 2013), focusing on radially symmetric (Fourier-symmetric) profiles clarifies the mechanism of bubbling in critical embeddings.
• Unified treatment of regularity: Both integer and fractional smoothness orders can be handled within a single framework.

7. Summary

The Fourier-Symmetric Sobolev space, characterized via quadratic multiscale expressions that depend solely on the metric and measure, provides a robust and unifying theory for function regularity. It is symmetric with respect to the Fourier transform and can be extended to general metric measure spaces. Notably, it recovers classical results for $\mathbb{R}^n$ and equips analysts with tools for general geometric settings, making it persuasive for applications in harmonic analysis, geometry, PDE, and beyond.