Riesz Potentials in Analysis & PDEs

• Riesz Potentials are integral operators defined via convolution with the kernel |x-y|^(α-d), generalizing classical Newtonian and Coulomb potentials.
• They provide sharp mapping properties between L^p and Sobolev spaces, underpinning results such as the Hardy–Littlewood–Sobolev theorem and fractional embeddings.
• Their extension to metric measure spaces and non-smooth domains facilitates advanced applications in PDE analysis, numerical methods, and potential theory.

A Riesz potential is a classical integral operator fundamental in harmonic analysis, potential theory, functional inequalities, and the study of various partial differential equations. For a locally integrable function or measure on Euclidean or broader metric spaces, the Riesz potential of order $\alpha$, $0 < \alpha < d$, generalizes the Newtonian and Coulomb potentials, extends to non-Euclidean contexts, and provides sharp mapping properties and a deep connection to capacities, regularity theory, and spectral analysis. Its unifying kernel, $|x-y|^{\alpha - d}$ (or variants), confers powerful convolution, multiplier, and energy structures, persisting across settings from classical $\mathbb{R}^n$ to stratified Lie groups, metric measure spaces, non-smooth domains, and even spaces with nonmeasurable functions.

1. Core Definition and Fundamental Properties

Let $f$ be a (locally) integrable function on $\mathbb{R}^d$ and $0 < \alpha < d$. The Riesz potential $I_\alpha f$ is defined by convolution: $$I_\alpha f(x) = C_{d,\alpha} \int_{\mathbb{R}^d} |x-y|^{\alpha-d} f(y) \, dy,$$ with the normalization (for $d \ge 1$): $$C_{d,\alpha} = \frac{\Gamma\left(\frac{d-\alpha}{2}\right)}{\pi^{d/2} 2^\alpha \Gamma\left(\frac{\alpha}{2}\right)}.$$ This kernel is homogeneous of degree $\alpha-d$, and $I_\alpha$ is the (left-)inverse of the fractional Laplacian $(-\Delta)^{\alpha/2}$ in the sense that $$\widehat{I_\alpha f}(\xi) = |\xi|^{-\alpha} \widehat{f}(\xi)$$ (Herrmann, 2013, Rubin, 2011, Nuutinen et al., 2015).

Specializations:

• For $\alpha = 2$ and $d \ge 3$, $I_2$ gives the Newton/Coulomb potential fundamental to classical potential theory (Herrmann, 2013, Burchard et al., 2020).
• For $\alpha = d$, one recovers logarithmic potentials: $u(x) = \int \log \frac{1}{|x-y|} f(y) dy$ (Pritsker et al., 2013, Cufí et al., 2016).
• On metric measure spaces, Riesz potentials generalize to operators of the form $$I_\alpha f(x) = \int_X \frac{f(y)}{\mu(B(x, d(x, y)))^{1-\alpha}} d\mu(y)$$ where $(X,d,\mu)$ is metric and $\mu$ doubling or upper-doubling (Nuutinen et al., 2015, Iaffei et al., 2013).

Fourier multiplier characterization, scaling properties, and convolutional structure enable direct transfer to functional and PDE contexts.

2. Mapping Properties, Regularity, and Sharp Inequalities

Riesz potentials map $L^p$ spaces into $L^q$ or Sobolev spaces under sharp ranges:

• Hardy–Littlewood–Sobolev Theorem: For $1 < p < \frac{d}{\alpha}$, $q = \frac{dp}{d-\alpha p}$,

$$I_\alpha : L^p(\mathbb{R}^d) \to L^q(\mathbb{R}^d) \quad \text{is bounded}$$

and the bound is sharp (Garg et al., 2014, Breit et al., 14 Jan 2025).

• Fractional Sobolev Embedding: $I_\alpha$ is equivalent to the mapping $L^2 \to H^{\alpha/2}$ on flat or smooth manifolds, and in polygonal boundaries with singularities retains the same Sobolev shift (Claeys et al., 2021).
• Endpoint and Beyond $L^1$ Barrier: At $p=1$, only the weak-type estimate holds in general:

$$I_\alpha : L^1(\mathbb{R}^d) \to L^{d/(d-\alpha),\infty}(\mathbb{R}^d),$$

but for vector-valued $F$ satisfying a linear "co-canceling" constraint (e.g., divergence-free), a strong estimate is restored (Breit et al., 14 Jan 2025, Breit et al., 6 Dec 2025):

$$I_\alpha : L^1(\mathbb{R}^n) \cap \{\operatorname{div} F = 0\} \to L^{n/(n-\alpha),1}.$$

• Regularity Estimates: For $p > d/\alpha$, $I_\alpha$ confers nearly-Lipschitz regularity. For the critical case $p = d/\alpha$, one obtains a log-Lipschitz modulus; Poisson’s equation solutions inherit optimal Hölder or log-Hölder continuity (Garg et al., 2014).

In geometric settings (e.g., polygonal $\partial\Omega$), the gain in Sobolev regularity persists up to singularities, as proved using Mellin transform methods (Claeys et al., 2021).

3. Riesz Potentials in Metric Measure and Quasi-Metric Spaces

The classical definition extends via: $$I_\alpha f(x) = \int_X \frac{f(y)}{\mu(B(x, d(x,y)))^{1-\alpha}} d\mu(y)$$ for $0< \alpha <1$ with $\mu$ doubling or "upper doubling" (Nuutinen et al., 2015, Iaffei et al., 2013). Mapping theorems hold under sharp conditions:

• Upper-doubling of the measure and lower-type growth for the dominating function $\lambda$ ensure boundedness $L^p \to L^q$ (Iaffei et al., 2013).
• For two-component metric spaces joined at a point and with different Ahlfors exponents, Riesz potentials still respect the local dimension with precise $L^p \to L^q$ mapping (Iaffei et al., 2013).
• Grand Morrey and Morrey–type spaces (even nonhomogeneous) admit Riesz bounds provided the classical bound holds uniformly as exponents vary, via a reduction lemma (Kokilashvili et al., 2012).

Riesz capacity on such spaces, defined via the infimum of $\|f\|_{L^p}^p$ over $f$ with $I_\alpha f \ge 1$ on $E$, has monotonicity, countable subadditivity, outer regularity, and links to modified Hausdorff content (Nuutinen et al., 2015).

4. Role in Potential Theory, Capacities, and Fine Properties

Riesz potentials underpin the notion of Riesz energy and Riesz $p$-capacity, and are central to:

• Energy minimization: The functionals $$\mathcal{E}_\alpha(E) = \iint_{E \times E} |x-y|^{\alpha-d} dx\,dy$$ lead to notions of equilibrium measures and uniqueness of energy minimizers (balls) with established quantitative stability (Burchard et al., 2020).
• Capacities and Exceptional Sets: Differentiability in the capacity sense for Riesz potentials is characterized by vanishing $(d-1)$-density and existence of principal values of the vector Riesz transform; exceptions are rare (sets of zero $C^1$-harmonic capacity or appropriate Hausdorff measures in dimension two) (Cufí et al., 2016).
• Hausdorff Content and Nonmeasurable Functions: Using Choquet-type integrals with respect to Hausdorff content, Riesz potentials are extended to arbitrary functions, removing measurability obstacles and establishing sharp $N\!L^p\rightarrow N\!L^q$ bounds (Harjulehto et al., 2024).

The fine structure of exceptional sets and links to analytic, Wiener, and $C^1$-harmonic capacity has deep implications for function regularity and singular integrals (Cufí et al., 2016).

5. Applications: PDEs, Inversion, Numerical Analysis, and Physics

Linear and Nonlinear PDEs:

• gradient bounds for parabolic or quasilinear equations via caloric or "intrinsic" Riesz potentials, with regularity inherited from the data via potential estimates (Kuusi et al., 2013).
• For fractional Laplacians, $(-\Delta)^s$, the Riesz potential $(-\Delta)^{-s}$ provides explicit inverses and is fundamental to nonlocal operator theory and fractional boundary value problems. Explicit formulas exist for its action on broad classes of orthogonal polynomials and Bessel functions, directly supporting spectral schemes (Gutleb et al., 2023).

Inversion and Integral Geometry:

• The Riesz potential is inverted by explicit convolution and wavelet-integral formulas, and via the Radon–John transform identity, serves as the analytic core for tomographic reconstruction (Rubin, 2011).

Numerical and Computational methods:

• Riesz potentials supply sparse, spectrally accurate bases for high-dimensional fractional PDEs. Closed-form expressions in Jacobi, Hermite, Laguerre, and Zernike bases facilitate fast solvers (Gutleb et al., 2023).

Physics and Generalized Potentials:

• Fractional Riesz potentials interpolate smoothly between Coulomb and Yukawa (screened) potentials as fractional exponent varies, unifying various cluster-energy contributions in nuclear shell and pairing models. Macroscopic energy scalings adjust continuously via the choice of $\alpha$ (Herrmann, 2013).

6. Extensions: Generalized Spaces, Trace Principles, and Co-Canceling Constraints

Riesz potentials extend to broad function space frameworks and operator classes:

• Herz and Lorentz–Herz spaces: Trace inequalities, Gagliardo–Nirenberg–Sobolev inequalities, and optimal embeddings for Riesz potentials are fully characterized, with exact parameter constraints for optimality (Bhat et al., 2024).
• Co-canceling constraints: In the presence of a co-canceling differential constraint (e.g., divergence-free), sharp $L^1\to L^{n/(n-\alpha),1}$ estimates, and more generally, bounds in Orlicz and Lorentz–Zygmund targets are restored, filling the classical $p=1$ gap (Breit et al., 14 Jan 2025, Breit et al., 6 Dec 2025).
• General metric measure spaces: Under upper (not necessarily exact) doubling, all classical $L^p \to L^q$ mapping and capacity-theory persists (Iaffei et al., 2013, Nuutinen et al., 2015).

7. Specialized and Non-Euclidean Riesz Potentials

• Polygonal and Singular Geometries: Even on non-smooth domains, e.g., polygons with corners, the Sobolev-gain (from $L^2$ to $H^{1/2}$) is preserved, with the Mellin transform providing symbol calculus adapted to corner singularities (Claeys et al., 2021).
• Lie Groups and Carnot groups: On Heisenberg-type (H-type) groups, Riesz potentials respect the intrinsic nonisotropic geometry, have explicit integral representations in terms of the homogeneous gauge, and retain superposition principles for $p$-sub(super)harmonicity (Garofalo et al., 2010).

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By acting as analytical bridges across classical analysis, geometric measure theory, PDEs, and numerical mathematics, Riesz potentials and their extensions remain central—providing both sharp functional estimates and nuanced insight into the regularity and structure of functions, measures, and solutions in diverse mathematical and applied contexts.