Discrete Riesz Potentials in Graphs and Lattices

• Discrete Riesz Potentials are discrete analogues of classical fractional integrals, defined via summation kernels on lattices, trees, and graphs.
• They extend potential theory by adapting mapping properties, boundedness, and sharp exponent conditions from continuous to discrete settings using weighted, Morrey, and variable exponent spaces.
• Applications include energy minimization, polarization problems, and sensor placement, with rigorous analytical extensions via analytic continuation and optimal scaling laws.

Discrete Riesz Potentials are discrete analogues of the classical Riesz potential (fractional integral operators), tailored to settings such as integer lattices, homogeneous trees, or general graphs. They are defined via summation kernels with singularities reflecting their continuous counterparts but adapted for discrete domains. Discrete Riesz potentials play a foundational role in potential theory on infinite graphs, discrete harmonic analysis, and discrete energy minimization, often preserving the structural analogies and key mapping properties observed in the continuous theory.

1. Definitions and Fundamental Kernel Constructions

The discrete Riesz potential operator $I_\alpha$ on $\mathbb{Z}^n$ (with $0 < \alpha < n$) is given by convolution with the kernel $|x-y|^{-(n-\alpha)}$, where $|x-y|$ denotes the Euclidean distance in $\mathbb{Z}^n$:

$$(I_\alpha f)(x) = \sum_{y \in \mathbb{Z}^n} f(y) |x - y|^{-(n-\alpha)}, \qquad x \in \mathbb{Z}^n$$

The definition generalizes to high-dimensional discrete settings, weighted sums, and structured domains such as regular trees or periodic lattices. On homogeneous trees $T_q$, the discrete Riesz potential aligns with the Green kernel of the Laplacian and depends only on the tree distance $d_T(x,y)$:

$$I_1 \mu(x) = \sum_{y \in T} G_T(x, y)\, \mu(y), \qquad G_T(x, y) = \frac{q}{q - 1} q^{-d_T(x, y)}$$

On periodic lattices $\Lambda \subset \mathbb{R}^d$, for $0 < s < d$, the naive sum diverges, necessitating analytic continuation to define the renormalized periodic Riesz kernel $K_s^{\mathrm{per}}(x, y)$ via the Epstein-Hurwitz zeta function (Hardin et al., 2014):

$$K_s^{\mathrm{per}}(x, y) = \zeta_\Lambda(s; x - y) + \frac{2\pi^{d/2}}{(d-s)\Gamma(s/2)}$$

This construction ensures well-defined long-range interactions in discrete energy minimization contexts.

2. Discrete Riesz Potentials and Associated Function Spaces

Mapping properties of discrete Riesz potentials closely mirror the continuous theory but require specialized function spaces and adaptation of analytic machinery:

• $\ell^q(\mathbb{Z}^n)$ spaces: Standard sequence spaces equipped with $q$-norms.
• Discrete Hardy spaces $H^p(\mathbb{Z}^n)$ for $0 < p \leq 1$: Defined via discrete maximal functions or atomic decompositions; key for endpoint estimates (Rocha, 2024, Rocha, 28 Aug 2025).
• Weighted Lebesgue spaces $\ell^p_w(\mathbb{Z}^d)$: Incorporate Muckenhoupt $A_p$ weights; essential for inequalities under non-uniform densities (Hao et al., 2023).
• Discrete Morrey spaces: Norms depend on local averages and variable scaling, enabling fine-grained control over local and global behavior (Hao et al., 2023).
• Variable exponent spaces $\ell^{p(\cdot)}(\mathbb{Z})$: Accommodate spatially varying integrability, used in modern applications and extrapolation theory (Rocha, 2024).

Atomic decompositions in $H^p$ are critical: (p,∞,N_p)-atoms are functions supported on cubes, with controlled $\ell^\infty$ norm and cancellation of moments up to order $N_p = \lfloor n(p^{-1} - 1) \rfloor$.

3. Boundedness, Sharp Exponents, and Maximal Operators

Discrete Riesz potentials exhibit boundedness analogous to the Hardy-Littlewood-Sobolev theorem. For $0 < \alpha < n$, $0 < p \leq 1$, and $1/q = 1/p - \alpha/n$, the mapping

$$I_\alpha: H^p(\mathbb{Z}^n) \to \ell^q(\mathbb{Z}^n)$$

is bounded, with operator norm depending only on $n$, $\alpha$, and $p$ (Rocha, 2024, Rocha, 28 Aug 2025). The atomic approach uses uniform estimates on atoms and kernel decay outside localized cubes, with sharpness dictated by cancellation order and the scaling relation; failure occurs if $p$ is too small or $q \leq 1$ (Rocha, 2024).

Fractional maximal operators

$$(M_\alpha f)(x) = \sup_{Q \ni x} |Q|^{\alpha/n - 1} \sum_{y \in Q} |f(y)|$$

dominate discrete Riesz potentials and admit similar boundedness on weighted spaces for appropriate $A_{p,q}$ classes and scaling relations (Hao et al., 2023). On variable exponent spaces, strong and vector-valued inequalities hold provided Hardy-Littlewood maximal operators are bounded on suitable dual spaces, and the Rubio de Francia algorithm ensures extrapolation of boundedness (Rocha, 2024).

4. Discrete Potential Theory on Trees and Comparative Analysis

On homogeneous trees $T_q$, discrete Riesz potentials are fundamentally connected to the random walk Laplacian and the Green kernel, which decays exponentially in tree distance. Subharmonic functions $u$ on $T_q$ have an associated Riesz measure $\mu_u(x) = \Delta_T u(x)$, and potential theory parallels the Poincaré disk:

• Existence of harmonic majorant $\iff$ finiteness of the first boundary moment $\sum_{y \in T} d_T(y, \partial T)\, \mu_u(y)$ (Boiko et al., 2014).
• Boundary growth control leads to quantitative moment conditions for Riesz measures and sharp characterizations of majorant existence.
• Extended moment conditions via auxiliary functions $\Phi, \Psi$ quantify finer behaviors at the boundary.

This correspondence demonstrates that classical potential-theoretic tools (moment criteria, harmonic majorants, Green functions) transfer directly to discrete settings with appropriate modifications.

5. Discrete Energy Minimization and Polarization Problems

Discrete Riesz potentials underpin a wide array of energy minimization and polarization problems. For a compact set $A \subset \mathbb{R}^p$, the discrete Riesz s-potential for an $N$-point configuration is

$$U_s(y; \omega_N) = \sum_{i=1}^N \frac{1}{|y - x_i|^s}$$

The polarization (Chebyshev constant) $P_s(A; N)$ maximizes the minimal potential over all $N$-point sets. Asymptotic results for rectifiable $A$ and $s \geq d$ yield exact scaling laws:

• For hypersingular case $s > d$, $$P_s(A; N) \sim \sigma_{s, d} [\mathcal{H}_d(A)]^{-s/d} N^{s/d}$$
• Weak-* limit distributions of point configurations converge to densities related to Hausdorff measure and weighting (Borodachov et al., 2016)
• Periodic problems utilize renormalized kernels and analytic continuation, with minimal energy scaling governed by Epstein-Hurwitz zeta functions and explicit constants (Hardin et al., 2014)

Applications span sensor placement, discrepancy theory, polynomial inequalities, and physical models such as generalizations of Thomson's problem.

6. Weighted, Variable-Exponent, and Morrey Space Extensions

The theory of discrete Riesz potentials extends to weighted and variable-exponent settings:

• Weighted Lebesgue estimates: Boundedness of $I_a: \ell^p_w \to \ell^q_w$ for $w$ in $A(p, q)$ as per discrete Muckenhoupt classes (Hao et al., 2023)
• Discrete Morrey spaces: Norms control local averages; boundedness requires reverse-doubling and weight scaling (Hao et al., 2023)
• Variable-exponent $\ell^{p(\cdot)}$ spaces: Main theorem gives $I_a: \ell^{p(\cdot)} \to \ell^{q(\cdot)}$ boundedness whenever Hardy-Littlewood maximal is bounded on the dual space, leveraging weighted inequalities and Rubio de Francia iteration (Rocha, 2024)
• Vector-valued inequalities for fractional maximal operators: Precise control over sequences of functions and their maximal averages (Rocha, 2024)

These generalizations permit fine-scale analysis, accommodate non-uniform density, and support advanced extrapolation and interpolation results.

7. Structural Analogies, Sharpness, and Open Directions

Discrete Riesz potentials retain the harmonic, moment, and potential-theoretic structures of their continuous analogues. The range of exponents is sharp: boundedness breaks down if the cancellation is insufficient or the parameters fall outside critical scaling. All central machinery—maximal operator domination, atomic norm control, kernel summability—translates from Euclidean to discrete models. Extensions to graphs, trees, and weighted or variable-exponent settings continue to be developed, with rigorous criteria for mapping properties and optimal configurations (Boiko et al., 2014, Hao et al., 2023, Rocha, 2024, Rocha, 28 Aug 2025, Borodachov et al., 2016, Hardin et al., 2014, Rocha, 2024).

Research continues into the sharpness of constants, extension to non-lattice graphs, fine-scale moment conditions, and connections with probabilistic and physical models, illustrating the persistent relevance and flexibility of discrete Riesz potential theory across analysis, combinatorics, and applied mathematics.