Discrete Riesz Potential in Lattice Analysis

• Discrete Riesz potential is a fractional integral operator defined on ℤⁿ using a kernel |i−j|^(−(n−α)) for 0<α<n, mirroring continuous Riesz potentials.
• It demonstrates boundedness from discrete Hardy spaces Hᵖ to Hᑫ with the sharp scaling relation 1/q = 1/p − α/n, extending classical results.
• Atomic decomposition, maximal function estimates, and exotic endpoint treatments illustrate its adaptation of continuous fractional integration methods to discrete settings.

The discrete Riesz potential is a fractional integral operator defined on discrete structures, most prominently on the lattice $\mathbb{Z}^n$, by summing against the kernel $|i-j|^{-(n-\alpha)}$ for $0<\alpha<n$. In the lattice setting, it serves as a discrete analogue of the continuous Riesz potential on $\mathbb{R}^n$, and its analysis is closely tied to discrete Hardy spaces, discrete Lebesgue and Morrey spaces, variable exponent sequence spaces, and fractional maximal operators (Rocha, 28 Aug 2025). Related literature also uses closely connected terminology for endpoint operators arising from fractional discrete Laplacians, for Green-potential constructions on trees, and for pairwise interaction functionals such as discrete Riesz energy; these usages are linked by kernel-based summation, but they address distinct analytical and geometric questions (Li et al., 1 Mar 2026).

1. Lattice definition and core operator-theoretic setting

On $\mathbb{Z}^n$, the discrete Riesz potential $I_\alpha$ is defined for $0<\alpha<n$ by

$$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$

for $j \in \mathbb{Z}^n$ (Rocha, 28 Aug 2025). In dimension one, the same definition appears in the form

$$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$

with $|i-j|^{-(n-\alpha)}$0 (Rocha, 2024).

This operator is presented as the discrete analogue of the continuous Riesz potential

$|i-j|^{-(n-\alpha)}$1

and the correspondence extends to the parameter relation

$|i-j|^{-(n-\alpha)}$2

described as the classical scaling relation (Rocha, 28 Aug 2025). In the discrete Hardy-space framework, the relevant source space is

$|i-j|^{-(n-\alpha)}$3

where $|i-j|^{-(n-\alpha)}$4, $|i-j|^{-(n-\alpha)}$5, and $|i-j|^{-(n-\alpha)}$6 is a suitable discrete scaling (Rocha, 28 Aug 2025).

The discrete Hardy-space formulation is necessary because, as stated in the comparative discussion with the continuous theory, $|i-j|^{-(n-\alpha)}$7 and $|i-j|^{-(n-\alpha)}$8 are not comparable for $|i-j|^{-(n-\alpha)}$9, necessitating atomic analysis (Rocha, 28 Aug 2025). This places the operator within the standard framework of fractional integration in harmonic analysis, but with summation, counting arguments, and discrete cubes replacing integration, measure estimates, and Euclidean balls.

2. Boundedness on discrete Hardy spaces

A central result is the $0<\alpha<n$0 boundedness theorem. For $0<\alpha<n$1, the discrete Riesz potential $0<\alpha<n$2 is a bounded operator

$0<\alpha<n$3

whenever $0<\alpha<n$4 and

$0<\alpha<n$5

that is, there exists a constant $0<\alpha<n$6, independent of $0<\alpha<n$7, such that

$0<\alpha<n$8

The paper emphasizes that this holds on the full range $0<\alpha<n$9, with no restriction on the dimension other than $\mathbb{R}^n$0 (Rocha, 28 Aug 2025).

This extends earlier work that established boundedness only for $\mathbb{R}^n$1, and the significance of the extension is stated explicitly: it allows the theory and tools of Hardy spaces in the discrete setting to be fully parallel to the continuous setting, including the endpoint $\mathbb{R}^n$2 (Rocha, 28 Aug 2025). The same source states that the extension avoids the lower bound $\mathbb{R}^n$3, which had been a barrier in earlier work, and attributes this to the use of the maximal characterization rather than only molecular decomposition tied to Riesz transforms.

A related earlier result proved that

$\mathbb{R}^n$4

for $\mathbb{R}^n$5, $\mathbb{R}^n$6, and $\mathbb{R}^n$7, giving the boundedness

$\mathbb{R}^n$8

as the natural discrete fractional integration estimate on $\mathbb{R}^n$9 (Rocha, 2024). Taken together, these results indicate a progression from Hardy-to-$\mathbb{Z}^n$0 control to Hardy-to-Hardy control under the same scaling.

3. Atomic decomposition, maximal functions, and proof architecture

The Hardy-space theory used for discrete Riesz potentials is built on atomic decomposition. Every $\mathbb{Z}^n$1 for $\mathbb{Z}^n$2 can be represented as

$\mathbb{Z}^n$3

with $\mathbb{Z}^n$4, where the atoms satisfy uniform size and moment conditions (Rocha, 28 Aug 2025). The atomic size and moment condition are stated as follows: support in a cube $\mathbb{Z}^n$5, $\mathbb{Z}^n$6, and vanishing moments up to order

$\mathbb{Z}^n$7

(Rocha, 28 Aug 2025).

For the earlier $\mathbb{Z}^n$8 theorem, the atoms are described as $\mathbb{Z}^n$9-atoms supported on a discrete cube $I_\alpha$0, satisfying

$I_\alpha$1

together with vanishing moments

$I_\alpha$2

where $I_\alpha$3 (Rocha, 2024).

A key uniform estimate in the Hardy-to-Hardy argument is that for any $I_\alpha$4-atom $I_\alpha$5,

$I_\alpha$6

with $I_\alpha$7 independent of the atom (Rocha, 28 Aug 2025). The proof then aggregates these atomic estimates using $I_\alpha$8-control of coefficients. For $I_\alpha$9, subadditivity of the quasi-norm is used; for $0<\alpha<n$0, Minkowski’s inequality applies (Rocha, 28 Aug 2025).

The technical mechanism combines local and far-region estimates. In the $0<\alpha<n$1 result, the strategy is to reduce to atoms, estimate the output in $0<\alpha<n$2 for atoms, split into a local region and a region away from the atom, and then use Taylor expansion together with vanishing moments so that the remainder decays fast enough to yield summability in $0<\alpha<n$3 (Rocha, 2024). The same source states that discrete summability estimates rely on the fact that for $0<\alpha<n$4,

$0<\alpha<n$5

and that the discrete fractional maximal function is used in the far-region estimate.

This proof architecture mirrors the continuous theory. The available summaries explicitly state that boundedness results for Riesz potentials on continuous Hardy spaces hinge on similar relationships of parameters and are closely tied to molecular or atomic decompositions and maximal functions, while the discrete case requires careful handling of cubes, counting arguments, and adaptation of maximal inequality proofs from the continuous to the discrete setting (Rocha, 28 Aug 2025).

4. Weighted, Morrey, and variable exponent sequence-space theories

Beyond Hardy spaces, the discrete Riesz potential has been studied on several sequence-space scales. On discrete weighted Lebesgue spaces, for $0<\alpha<n$6, $0<\alpha<n$7, and $0<\alpha<n$8, if $0<\alpha<n$9, then

$$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$0

(Hao et al., 2023). The same work extends the boundedness to discrete weighted Morrey spaces. Under $$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$1, $$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$2, $$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$3, and $$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$4, for every $$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$5,

$$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$6

The proof uses the boundedness of the discrete fractional Hardy-Littlewood maximal operator, pointwise domination by $$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$7, and a discrete version of Whitney decomposition (Hao et al., 2023).

A related Morrey-space theory in arbitrary dimension defines

$$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$8

with $$(I_{\alpha} b)(j) = \sum_{i \in \mathbb{Z}^n \setminus \{j\}} \frac{b(i)}{|i-j|^{n-\alpha}}$$9, and proves boundedness on discrete Morrey spaces via a discrete Fefferman-Stein inequality and a Hedberg-type estimate (Gunawan et al., 2018). In that setting, the proof yields

$j \in \mathbb{Z}^n$0

which is explicitly described as the discrete analogue of Hedberg’s inequality (Gunawan et al., 2018).

Variable exponent sequence spaces provide another extension. For $j \in \mathbb{Z}^n$1, if $j \in \mathbb{Z}^n$2, $j \in \mathbb{Z}^n$3, and

$j \in \mathbb{Z}^n$4

with the discrete Hardy-Littlewood maximal operator bounded on $j \in \mathbb{Z}^n$5, then

$j \in \mathbb{Z}^n$6

for every $j \in \mathbb{Z}^n$7 (Rocha, 2024). The proof is said to apply discrete weighted norm inequalities and the Rubio de Francia algorithm.

These results collectively show that the discrete Riesz potential admits a sequence-space theory paralleling continuous harmonic analysis: weighted inequalities, Morrey estimates, maximal-operator domination, and variable exponent extensions all appear in discrete form (Hao et al., 2023).

5. Endpoint and exotic variants from the fractional discrete Laplacian

An endpoint phenomenon appears in the study of the derivative of the fractional discrete Laplacian at $j \in \mathbb{Z}^n$8. For the multidimensional discrete Laplacian $j \in \mathbb{Z}^n$9 on $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$0, the right hand derivative of $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$1 at $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$2 is described as an exotic discrete Riesz potential, namely, the endpoint case: the order is $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$3 in Stein-Wainger sense; for $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$4, the corresponding derivative is also an exotic discrete Riesz potential with an additional corrector (Li et al., 1 Mar 2026).

For $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$5, the derivative is

$$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$6

and the kernel $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$7 is identified as the discrete Stein-Wainger Riesz kernel of order zero (Li et al., 1 Mar 2026). The same source explains that this operator is called exotic because the kernel is not a standard singular integral and the sum conditionally converges due to the $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$8 decay.

For $$(I_\alpha a)(j) = \sum_{i \in \mathbb{Z}\setminus\{j\}} \frac{a(i)}{|i-j|^{1-\alpha}}, \qquad j \in \mathbb{Z},$$9, the derivative takes the form

$|i-j|^{-(n-\alpha)}$00

where

$|i-j|^{-(n-\alpha)}$01

The kernel decays like $|i-j|^{-(n-\alpha)}$02 as $|i-j|^{-(n-\alpha)}$03, which the paper places exactly at the threshold for non-integrability (Li et al., 1 Mar 2026).

This endpoint theory is explicitly connected to the logarithmic Laplacian of Chen-Weth in the continuous setting and is presented as an extension of that theory to the discrete lattice (Li et al., 1 Mar 2026). A plausible implication is that the term “discrete Riesz potential” now spans both the classical fractional orders $|i-j|^{-(n-\alpha)}$04 and an endpoint order $|i-j|^{-(n-\alpha)}$05 that requires separate normalization and corrector terms.

6. Analogies, related structures, and distinct usages

The discrete Riesz potential on $|i-j|^{-(n-\alpha)}$06 is repeatedly framed as a mirror of the continuous Riesz potential on $|i-j|^{-(n-\alpha)}$07. The available summaries state that the discrete operator and discrete Hardy spaces mirror their continuous analogues in definition, scaling, and essential estimates, and that the full-range $|i-j|^{-(n-\alpha)}$08 theorem is the discrete analogue of the classical theorem for the full Hardy range $|i-j|^{-(n-\alpha)}$09 (Rocha, 28 Aug 2025).

A different discrete setting appears on homogeneous trees. There, the Green potential

$|i-j|^{-(n-\alpha)}$10

is described as the tree-side Riesz potential in a comparative study between the unit disk and the homogeneous tree (Boiko et al., 2014). The same work emphasizes a translation dictionary between disk and tree potential theory, with Riesz decomposition

$|i-j|^{-(n-\alpha)}$11

for subharmonic functions on the tree (Boiko et al., 2014). This is not the same operator as the lattice fractional integral $|i-j|^{-(n-\alpha)}$12, but it belongs to the same broad kernel-potential tradition.

Another terminological branch concerns discrete Riesz energy on finite metric spaces. For a finite metric space $|i-j|^{-(n-\alpha)}$13, the discrete Riesz energy function is

$|i-j|^{-(n-\alpha)}$14

The paper on identification of finite circular metric spaces proves that two finite metric spaces have the same discrete Riesz energy if and only if they have the same multiset of edge lengths $|i-j|^{-(n-\alpha)}$15 (Kodama et al., 2024). This is analytically distinct from the operator $|i-j|^{-(n-\alpha)}$16: it is a pairwise interaction functional rather than a mapping between sequence spaces.

Related asymptotic optimization problems also use Riesz kernels on point configurations. For weighted $|i-j|^{-(n-\alpha)}$17-point Riesz $|i-j|^{-(n-\alpha)}$18-polarization on $|i-j|^{-(n-\alpha)}$19-rectifiable compact subsets of $|i-j|^{-(n-\alpha)}$20, one studies

$|i-j|^{-(n-\alpha)}$21

and obtains asymptotics and weak$|i-j|^{-(n-\alpha)}$22 limiting distributions of optimal configurations as $|i-j|^{-(n-\alpha)}$23 (Borodachov et al., 2016). Although this is not a lattice operator theory, it is part of the broader Riesz-potential landscape.

These distinctions matter. A common misconception is to treat all “discrete Riesz” objects as instances of the same theory. The source material shows instead that the phrase can refer to at least three non-equivalent objects: a fractional integral operator on lattices, an endpoint exotic kernel arising from discrete fractional Laplacians, and pairwise energy or polarization functionals on finite or geometric configurations (Li et al., 1 Mar 2026).

7. Significance and current analytical position

The main achievement of the recent Hardy-space theory is stated plainly: the boundedness of the discrete Riesz potential $|i-j|^{-(n-\alpha)}$24 from $|i-j|^{-(n-\alpha)}$25 to $|i-j|^{-(n-\alpha)}$26, with the sharp scaling relation

$|i-j|^{-(n-\alpha)}$27

now holds for the entire range $|i-j|^{-(n-\alpha)}$28, $|i-j|^{-(n-\alpha)}$29 (Rocha, 28 Aug 2025). The same summary states that this solidifies the analogy between discrete and continuous harmonic analysis at the level of Hardy spaces and fractional operators, both foundational and for further developments or applications in discrete analysis.

The broader body of results indicates that discrete fractional integration has reached a level of structural maturity comparable to several classical continuous theories. There are Hardy-space estimates (Rocha, 28 Aug 2025), Hardy-to-$|i-j|^{-(n-\alpha)}$30 estimates (Rocha, 2024), weighted Lebesgue and Morrey inequalities (Hao et al., 2023), discrete Morrey-space bounds via maximal operators and Fefferman-Stein inequalities (Gunawan et al., 2018), and variable exponent extensions via weighted norm inequalities and Rubio de Francia iteration (Rocha, 2024). This suggests a stable methodological core: atomic decomposition, maximal function characterization, fractional maximal operator control, and discrete analogues of covering and interpolation arguments.

At the same time, the endpoint order-zero theory shows that the discrete setting has phenomena that are not merely replicas of the continuous case. The need for exotic kernels, conditional summation, and corrector terms in the derivative-at-zero theory of the fractional discrete Laplacian marks a genuinely discrete layer of complexity (Li et al., 1 Mar 2026). In that sense, the discrete Riesz potential is both an analogue of a classical operator and a source of specifically lattice-based analytical structures.