Riesz Potential Inequalities

• Riesz potential inequalities are sharp, scale-sensitive bounds for fractional integration operators that underpin Sobolev embeddings, isoperimetric inequalities, and various weighted estimates.
• They extend classical results through weighted, two-weight, and rearrangement-invariant frameworks, incorporating log-bump and Hardy type conditions for optimal boundedness.
• Recent advancements include endpoint L1 estimates, exponential integrability in Adams–Moser–Trudinger type inequalities, and extensions to Dunkl analysis and manifold settings.

The Riesz potential inequalities comprise a spectrum of sharp, scale-sensitive bounds for fractional integration operators, fundamental to harmonic analysis, PDE theory, and geometric measure theory. The classical Riesz potential on $\mathbb{R}^n$ is defined by

$$I_{\alpha}f(x) = \int_{\mathbb{R}^n} |x-y|^{\alpha-n} f(y)\,dy,\quad 0<\alpha<n,$$

and its boundedness properties, extensions under weights, generalizations to rearrangement-invariant spaces, and connections to Sobolev embeddings and isoperimetric inequalities are central to modern functional analysis.

1. Classical and Weighted Riesz Potential Inequalities

The foundational result is the strong-type Sobolev embedding: for $1 < p < n/\alpha$, $q = np/(n-\alpha p)$,

$$\|I_\alpha f\|_{L^q(\mathbb{R}^n)} \leq C \|f\|_{L^p(\mathbb{R}^n)},$$

with the endpoint $p=1$ yielding only weak-type estimates:

$$\|I_\alpha f\|_{L^{n/(n-\alpha),\infty}(\mathbb{R}^n)} \leq C \|f\|_{L^1(\mathbb{R}^n)}.$$

Weighted versions involve the Muckenhoupt $A_{p,q}$ condition:

$$[w]_{A_{p,q}} = \sup_Q \left( \int_Q w^q \right)^{1/q} \left( \int_Q w^{-p'} \right)^{1/p'} < \infty,$$

where $Q$ ranges over cubes in $\mathbb{R}^n$ and $p, q$ satisfy the same scaling $1/p-1/q = \alpha/n$ (Cruz-Uribe et al., 2012).

Two-weight inequalities and sharp mixed $A_p$–$A_\infty$ conditions were established, with improvements for both weak and strong types, and further sharp “log-bump” sufficiency conditions involving Orlicz–Luxemburg norms and separated bump conditions (Cruz-Uribe et al., 2012). For instance,

$$\|I_\alpha (f \sigma)\|_{L^{q,\infty}(u)} \lesssim [u, \sigma]_{A_{s(p)}}^{1/q} [u]_{A_\infty'}^{1/p'} \|f\|_{L^p(\sigma)},$$

where separated log-bump conditions yield the optimal range near the Sobolev scaling.

On spaces with non-doubling measures, good–$\lambda$ inequalities and related strong-type $L^p$–$L^q$ estimates for $w \in A_{p,q}(\mu)$ extend the theory beyond classical Lebesgue measure (Bhandari, 2021).

2. Rearrangement-Invariant and Endpoint Inequalities

For rearrangement-invariant (r.i.) Banach function spaces $X(\mathbb{R}^n)$ and $Y(\mathbb{R}^n)$, Riesz potential inequalities are characterized by Hardy-type one-dimensional inequalities. For a co-canceling (or canceling) homogeneous differential operator $L(D)$ of order $k<n$, Van Schaftingen, Raiţă, Hernandez, and Spector established that for $F \in X$ with $L(D)F=0$,

$$\|I_\alpha F\|_{Y(\mathbb{R}^n)} \leq C \|F\|_{X(\mathbb{R}^n)}$$

if and only if the associated 1D Hardy estimate holds:

$$\left\| \int_s^\infty r^{-1+\alpha/n}f(r)\,dr \right\|_{Y(0,\infty)} \leq C \|f\|_{X(0,\infty)}.$$

This framework encompasses Lebesgue, Orlicz, Lorentz-Zygmund, Orlicz-Lorentz, and borderline spaces such as $L^1(\log L)^r$ under differential constraints, including the cases where unweighted $I_\alpha : L^1 \to L^{n/(n-\alpha)}$ fails without constraints (Breit et al., 6 Dec 2025, Breit et al., 14 Jan 2025).

The endpoint $L^1$-estimate for scalar potentials fails in the absence of constraints but is restored under co-canceling conditions (e.g., divergence-free fields), and further, for $L^1(\log L)^r$ type spaces, with optimal targets in Lorentz-Zygmund scales (Breit et al., 6 Dec 2025).

3. Exponential Integrability and Adams–Moser–Trudinger Inequalities

At the critical Sobolev index ($n/\alpha$), Riesz potentials saturate Sobolev embedding, and their integrability is of exponential order. The sharp Adams inequality on $\mathbb{R}^n$ is

$$\sup_{\|f\|_{L^{n/\alpha}} + \|I_\alpha f\|_{L^{n/\alpha}} \leq 1} \int_{\mathbb{R}^n} \exp\left( \frac{1}{A_\alpha} |I_\alpha f(x)|^{n/(n-\alpha)} \right) dx < \infty,$$

with the optimal constant $A_\alpha = |S^{n-1}|$ (Fontana et al., 2017). Analogous Moser–Trudinger inequalities hold for critical Sobolev spaces and extend to Riesz-subcritical kernels and domains, under explicit integrability and rearrangement conditions (Fontana et al., 2019).

Generalizations include nonhomogeneous measure spaces, kernels with better decay at infinity, and applications to pseudo-differential operators, as well as to the hyperbolic space and domains with suitable Poincaré inequality (Fontana et al., 2019).

4. Potential Inequalities on General Function Spaces and Structures

Riesz potential inequalities have been established fully on:

• Orlicz, Orlicz–Morrey, and Generalized Orlicz–Morrey Spaces: Characterized by sharp Zygmund-type integral inequalities, both for the potential and for commutators with BMO functions, with necessity and sufficiency (Guliyev et al., 2013).
• Herz and Herz–Morrey–Hardy Spaces: Including constant and variable exponent settings, with atomic and shell-wise decomposition techniques. Trace and Sobolev-type inequalities follow, and parameter optimality is demonstrated by precise counterexamples (Bhat et al., 9 Mar 2024, Gurbuz, 21 Nov 2024).
• Dunkl Analysis: The Dunkl–Riesz potential replaces translation and Lebesgue measure by the Dunkl analogue, with weighted $(L^p,L^q)$ boundedness characterized by precise rearrangement conditions, and Sobolev embeddings with the Dunkl-dimension $2\gamma+d$ (Abdelkefi et al., 2013).

5. Pointwise Gradient and Operator Inequalities

Sharp pointwise gradient estimates for Riesz potentials relate the potential, its gradient, and lower-order “contiguous” potentials via variational and hypergeometric techniques. For $f$ bounded and supported away from $x$:

$$|\nabla I_\alpha[f](x)|^2 \leq C N_\alpha(I_\alpha[f](x), I_{\alpha-2}[f](x)),$$

where $N_\alpha$ is an explicit function arising from extremal variational principles (Tkachev, 2018).

For a large class of rough, non-smooth operators $T$ (including maximal, spherical, and principal value operators), pointwise bounds of the form

$|Tf(x)| \leq C I_1(|\nabla f|)(x)$

yield, via interpolation and weighted theory, Sobolev-type inequalities identical to those for the Riesz potential (Hoang et al., 2023).

6. Isoperimetric and Reverse Inequalities

Riesz potential operators on $L^2(\Omega)$ exhibit sharp isoperimetric inequalities: for the Schatten $p$-norm, the ball maximizes the norm among all domains of fixed measure,

$$\|I_\alpha\|_{p,\Omega} \leq \|I_\alpha\|_{p,B},\quad p > p_0 = d/\alpha,$$

and similar extremal properties hold for all eigenvalues (Rayleigh-Faber-Krahn, Hong-Krahn-Szegö) (Rozenblum et al., 2015).

Reverse triangle inequalities for Riesz potentials, linking sums of infima of potentials to the infimum of sums, are governed by sharp constants encoding the geometry and equilibrium measure of $E \subset \mathbb{R}^n$, with the farthest distance function representation as a Riesz potential being central (Pritsker et al., 2013).

7. Riesz Potential Inequalities on Manifolds and Further Directions

On manifolds with quadratic decay of curvature, Riesz potential inequalities are established via heat kernel and parametrix technology. Riesz and reverse Riesz transforms, Hardy, and weighted Sobolev inequalities are proven equivalent under such geometric conditions, with precise Lorentz–type endpoint bounds (He, 19 Mar 2025). The harmonic annihilation method allows for the handling of leading asymptotic terms in the potential kernel.

Further directions concern non-Euclidean settings (e.g., Heisenberg groups), endpoint $L^1$ theory for non-convolution operators, trace inequalities, and compactness/isoperimetric phenomena for fractional potentials.

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Table: Key Types of Riesz Potential Inequalities

Context/Space	Type/Estimate	Reference
Classical weighted Lebesgue	$L^p(w^p) \to L^q(w^q)$ via $A_{p,q}$, log-bump conditions	(Cruz-Uribe et al., 2012)
Rearrangement-invariant, $L^1$	Co-canceling: $I_\alpha F: X \to Y$ iff Hardy 1D estimate	(Breit et al., 6 Dec 2025)
Adams/Moser–Trudinger	$\int \exp( c |I_\alpha f|^{n/(n-\alpha)})$ bounded (critical index)	(Fontana et al., 2017)
Orlicz–Morrey/Herz	$$I_\gamma: M_{\Phi,\varphi_1} \to M_{\Psi,\varphi_2}$$ iff Zygmund cond	(Guliyev et al., 2013)
Dunkl analysis	Rearrangement-based $L^p \to L^q$ with Dunkl measure	(Abdelkefi et al., 2013)
Lorentz endpoint, Manifold	$I_\alpha: L^{\nu,1} \to L^{\nu,\infty}$, geometric Hardy equivalence	(He, 19 Mar 2025)

• "One and two weight norm inequalities for Riesz potentials" (Cruz-Uribe et al., 2012).
• "Riesz potential estimates under co-canceling constraints" (Breit et al., 6 Dec 2025).
• "Adams inequalities for Riesz subcritical potentials" (Fontana et al., 2019).
• "Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on R^n" (Fontana et al., 2017).
• "Reverse Triangle Inequalities for Riesz Potentials and Connections with Polarization" (Pritsker et al., 2013).
• "Isoperimetric inequalities for Schatten norms of Riesz potentials" (Rozenblum et al., 2015).
• "Pointwise estimates for rough operators with applications to Sobolev inequalities" (Hoang et al., 2023).
• "An $L^1$-type estimate for Riesz potentials" (Schikorra et al., 2014).
• "Some properties of the Riesz potentials in Dunkl analysis" (Abdelkefi et al., 2013).
• "Riesz and reverse Riesz on Manifolds with Quadratically Decaying Curvature" (He, 19 Mar 2025).
• "On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces" (Guliyev et al., 2013).
• "Some Inequalities for Riesz Potential on Homogeneous Variable Exponent Herz-Morrey-Hardy Spaces" (Gurbuz, 21 Nov 2024).
• "Trace Principle for Riesz Potentials on Herz-Type Spaces and Applications" (Bhat et al., 9 Mar 2024).
• "Good lambda inequalities for non-doubling measures in $\mathbb{R}^n$" (Bhandari, 2021).
• "Sharp pointwise gradient estimates for Riesz potentials with a bounded density" (Tkachev, 2018).