The two-weight fractional Poincaré-Sobolev sandwich

Abstract: We establish a two-weight fractional Poincaré-Sobolev sandwich, consisting of a two-weight fractional Poincaré-Sobolev inequality and a two-weight embedding from the first-order Sobolev space to a Triebel-Lizorkin space defined via a difference norm. Our constants are asymptotically sharp as the fractional parameter approaches $1$. Our results are new even in the one-weight case. For each inequality we give explicit quantitative dependence on Muckenhoupt weight characteristics and treat both subcritical and critical regimes, the former via elementary methods and the latter via sparse domination. As one of our main tools, we establish a new sparse domination result for Triebel-Lizorkin difference norms. Our methods unify, simplify and significantly extend various earlier approaches.

• The paper presents a sharp quantitative two-weight theory for fractional Poincaré–Sobolev inequalities, tracking constants and weight dependencies as the smoothness parameter approaches 1.
• It employs innovative sparse domination and dyadic telescoping methods to bound oscillation and fractional difference norms in weighted settings.
• The results extend classical one-weight cases and have key implications for regularity in degenerate elliptic PDEs and nonlocal-to-local transitions.

The Two-Weight Fractional Poincaré–Sobolev Sandwich: An Authoritative Summary

Introduction and Problem Setting

The paper "The two-weight fractional Poincaré–Sobolev sandwich" (2604.08416) develops a sharp, quantitative two-weight theory for fractional Poincaré–Sobolev inequalities and related embeddings. The focus is on generalizing classical and fractional Poincaré–Sobolev inequalities from the setting of scalar weights to a two-weight scenario, with full tracking of constants and weight characteristic dependencies, particularly as the fractional smoothness parameter approaches the integer order limit. These results yield not only new insights in the two-weight setting but also extend and sharpen the one-weight case.

The primary objects are weighted Lebesgue spaces $L^p_\sigma(Q)$, weighted Triebel–Lizorkin spaces $F^{s, \sigma}_{p,r}(Q)$ (with definitions through difference quotient seminorms), and the analysis is centered on cubes $Q \subseteq \mathbb{R}^d$. The main results sandwich the fractional seminorm between oscillation and gradient norms, reflecting fine regularity properties in weighted frameworks.

Main Results

The "Sandwich" and Its Constituents

The central inequalities concern bounding the oscillation

$\|f - \langle f \rangle_Q\|_{L^q_\omega(Q)}$

both above and below by the weighted fractional difference norm

$$[F]_{F^{s,\sigma}_{p,r}(Q)} = \left( \int_Q \left( \int_Q \frac{|f(x) - f(y)|^r}{|x - y|^{d + s r}} dy \right)^{p/r} \sigma(x)^p dx \right)^{1/p}$$

and the gradient norm $|\nabla f|_{L^p_\sigma(Q)}$, ultimately establishing the two-sided "sandwich": $$\|f - \langle f \rangle_Q\|_{L^{p_0}_{\sigma_0}(Q)} \lesssim (1 - s)^{1/r} [f]_{F^{s,\sigma_s}_{p_s, r}(Q)} \lesssim |\nabla f|_{L^{p_1}_{\sigma_1}(Q)}$$ with constants that are asymptotically sharp as $s \to 1$, i.e., with sharp Bourgain–Brezis–Mironescu (BBM) factors.

Two-Weight Framework and Novelty

All results are established in a two-weight framework involving Muckenhoupt-type balance conditions between the input and output weights (i.e., $(\omega, \sigma)$ in $A_{p,q}^{\alpha}(Q)$), and explicit dependencies on the weight characteristics are derived. The paper treats two regimes:

• Subcritical: positive scaling deficit $F^{s, \sigma}_{p,r}(Q)$0 allows straightforward dyadic telescoping and elementary proofs
• Critical: scaling-invariant case $F^{s, \sigma}_{p,r}(Q)$1 requires sharp sparse domination and refined harmonic analysis, yielding delicate dependence on $F^{s, \sigma}_{p,r}(Q)$2 characteristics

This program unifies, extends, and sharpens several previous approaches, and many cases (notably those with $F^{s, \sigma}_{p,r}(Q)$3 or $F^{s, \sigma}_{p,r}(Q)$4) are entirely new even in the one-weight setting.

Technical Tools and Approaches

Weight Classes and Sparse Domination

The core weight conditions rely on local two-weight $F^{s, \sigma}_{p,r}(Q)$5 Muckenhoupt characteristics, which subsume the classical $F^{s, \sigma}_{p,r}(Q)$6 case. In the critical case, additional $F^{s, \sigma}_{p,r}(Q)$7 control is necessary.

Crucial to the argumentation is sparse domination: oscillation and fractional difference operators are pointwise dominated (either deterministically or by sparse operators), both in classical and in the fractional nonlocal situations. For the embedding between Sobolev and Triebel–Lizorkin spaces, the paper establishes a new sparse domination theorem for difference seminorms.

Dyadic and telescoping decompositions are ubiquitous in the subcritical case, while for the critical case, the proofs rely on weighted norm inequalities for the associated sparse operators.

Main Theorems

1. Two-Weight Poincaré–Sobolev Inequality

A sharp local inequality is proved for

$F^{s, \sigma}_{p,r}(Q)$8

where $F^{s, \sigma}_{p,r}(Q)$9 is singular as $Q \subseteq \mathbb{R}^d$0. In the critical regime, the authors include optimal dependence on $Q \subseteq \mathbb{R}^d$1.

2. Two-Weight Fractional Poincaré–Sobolev Inequality

In the nonlocal fractional difference norm setting, they establish

$Q \subseteq \mathbb{R}^d$2

with asymptotically sharp constant as $Q \subseteq \mathbb{R}^d$3. The result is fully new for $Q \subseteq \mathbb{R}^d$4 or $Q \subseteq \mathbb{R}^d$5.

3. Two-Weight Sobolev–Triebel–Lizorkin Embedding

A two-weight, sharp BBM-formula embedding is provided: $Q \subseteq \mathbb{R}^d$6 with explicit, tracked dependence on parameters and weight characteristics.

Quantitative Dependence

For all main results, quantitative dependence on $Q \subseteq \mathbb{R}^d$7 and endpoint $Q \subseteq \mathbb{R}^d$8 characteristics is explicitly described. The constants are not just uniform but have optimal or near-optimal dependence in various parameters, including the scaling deficit $Q \subseteq \mathbb{R}^d$9, the exponents $\|f - \langle f \rangle_Q\|_{L^q_\omega(Q)}$0, and the localization imposed by the two-weight balancing.

Theoretical and Practical Implications

These results have direct implications for the regularity theory of degenerate and weighted elliptic PDEs, notably in situations where the underlying space is not homogeneous or self-similar, and the nonlocal phenomena (as in the fractional case) interact subtly with weighted degeneracies. Explicit dependence on weight characteristics is critical in quantitative regularity, unique continuation, and the analysis of limiting passage from nonlocal to local equations (BBM-type asymptotics).

On the harmonic analysis side, the novel sparse domination principles for difference norms open pathways to further multidimensional and multi-parameter generalizations, and the techniques are amenable to adaptations in metric measure space, function space theory, and higher-order embedding settings.

There is substantial scope for extending these results to:

• Higher-order derivatives, via polynomial subtractions (prototypes for higher-order BBM-theory)
• General Banach function spaces, exploiting vector-valued and maximal function theory
• John domains or more general domains with appropriate geometric measure theory support
• Global inequalities via exhaustion of $\|f - \langle f \rangle_Q\|_{L^q_\omega(Q)}$1 and associated limiting formulations

Moreover, the sharp BBM-factor inclusion makes these estimates foundational for the study of $\|f - \langle f \rangle_Q\|_{L^q_\omega(Q)}$2-convergence and the passage between nonlocal and local frameworks.

Open Questions and Future Directions

• BBM-type formulas in two-weight/endpoint settings: The results suggest new limit theorems for weighted Triebel–Lizorkin norms as $\|f - \langle f \rangle_Q\|_{L^q_\omega(Q)}$3.
• Global and endpoint estimates: Extending the sharp constants and tracking for global and limiting scenarios, and optimal weak-type inequalities.
• Higher-order and vector-valued generalizations: Can the methodology (especially sparse domination for difference norms) be pushed to cover higher-order cases in full generality?
• Maximal and potential operator connections: How do these sandwich inequalities interact with the latest in two-weight boundedness of nonlocal maximal and potential operators?

Conclusion

This paper presents a comprehensive two-weight theory for fractional Poincaré–Sobolev inequalities, embedding theorems, and related estimates with precise tracking of constants, weight characteristics, and sharp asymptotics as the smoothness parameter approaches the local case. Many results are new even for classical one-weight cases and offer a blueprint for further development in the analysis of weighted and fractional function space embeddings, their applications in PDE theory, and the fine properties of functions in degenerate, nonlocal, or irregular contexts. The underlying harmonic analytic machinery, most notably the sparse domination for difference seminorms, is a notable technical contribution with broad applicability.

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References

• (2604.08416) Emiel Lorist, Carel Wagenaar. "The two-weight fractional Poincaré–Sobolev sandwich". (2604.08416)