Recovering functions via doubly homogeneous nonlocal gradients

TL;DR

This work develops a comprehensive framework for recovering functions from a class of doubly homogeneous nonlocal gradients $\mathcal{G}$ with kernels $g$ exhibiting distinct near-zero and far-field scaling. A central result is a representation formula $u = V * \mathcal{G}u$, where $V$ is constructed from a kernel $\omega$ solving $\omega*g = |x|^{2-d}$, enabling inversion of the nonlocal gradient under precise regularity and Fourier-positivity assumptions. The authors derive Sobolev-type embeddings in both Lebesgue and Lorentz spaces for $\mathcal{G}$, including two-scale kernels, and establish decay/regularity properties for the representing kernel $V$. They also analyze radial homogeneous distributions and their Fourier transforms, proving rigidity results for order-zero cases and positivity results for the Fourier transform under natural monotonicity assumptions. The paper concludes with Brezis-inspired open questions, suggesting avenues for future work in nonlocal gradient inversion and endpoint estimates.

Abstract

We investigate a class of nonlocal gradients featuring distinct homogeneities at zero and infinity. We establish a representation formula for such doubly homogeneous operators and derive associated Sobolev-type inequalities. We also propose open questions linked to our results, suggesting directions for future research inspired by the work of Haim Brezis.

Theorems & Definitions (61)

• Definition 1
• Theorem 2
• Theorem 3
• Theorem 7
• Theorem 8
• Theorem 9
• Theorem 10
• Proposition 12
• Lemma 13
• proof : Proof of Lemma \ref{['lemmaQEstimate']}