Recovering functions via doubly homogeneous nonlocal gradients

2025/11/03 by Buccheri, Stefano, Ponce, Augusto C.
#26A33 #42A38 #47A67 #47G20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2511.01606

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.

Citations

• A fractional approach to strain-gradient plasticity: beyond core-radius of discrete dislocations
• Nonlocal gradients: Fundamental theorem of calculus, Poincaré inequalities and embeddings
• Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings
• An Optimal Sobolev Embedding for L¹
• On a new class of fractional partial differential equations II
• On a new class of fractional partial differential equations

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