Recovering functions via doubly homogeneous nonlocal gradients

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.

Dedicated to the memory of Haim Brezis, with deep admiration and gratitude

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