Interior gradient Hölder regularity for singular fractional p-Laplacian equations

Establish interior C^{1,\alpha} regularity for bounded weak solutions of singular fractional p-Laplacian equations throughout the full relevant fractional-order and singular parameter range, beyond the near-local orders treated in the paper.

Background

The paper studies bounded weak solutions of the homogeneous singular fractional p-Laplace equation with 1<p<2. Its main theorem proves interior C^{1,\alpha} regularity for every exponent below an admissible local p-harmonic gradient exponent when the fractional order s is sufficiently close to one. This constitutes only a partial result toward the broader regularity problem.

The unresolved issue is the general interior C^{1,\alpha} theory for the pure singular fractional p-Laplacian, particularly away from the near-local regime. The difficulty arises from the singularity of the flux t\mapsto |t|^{p-2}t near zero and the resulting lack of uniform ellipticity after affine normalization.