Extremals for singular Onofri type inequality in weighted Sobolev spaces

@misc{borgia2026extremals,
  title = {{Extremals for singular Onofri type inequality in weighted Sobolev spaces}},
  author = {Natalino Borgia and Silvia Cingolani and Rubén Fiñana Aranega and Gabriele Mancini},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.15195},
  eprint = {2610.15195},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.15195},
}

TY  - UNPB
TI  - Extremals for singular Onofri type inequality in weighted Sobolev spaces
AU  - Natalino Borgia
AU  - Silvia Cingolani
AU  - Rubén Fiñana Aranega
AU  - Gabriele Mancini
PY  - 2026
DA  - 2026/10/10
PB  - arXiv
DO  - 10.48550/arxiv.2610.15195
UR  - https://arxiv.org/abs/2610.15195
AB  - The N-dimensional Euclidean Onofri inequality was proved by Del Pino and Dolbeault for smooth compactly supported functions in ℝ^N with N ≥ 2. In this paper, we consider the singular version introduced by Lam and Lu, and extend it to a natural weighted Sobolev space. Next, we establish a sharp singular Carleson-Chang type inequality and prove its equivalence with the singular N-dimensional Euclidean Onofri inequality. Furthermore, we prove the existence of nontrivial, non-constant optimizers for the functional associated with the singular Euclidean Onofri inequality, answering open questions left by Del Pino and Dolbeault in the non-singular case, and by Lam and Lu in the case with singularities. To this end, we exploit the classification result obtained for a singular Liouville-type equation involving the N-Laplacian. We also prove the conformal invariance of the functional, establish the sharpness of the singular N-dimensional Euclidean Onofri inequality in terms of optimal constants, and state several consequences, including a sharp Poincaré inequality.
ER  -