Fractional elementary symmetric functions and Hessian estimates in the plane
Abstract
We prove an endpoint Pucci maximum principle for domains in ℝ2, formulated using a generalization of elementary symmetric polynomials to noninteger orders. As a result, we sharpen an earlier result of Astala, Iwaniec and Martin, and deduce a new Sobolev inequality for q-Hessians where 1 < q < 2. The fractional elementary symmetric functions we use are derived through fractional differentiation. Their signed singular value extension coincides with the Burkholder functional introduced by Burkholder and gives us access to a local Burkholder inequality due to Astala, Iwaniec, Prause and Saksman, leading to a different proof of the Pucci maximum principle from the one in that used quasiconformal maps. We include an elementary real-variable proof of this local Burkholder inequality, obtained by combining a few known techniques from existing literature.
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BibTeX
@misc{memana2026fractional,
title = {{Fractional elementary symmetric functions and Hessian estimates in the plane}},
author = {Gautam Neelakantan Memana and Neil S. Trudinger and Po-Lam Yung},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19410},
eprint = {2610.19410},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19410},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fractional elementary symmetric functions and Hessian estimates in the plane
AU - Gautam Neelakantan Memana
AU - Neil S. Trudinger
AU - Po-Lam Yung
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.19410
UR - https://arxiv.org/abs/2610.19410
AB - We prove an endpoint Pucci maximum principle for domains in ℝ^2, formulated using a generalization of elementary symmetric polynomials to noninteger orders. As a result, we sharpen an earlier result of Astala, Iwaniec and Martin, and deduce a new Sobolev inequality for q-Hessians where 1 < q < 2. The fractional elementary symmetric functions we use are derived through fractional differentiation. Their signed singular value extension coincides with the Burkholder functional introduced by Burkholder and gives us access to a local Burkholder inequality due to Astala, Iwaniec, Prause and Saksman, leading to a different proof of the Pucci maximum principle from the one in that used quasiconformal maps. We include an elementary real-variable proof of this local Burkholder inequality, obtained by combining a few known techniques from existing literature.
ER -