Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients
Abstract
We study the representation of bilinear forms associated with Riesz fractional gradients by symmetric integral kernels. Motivated by [SS18, Open Problem 1.10], we give a complete description of the constant coefficient case. We prove that any symmetric representing kernel is uniquely determined and we derive its explicit formula in terms of the constant matrix A. This yields a sharp characterization of the matrices for which the kernel is either nonnegative or uniformly positive. In particular, [SS18, Open Problem 1.10] has a negative answer (even for constant matrices). We also identify a class of nonconstant antisymmetric divergence free perturbations that are invisible to the fractional bilinear form and hence leave the representing kernel unchanged. The general case of nonconstant matrices exhibits a substantially different behavior and will be the main subject of our forthcoming paper [DSV26].
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BibTeX
@misc{dipierro2026kernel,
title = {{Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients}},
author = {Serena Dipierro and Caterina Sportelli and Enrico Valdinoci},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.20110},
eprint = {2610.20110},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.20110},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Kernel representations of fractional gradient bilinear forms. Part I. Constant coefficients
AU - Serena Dipierro
AU - Caterina Sportelli
AU - Enrico Valdinoci
PY - 2026
DA - 2026/10/14
PB - arXiv
DO - 10.48550/arxiv.2610.20110
UR - https://arxiv.org/abs/2610.20110
AB - We study the representation of bilinear forms associated with Riesz fractional gradients by symmetric integral kernels. Motivated by [SS18, Open Problem 1.10], we give a complete description of the constant coefficient case. We prove that any symmetric representing kernel is uniquely determined and we derive its explicit formula in terms of the constant matrix A. This yields a sharp characterization of the matrices for which the kernel is either nonnegative or uniformly positive. In particular, [SS18, Open Problem 1.10] has a negative answer (even for constant matrices). We also identify a class of nonconstant antisymmetric divergence free perturbations that are invisible to the fractional bilinear form and hence leave the representing kernel unchanged. The general case of nonconstant matrices exhibits a substantially different behavior and will be the main subject of our forthcoming paper [DSV26].
ER -