PreprintCC BY 4.0arXiv
Heat flow and the fundamental solution of the Baouendi-Grushin operator
Abstract
For the Baouendi-Grushin operator ℒγ= Δx + |x|2γΔy in Rm+k, with γ>0, we prove that the time integral of the heat kernel with pole at the origin equals Cγ,Q Nγ2−Q, where Nγ is the gauge and Q the homogeneous dimension. When γnot=1 no closed form of the heat kernel is known. After a partial Fourier transform in y, the time integral of the heat kernel becomes the Green function of an anharmonic oscillator, which solves a generalized modified Bessel equation, and the gauge appears when the inverse Fourier transform is computed as a Hankel transform. In particular, the heat flow provides a new proof that Cγ,Q Nγ2−Q is a fundamental solution of −ℒγ with pole at the origin.
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BibTeX
@misc{garofalo2026heat,
title = {{Heat flow and the fundamental solution of the Baouendi-Grushin operator}},
author = {Nicola Garofalo},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01336},
eprint = {2611.01336},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01336},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Heat flow and the fundamental solution of the Baouendi-Grushin operator
AU - Nicola Garofalo
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01336
UR - https://arxiv.org/abs/2611.01336
AB - For the Baouendi-Grushin operator ℒ_γ= Δ_x + |x|^2γΔ_y in R^m+k, with γ>0, we prove that the time integral of the heat kernel with pole at the origin equals C_γ,Q N_γ^2−Q, where N_γ is the gauge and Q the homogeneous dimension. When γnot=1 no closed form of the heat kernel is known. After a partial Fourier transform in y, the time integral of the heat kernel becomes the Green function of an anharmonic oscillator, which solves a generalized modified Bessel equation, and the gauge appears when the inverse Fourier transform is computed as a Hankel transform. In particular, the heat flow provides a new proof that C_γ,Q N_γ^2−Q is a fundamental solution of −ℒ_γ with pole at the origin.
ER -