PreprintCC BY 4.0arXiv
Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type
Abstract
We determine the sharp Lp ranges of the Riesz transform and the reverse Riesz inequality for a class of power-degenerate elliptic operators of Grushin type. When the degenerate variable has dimension at least two, the Riesz transform is bounded throughout the reflexive range. In dimension one, a capacity-driven transition occurs: weak degeneracy produces a sharp finite upper endpoint, whereas strong degeneracy separates the two half-lines and restores the full range. The reverse inequality displays a different sharp phenomenon. In the weakly degenerate one-dimensional regime it holds throughout the reflexive range except at a single Hardy-critical exponent, while in all remaining regimes it holds for every 1<p<∞.
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BibTeX
@misc{he2026sharp,
title = {{Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type}},
author = {Dangyang He},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.15599},
eprint = {2607.15599},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.15599},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Sharp Riesz and reverse Riesz inequalities for degenerate elliptic operators of Grushin type
AU - Dangyang He
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2607.15599
UR - https://arxiv.org/abs/2607.15599
AB - We determine the sharp L^p ranges of the Riesz transform and the reverse Riesz inequality for a class of power-degenerate elliptic operators of Grushin type. When the degenerate variable has dimension at least two, the Riesz transform is bounded throughout the reflexive range. In dimension one, a capacity-driven transition occurs: weak degeneracy produces a sharp finite upper endpoint, whereas strong degeneracy separates the two half-lines and restores the full range. The reverse inequality displays a different sharp phenomenon. In the weakly degenerate one-dimensional regime it holds throughout the reflexive range except at a single Hardy-critical exponent, while in all remaining regimes it holds for every 1<p<∞.
ER -