Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator
Abstract
We study the partially fractional Grushin-type operator ℒα,βu=(−Δx)αu−(1+β)2|x|2βΔy u, (x,y)∈ℝn×ℝm, 0<α<1, β≥ 0, which couples fractional diffusion in x with degenerate local diffusion in y. Working with the anisotropic dilations (x,y)↦(ρx,ρα+βy) and an adapted quasi-distance, we prove a uniform fractional Grushin-Poincaré inequality on anisotropic cylinders and, via a Riesz-type potential built from a telescoping sum over dyadic cylinders, a global Sobolev inequality at the sharp critical exponent 2*α,β=2Qα,β/(Qα,β−2α), where Qα,β=n+(α+β)m. We then establish a concentration-compactness principle for the associated energy space. For β>0 the weight |x|2β destroys translation invariance in x, and we show that concentration is confined to the degenerate hyperplane $ Σ= \{(0,y):y\in\mathbb{R}^m\}$; this localization yields the existence of extremals for the sharp constant Sα,β(n,m).
Related
- Extremals and Thresholds for Critical Singular Anisotropic Moser-Trudinger Inequalities
- Sharp Weighted Sobolev Trace Inequalities on Convex Cones and Rigidity of Extremals
- Extremals for Sharp Trace Gagliardo--Nirenberg--Sobolev Inequalities on Convex Cones
- Extremals for Trudinger-Moser inequalities via polygonal profiles
- Unique Positive Extremals and Sharp Stability Constants for Singular Moser-Trudinger Inequalities
BibTeX
@misc{malhotra2026critical,
title = {{Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator}},
author = {S. Malhotra and L. Sharma and K. Sreenadh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09001},
eprint = {2610.09001},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09001},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator
AU - S. Malhotra
AU - L. Sharma
AU - K. Sreenadh
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.09001
UR - https://arxiv.org/abs/2610.09001
AB - We study the partially fractional Grushin-type operator ℒ_α,βu=(−Δ_x)^αu−(1+β)^2|x|^2βΔ_y u, (x,y)∈ℝ^n×ℝ^m, 0<α<1, β≥ 0, which couples fractional diffusion in x with degenerate local diffusion in y. Working with the anisotropic dilations (x,y)↦(ρx,ρ^α+βy) and an adapted quasi-distance, we prove a uniform fractional Grushin-Poincaré inequality on anisotropic cylinders and, via a Riesz-type potential built from a telescoping sum over dyadic cylinders, a global Sobolev inequality at the sharp critical exponent 2^*_α,β=2Q_α,β/(Q_α,β−2α), where Q_α,β=n+(α+β)m. We then establish a concentration-compactness principle for the associated energy space. For β>0 the weight |x|^2β destroys translation invariance in x, and we show that concentration is confined to the degenerate hyperplane $ Σ= \{(0,y):y\in\mathbb{R}^m\}$; this localization yields the existence of extremals for the sharp constant S_α,β(n,m).
ER -