Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator

Abstract

We study the partially fractional Grushin-type operator \begin{equation} \mathcal L_{α,β}u=(-Δ_x)^αu-(1+β)^2|x|^{2β}Δ_y u, \qquad (x,y)\in\mathbb{R}^n\times\mathbb{R}^m,\quad 0<α<1,\ β\ge 0, \end{equation} which couples fractional diffusion in $x$ with degenerate local diffusion in $y$. Working with the anisotropic dilations $(x,y)\mapsto(ρx,ρ^{α+β}y)$ 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 $S_{α,β}(n,m)$.