Critical Sobolev Inequalities and Extremals for a Partially Fractional Grushin Operator

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 $(x,y)\mapsto (\rho x,{\rho }^{\alpha +\beta }y)$ 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 $2_{\alpha ,\beta }^{\ast }=2Q_{\alpha ,\beta }\mathrm{/}(Q_{\alpha ,\beta }-2\alpha )$, where $Q_{\alpha ,\beta }=n+(\alpha +\beta )m$. We then establish a concentration-compactness principle for the associated energy space. For $\beta >0$ the weight $\mathrm{\mid }x{\mathrm{\mid }}^{2\beta }$ destroys translation invariance in $x$, and we show that concentration is confined to the degenerate hyperplane $\mathrm{\Sigma }=\{(0,y):y\in {\mathbb{R}}^m\}$; this localization yields the existence of extremals for the sharp constant $S_{\alpha ,\beta }(n,m)$.