Weighted Gagliardo--Nirenberg Inequalities of Grushin Type: Symmetries, Compactness, and Optimizers

In this paper, we study a family of weighted Gagliardo–Nirenberg interpolation inequalities associated with Grushin–type operators on ${\mathbb{R}}^n\times {\mathbb{R}}^m$. Specifically, we examine how to control the weighted norm $\mathrm{\parallel }\mathrm{\mid }x{\mathrm{\mid }}^{\beta }u{\mathrm{\parallel }}_{L^B}$ by balancing fractional derivatives $D_x^{s}u$ with weighted derivatives $\mathrm{\mid }x{\mathrm{\mid }}^b{D}_y^{r}u$. Using two–parameter scaling, spatial translations, and Littlewood–Paley decompositions, we map out exact parameter conditions and demonstrate that the scaling exponents are both necessary and optimal. For multi-parameter weighted extensions, we leverage Khintchine-type random sums to show why $Q\ge 2$ is strictly required. When $x$ and $y$ are radially symmetric, two key advantages emerge: the allowable parameter range for the weights expands, and compact embeddings of the energy space into $L^p$, $2<p<B$, are completely restored. We establish this compactness through two complementary tools, geometric covering arguments and a generalized three–norm version of Strauss’s lemma, that yields explicit pointwise weighted decay estimates. Finally, in the scale–invariant case $\beta =0$, a concentration–compactness analysis proves that the inequality’s sharp constant is achieved by an optimal function, with minimizing sequences remaining precompact up to the natural three-parameter group of symmetries.