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

Abstract: In this paper, we study a family of weighted Gagliardo--Nirenberg interpolation inequalities associated with Grushin--type operators on $\mathbb{R}ⁿ \times \mathbb{R}<sup>m$. Specifically, we examine how to control the weighted norm $||x|^βu |_{L<sup>B}$ by balancing fractional derivatives $D_x^s u$ with weighted derivatives $|x|^b D_y^r u$. 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 $L<sup>p$, $2 < p < B$, 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 $β=0$, 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.