发表机构
Nazarbayev University（纳扎尔巴耶夫大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究ℝⁿ×ℝᵐ上Grushin型算子相关的加权Gagliardo--Nirenberg插值不等式，确定参数条件与最优缩放指数，利用径向对称性恢复紧嵌入，证明β=0时存在最优解。
AI 中文摘要
本文研究与ℝⁿ×ℝᵐ上Grushin型算子相关的一族加权Gagliardo--Nirenberg插值不等式，具体探讨如何通过平衡分数阶导数Dₓˢu与加权导数|x|ᵇDᵧʳu来控制加权范数|||x|ᵦu||_{Lᴮ}。利用双参数缩放、空间平移及Littlewood--Paley分解，我们确定了精确的参数条件，并证明缩放指数既是必要的也是最优的。对于多参数加权扩展，我们借助Khintchine型随机和说明为何严格要求Q≥2。当x和y具有径向对称性时，会出现两个关键优势：权重的允许参数范围扩大，且能量空间到Lᵖ（2<p<B）的紧嵌入完全恢复。我们通过两种互补工具建立该紧性：几何覆盖论证与Strauss引理的广义三范数版本，后者可得到显式逐点加权衰减估计。最后，在尺度不变情形β=0下，集中紧性分析证明该不等式的尖锐常数由最优函数达到，且极小序列在自然三参数对称群下仍为预紧的。
英文摘要
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 $\||x|^βu \|_{L^B}$ 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 $L^p$, $2 < p < B$, 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.
