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 ℝn × ℝm. Specifically, we examine how to control the weighted norm ||x|βu |LB by balancing fractional derivatives Dxs u with weighted derivatives |x|b Dyr 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 ≥ 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 Lp, 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.
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BibTeX
@misc{esfahani2026weighted,
title = {{Weighted Gagliardo--Nirenberg Inequalities of Grushin Type: Symmetries, Compactness, and Optimizers}},
author = {Amin Esfahani},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.11819},
eprint = {2610.11819},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.11819},
}
RIS
TY - UNPB
TI - Weighted Gagliardo--Nirenberg Inequalities of Grushin Type: Symmetries, Compactness, and Optimizers
AU - Amin Esfahani
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.11819
UR - https://arxiv.org/abs/2610.11819
AB - In this paper, we study a family of weighted Gagliardo--Nirenberg interpolation inequalities associated with Grushin--type operators on ℝ^n × 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 ≥ 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.
ER -