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Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications
Abstract
In this article, we first investigate the necessary and sufficient conditions on the ranges of s,s1,s2∈ R, 1≤ p1,p2,q≤ ∞, θ1,θ2≥ 0 for the validity of the fractional Gagliardo--Nirenberg inequalities |Dsu|Lq(Rd)≲ |Ds1u|Lp1(Rd)θ1|Ds2u|Lp2(Rd)θ2. Secondly, we consider the anisotropic Gagliardo--Nirenberg inequalities |u|Lq(Rd)≲ |u|Lp0(Rd)θ0 ∏j=1n|Dxjsju|Lpj(Rd)θj. We show the sharp conditions of these inequalities except ``one" case. Finally, we establish the profile decomposition associated with the above inequalities when pj = 2, 0≤ j≤ n. Using these results, we establish existence of extremizers and construct soliton solutions of relevant dispersive equations.
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BibTeX
@misc{chen2026fractional,
title = {{Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications}},
author = {Jie Chen},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.27841},
eprint = {2608.27841},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.27841},
}
RIS
TY - UNPB
TI - Fractional and anisotropic Gagliardo--Nirenberg inequalities with applications
AU - Jie Chen
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.27841
UR - https://arxiv.org/abs/2608.27841
AB - In this article, we first investigate the necessary and sufficient conditions on the ranges of s,s_1,s_2∈ R, 1≤ p_1,p_2,q≤ ∞, θ_1,θ_2≥ 0 for the validity of the fractional Gagliardo--Nirenberg inequalities |D^su|_L^q(R^d)≲ |D^s_1u|_L^p_1(R^d)^θ_1|D^s_2u|_L^p_2(R^d)^θ_2. Secondly, we consider the anisotropic Gagliardo--Nirenberg inequalities |u|_L^q(R^d)≲ |u|_L^p_0(R^d)^θ_0 ∏_j=1^n|D_x_j^s_ju|_L^p_j(R^d)^θ_j. We show the sharp conditions of these inequalities except ``one" case. Finally, we establish the profile decomposition associated with the above inequalities when p_j = 2, 0≤ j≤ n. Using these results, we establish existence of extremizers and construct soliton solutions of relevant dispersive equations.
ER -