2022/08/12 by Rodrigo Duarte, Duarte, Rodrigo, Jorge Drumond Silva +1 · 2 citations
Mathematics · #42B37 #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2208.06363
openalex publication_date 2022/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we prove weighted versions of the Gagliardo-Nirenberg interpolation inequality with Riesz as well as Bessel type fractional derivatives. We use a harmonic analysis approach employing several methods, including the method of domination by sparse operators, to obtain such inequalities for a general class of weights satisfying Muckenhoupttype conditions. We also obtain improved results for some particular families of weights, including power-law weights |x|α. In particular, we prove an inequality which generalizes both the Stein-Weiss inequality and the Caffarelli-Kohn-Nirenberg inequality. However, our approach is sufficiently flexible to allow as well for non-homogeneous weights and we also prove versions of the inequalities with Japanese bracket weights ⟨ x ⟩α=(1+|x|2)α/2.