Weighted Gagliardo-Nirenberg Interpolation Inequalities

Abstract: 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 $|x|^\alpha$. 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 $\langle x \rangle^\alpha=(1+|x|^2)^{\alpha/2}$.