论文标题
加权的Gagliardo-Nirenberg插值不平等
Weighted Gagliardo-Nirenberg Interpolation Inequalities
论文作者
论文摘要
在本文中，我们证明了与Riesz和Bessel型分数衍生物的Gagliardo-Nirenberg插值不平等的加权版本。我们使用一种使用多种方法的谐波分析方法，包括稀疏操作员的统治方法，以获得满足Muckenhoupttype条件的一般权重的这种不平等。我们还为某些特定的权重系列（包括幂律权重$ | x |^α$）获得了改进的结果。特别是，我们证明了一种不平等，它概括了Stein-Weiss不平等和Caffarelli-Kohn-Nirenberg的不平等。但是，我们的方法足够灵活以适合非均匀权重，我们还证明了日本支架重量的不平等版本$ \ langle x \ rangle x \ rangle^α=（1+ | x |^2）^{α/2} $。
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 $\langle x \rangle^α=(1+|x|^2)^{α/2}$.
