Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications

Abstract

Let $0 1$ be such that $ps 0$. 3- If $β\equiv \frac{N-ps}{2}$, as a consequence of the improved Hardy inequality, we obtain that for all $q<p$, there exists a positive constant $C(Ω)$ such that \begin{equation*} \int_{{\mathbb R}^N}\int_{{\mathbb R}^N} \dfrac{|u(x)-u(y)|^p}{|x-y|^{N+ps}|x|^β|y|^β} \,dy\,dx\ge C(Ω)\Big(\int_Ω \frac{|u(x)|^{p^*_{s,q}}}{|x|^{2β\frac{p^*_{s,q}}{p}}}\,dx\Big)^{\frac{p}{p^*_{s,q}}}, \end{equation*} for all $u\in \mathcal{C}^\infty_0(Ω)$ where $p^*_{s,q}=\frac{pN}{N-qs}$. \ Notice that the previous inequalities can be understood as the fractional extension of the Callarelli-Kohn-Nirenberg inequalities.