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

Abstract

Let $0<s<1$ and $p>1$ be such that $ps<N$. Assume that $\Omega$ is a bounded domain containing the origin. Staring from the ground state inequality by R. Frank and R. Seiringer we obtain: 1- The following improved Hardy inequality for $p\ge 2$ For all $q<p$, there exists a positive constant $C\equiv C(\Omega ,q,N,s)$ such that $$ \int_{{\mathbb R}^N}\int_{{\mathbb R}^N} \, \frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}\,dx\,dy - Λ_{N,p,s} \int_{{\mathbb R}^N} \frac{|u(x)|^p}{|x|^{p}}\,dx\geq C \int_Ω\dint_Ω\frac{|u(x)-u(y)|^p}{|x-y|^{N+qs}}dxdy $$ for all $u\in {\mathcal{C}}_0^{\infty }({\mathbb{R}}^N)$. Here ${\Lambda }_{N,p,s}$ is the optimal constant in the Hardy inequality. 2- Define $p_s^{\ast }=\frac{pN}{N-ps}$ and let $\beta <\frac{N-ps}{2}$, then \begin{equation*} \int_{{\mathbb R}^N}\int_{{\mathbb R}^N} \frac{|u(x)-u(y)|^p}{|x-y|^{N+ps}|x|^β|y|^β} \,dy\,dx\ge S(N,p,s,β)\Big(\int_{{\mathbb R}^N} \frac{|u(x)|^{p^*_{s}}}{|x|^{2β\frac{p^*_s}{p}}}\,dx\Big)^{\frac{p}{p^*_{s}}}, \end{equation*} for all $u\in {\mathcal{C}}_0^{\infty }(\Omega )$ where $S\equiv S(N,p,s,\beta )>0$. 3- If $\beta \equiv \frac{N-ps}{2}$, as a consequence of the improved Hardy inequality, we obtain that for all $q<p$, there exists a positive constant $C(\Omega )$ 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 $u\in {\mathcal{C}}_0^{\infty }(\Omega )$ where $p_{s,q}^{\ast }=\frac{pN}{N-qs}$. \ Notice that the previous inequalities can be understood as the fractional extension of the Callarelli-Kohn-Nirenberg inequalities.

24 pages, 01 figue