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 $$ \int_{ { \mathbb R}^N}\int_{ { \mathbb R}^N} \, \frac{ |u(x)-u(y)|^{ p}}{ |x-y|^{ N+ps}}\,dx\,dy - \Lambda_{ N,p,s} \int_{ { \mathbb R}^N} \frac{ |u(x)|^p}{ |x|^{ p}}\,dx\geq C \int_{ \Omega}\dint_{ \Omega}\frac{ |u(x)-u(y)|^p}{ |x-y|^{ N+qs}}dxdy $$ 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 $p^*_{ s}=\frac{ pN}{ N-ps}$ 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|^{ \beta}|y|^{ \beta}} \,dy\,dx\ge S(N,p,s,\beta)\Big(\int_{ { \mathbb R}^N} \frac{ |u(x)|^{ p^*_{ s}}}{ |x|^{ 2\beta\frac{ p^*_s}{ p}}}\,dx\Big)^{ \frac{ p}{ p^*_{ s}}}, \end{ equation*} for all $u\in \mathcal{ C}^\infty_0(\Omega)$ 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|^{ \beta}|y|^{ \beta}} \,dy\,dx\ge C(\Omega)\Big(\int_{ \Omega} \frac{ |u(x)|^{ p^*_{ s,q}}}{ |x|^{ 2\beta \frac{ p^*_{ s,q}}{ p}}}\,dx\Big)^{ \frac{ p}{ p^*_{ s,q}}}, \end{ equation*} for all $u\in \mathcal{ C}^\infty_0(\Omega)$ 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.
Categories: math.AP