Failure of Pólya-Szegő inequality for weighted nonlocal seminorm and on nonlocal C-K-N inequality
We create counterexamples to show that Pólya-Szegő type of inequality fails for the weighted nonlocal seminorm \begin{equation*} [u]_{W^{s,p, α}(\mathbb{R}^d)} := \left( \int_{\mathbb{R}^d} \int_{\mathbb{R}^d} \frac{|x|^{α_{1} p} |y|^{α_{2} p} |u(x)-u(y)|^{p}}{|x-y|^{d+sp}} \, dx \, dy \right)^{\frac{1}{p}} \ , \end{equation*} in all three regimes $α_1 +α_2 < 0, \, α_1 +α_2 = 0$, and $α_1 +α_2 >0$. These counterexamples are such that it completely settles the first two cases for all parameters. We also provide necessary and sufficient conditions on the parameters for $ [u]_{W^{s,p, α}(\mathbb{R}^d)} < \infty$, thus extending an earlier known result of [Serena Dipierro and Enrico Valdinoci, A density property for fractional weighted Sobolev spaces]. We also provide another proof of the fractional C-K-N type inequality, for most of the parameters (Hardy and Sobolev are included), using a familiar technique which is involved in the proof of fractional Sobolev inequality of [Eleonora Di Nezza, Giampiero Palatucci, and Enrico Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bulletin des sciences mathematiques] together with a symmetrization technique.