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 {equation*} [u]_{Ws,p, α(R^d)} := ( ∫R^d ∫R^d {|x|^{α₁ p} |y|^{α₂ p} |u(x)-u(y)|ᵖ}{|x-y|ᵈ⁺ˢᵖ} \, dx \, dy )1/p \ , {equation*} in all three regimes α₁ +α₂ < 0, \, α₁ +α₂ = 0, and α₁ +α₂ >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]_Ws,p, α(Rᵈ) < ∞, 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.