Higher order fractional weighted homogeneous spaces: characterization and finer embeddings

In this article, for N ≥ 2, s ∈ (1,2), p∈ (1, N/s), σ=s-1 and a ∈ [0, N-sp/2), we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space {align*} { D}s,p_a(R^N) := {C_c∞(R^N)}^{[·]s,p,a} where [u]s,p,a := ( R^N × R^N {| ∇ u(x) -∇ u(y) |^p}{|x-y |N+σ p} \, dx/|x|^a dy/|y|^a )1/p, {align*} and higher order fractional weighted homogeneous space {align*} {W}s,p_a(R^N):= \{u ∈ L_ap^*_s(R^N): \| ∇ u \|_{L_a^{p^*σ}(R^N)} + [u]s,p,a < ∞ \} {align*} with the weighted Lebesgue norm {align*} \| u \|_{L_a^{p^*α}(R^N)}:= ( ∫R^N { |u(x)|^{p^*α}}{|x|^{{2ap^*α}{p}}} \, {dx} )^{{1}{p^*α}}, where p^*α=Np/N-α p for α= s,σ. {align*} To achieve this, we prove that Cc∞(RN) is dense in Ws,pₐ(RN) with respect to [·]s,p,a, and [·]s,p,a is an equivalent norm on Ws,pₐ(RN). Further, we obtain a finer embedding of Ds,pₐ(RN) into the Lorentz space LNp/N-sp-2a, p(RN), where LNp/N-sp-2a, p(RN) Lₐp^*ₛ(RN).