Higher order fractional weighted homogeneous spaces: characterization and finer embeddings

Abstract

In this article, for $N\ge 2,s\in (1,2),p\in (1,\frac{N}{s}),\sigma =s-1$ and $a\in [0,\frac{N-sp}{2})$, we establish an isometric isomorphism between the higher order fractional weighted Beppo-Levi space \begin{align*} {\mathcal D}^{s,p}_a(\mathbb{R}^N) := \overline{\mathcal{C}_c^{\infty}(\mathbb{R}^N)}^{[\cdot]_{s,p,a}} \text{ where } [u]_{s,p,a} := \left( \iint_{\mathbb{R}^N \times \mathbb{R}^N} \frac{\left| \nabla u(x) -\nabla u(y) \right|^p}{\left|x-y \right|^{N+σp}} \, \frac{\mathrm{d}x}{|x|^a} \frac{\mathrm{d}y}{|y|^a} \right)^{\frac{1}{p}}, \end{align*} and higher order fractional weighted homogeneous space \begin{align*} \mathring{W}^{s,p}_a(\mathbb{R}^N):= \left\{u \in L_a^{p^*_s}(\mathbb{R}^N): \| \nabla u \|_{L_a^{p^*_σ}(\mathbb{R}^N)} + [u]_{s,p,a} < \infty \right\} \end{align*} with the weighted Lebesgue norm \begin{align*} \| u \|_{L_a^{p^*_α}(\mathbb{R}^N)}:= \left( \int_{\mathbb{R}^N} \frac{ |u(x)|^{p^*_α}}{|x|^{\frac{2ap^*_α}{p}}} \, {\mathrm{d}x} \right)^{\frac{1}{p^*_α}}, \text{ where } p^*_α=\frac{Np}{N-αp} \text{ for } α= s,σ. \end{align*} To achieve this, we prove that ${\mathcal{C}}_c^{\infty }({\mathbb{R}}^N)$ is dense in ${\mathring{W}}_a^{s,p}({\mathbb{R}}^N)$ with respect to $[\cdot {]}_{s,p,a}$, and $[\cdot {]}_{s,p,a}$ is an equivalent norm on ${\mathring{W}}_a^{s,p}({\mathbb{R}}^N)$. Further, we obtain a finer embedding of ${\mathcal{D}}_a^{s,p}({\mathbb{R}}^N)$ into the Lorentz space $L^{\frac{Np}{N-sp-2a},p}({\mathbb{R}}^N)$, where $L^{\frac{Np}{N-sp-2a},p}({\mathbb{R}}^N)⊊L_a^{p_s^{\ast }}({\mathbb{R}}^N)$.

24 pages

References in corpus (1)

• A First Course in Fractional Sobolev Spaces