Fractional Sobolev embedding

Question

Let $s\in (0,1)$ and $1<p<\infty$. Let $H^{s,p}(\mathbb{R}^n)=H^{s,p}$ the Bessel potential space, defined as the image of $L^p(\mathbb{R^n})$ by the Bessel potential. It is known that these spaces con be obtained by complex interpolation means by $$[L^p,W^{1,p}]_s=H^{s,p}(\mathbb{R}^n),$$ so I'm trying to prove all the known well facts about these spaces via only interpolation. In particular im trying to prove the sobolev embedding $$H^{s,p}\subset L^{p_s^*},$$ where $$p_s^*=\frac{np}{n-sp},$$ whenever $sp<n$. To prove it im using the interpolation propertie that states that if $T:E_0+E_1\to F_0+F_1$ is a bounded linear operator from $E_0 \to F_0$ and $E_1\to F_1$, then $T$ is a bounded linear operator from $[E_0,E_1]_s\to [F_0,F_1]_s$. To use this we have to differenciate cases for the value $p$. I have proved the cases $p\leq n$, so im interested in $sp<n$ and $p>n$. My idea is to define $$I:L^p+W^{1,p}\to L^p+C^{0,1-n/p};u\mapsto u,$$ since $W^{1,p}\subset C^{0,1-n/p}$ by the Morrey embedding. Now interpolating we have that $I$ is an inclusion operator from $H^{s,p}(\mathbb{R}^n)$ to $$[L^p,C^{0,1-n/p}]_s,$$ and since both spaces could be seen as Morrey-Campanato spaces, $L^p=\mathcal{L}^{p,0}$ and $C^{0,1-n/p}=\mathcal{L}^{p,p}$, we have the embedding $$[L^p,C^{0,1-n/p}]_s\subset \mathcal{L}^{p,sp},$$ so $$H^{s,p}(\mathbb{R}^n)\subset \mathcal{L}^{p,sp}(\mathbb{R}^n),$$ however that's not what i want. Is there any form of relating the space $\mathcal{L}^{p,sp}$ with $L^{p_s^*}(\mathbb{R}^n)?$ Or maybe a more suitable choice of the endpoints for the interpolation? Any idea will be very appreciated. Thanks in advance