In Theorem 6.5 of Hitchhiker’s guide to the fractional Sobolev spaces, it is stated that:
Let $s\in(0,1)$ and $p\in[1,+\infty)$ be such that $sp<n$. Then, there exists a positive constant $C$ depending only on $n,p,s$ such that, for any measurable and compactly supported function $f:\mathbb R^n\mapsto \mathbb R$, we have $$||f||^p_{L^{p^{\star}}(\mathbb R^n)}\le C\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}}dx\,dy \qquad (*)$$ where $p^{\star}:=\frac{np}{n-sp}$.
Then it states that
...by standard application of Holder inequality, it is trivial that $W^{s,p}(\mathbb R^n)$ is continuously embedded in $L^q(\mathbb R^n)$ for $q\in[p,p^\star]$, i.e. there exists constant $C'(n,p,s)$ such that $$||f||_{L^q(\mathbb R^n)}\le C'||f||_{W^{s,p}(\mathbb R^n)}$$
Unfortunately, I cannot see how $(*)$ would lead to the stated continuous embedding. As far as I know, by playing with Holder's inequality, we have that $$||f||_{L^q(\mathbb R^n)}\le||f||_{L^p(\mathbb R^n)}^{\alpha}||f||_{L^{p^\star}(\mathbb R^n)}^{\beta}$$
for some exponents $\alpha,\beta$ depending on $n,p,s,q$.
Question: Using $(*)$, how can one prove that there exists constant $C'(n,p,s)$ such that $$||f||_{L^q(\mathbb R^n)}\le C'||f||_{W^{s,p}(\mathbb R^n)}$$
Note that $$W^{s,p}(\mathbb R^n):=\left\{u\in L^p(\mathbb R^n):\frac{|u(x)-u(y)|}{|x-y|^{\frac np+s}}\in L^p(\mathbb R^n\times \mathbb R^n)\right\}$$
and
$$||u||_{W^{s,p}(\mathbb R^n)}=\left(\int_{\mathbb R^n}|u|^p dx+\int_{\mathbb R^n}\int_{\mathbb R^n}\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}dx\,dy\right)^{\frac 1p}$$
Let $X_{s,p} = \|f\|_{W^{s,p}} = (X_{0,p}^p + \dot{X}_{s,p}^p)^{1/p}$. You proved that $$ X_{0,q} = \|f\|_{L^q} ≤ X_{0,p}^α X_{0,p^*}^\beta $$ with $\alpha+\beta=1$. Since you also know by $(*)$ that $X_{0,p^*} ≤ C \,\dot X_{s,p}$, you get $$ X_{0,q} ≤ C^β\,X_{0,p}^α \, \dot X_{s,p}^\beta $$ and since $X_{0,p} ≤ X_{s,p}$ and $\dot X_{s,p} ≤ X_{s,p}$, you finally obtain $$ X_{0,q} ≤ C^β\,X_{s,p}^α \, X_{s,p}^\beta = C^β\,X_{s,p} $$ so $$ \|f\|_{L^q} ≤ C^β\,\|f\|_{W^{s,p}} $$