Let fractional Sobolev space $W^{s,p}(\mathbb{R}^N)\to L^q(\mathbb R^N)$ continuously for every $p\leq q\leq p^*$ where $p^*=\frac{Np}{N-p}$. This fact is proved in Theorem 6.5 of the following article:
Hitchhiker’s guide
to the fractional Sobolev spaces
(pdf link)
for $q=p^*$, i.e. for every $f\in W^{s,p}(\mathbb{R}^N)$, one has
$$
||f||_{L^{p^*}(\mathbb{R}^N)}^p\leq C\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|f(x)-f(y)|^p}{|x-y|^{N+sp}}\,dx dy.
$$
Now, the case of $q\in[p,p^*)$ "follows by Holder inequality" is written in the proof of Theorem 6.5 in the above article. I do not get how it follows from Holder inequality, since $\mathbb{R}^N$ is an unbounded domain. Can somebody kindly help, thanks.
Best Answer
Let $\|f\|_{W^{s,p}} := \|f\|_{L^p} + \|f\|_{\dot{W}^{s,p}}$, where $$\|f\|_{\dot{W}^{s,p}} := \iint_{\mathbb{R}^{2d}} \frac{|f(x)-f(y)|^p}{|x-y|^{d+sp}}\,\mathrm{d}x\,\mathrm{d}y$$
By Hölder's inequality, $$ \|f\|_{L^q} ≤ \|f\|_{L^p}^{\theta} \|f\|_{L^{p^*}}^{1-\theta} $$ with $θ = \frac{1/q - 1/p}{1/p^* - 1/p}$. As you write, by the Fractional Sobolev embedding, $$ \|f\|_{L^{p^*}} ≤ C_{d,s,p}\,\|f\|_{\dot{W}^{s,p}} $$ and so $$ \|f\|_{L^q} ≤ C_{d,s,p}^{1-\theta}\,\|f\|_{L^p}^{\theta} \|f\|_{\dot{W}^{s,p}}^{1-\theta}. $$ (This is sometimes called a Gagliardo-Nirenberg-Sobolev interpolation type inequality). Now by Young's inequality for the product, for any $a,b$ positive, it holds $a^{\theta}b^{1-\theta}≤ θ a + (1-\theta)b$, and so $a^{\theta}b^{1-\theta} ≤ a+b$. Therefore $$ \|f\|_{L^q} ≤ C_{d,s,p}^{1-\theta}\,(\|f\|_{L^p} +\|f\|_{\dot{W}^{s,p}}) = C_{d,s,p}^{1-\theta}\,\|f\|_{{W}^{s,p}}. $$