Whether the fractional Sobolev seminorm of any smooth function with compact support is finite

Question

Let $n\geqslant 1$, $p\in [1,\infty)$, and $s\in (0,1)$. Define the fractional Sobolev seminorm \begin{equation*} [f]_{\dot{W}^{s,p}(\mathbb{R}^n)} :=\Bigl[\frac{f(x)-f(y)}{|x-y|^{\frac{n}{p}+s}}\Bigr]_{L^p(\mathbb{R}^n\times\mathbb{R}^n)} =\Bigl(\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}}\, \mathrm{d}x\mathrm{d}y\Bigr)^\frac{1}{p}. \end{equation*}

My question is, for any $f\in C_c^\infty(\mathbb{R}^n)$ (i.e. the space of smooth functions with compact support), can we always expect $[f]_{\dot{W}^{s,p}(\mathbb{R}^n)}<\infty$?

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Remark: It seems that a paper needs this result to ensure the right hand side of an inequality is finite. Let $\mathrm{supp} f$ be the compact support of $f$. Then \begin{equation*} [f]_{\dot{W}^{s,p}(\mathbb{R}^n)}^p =2\int_{\mathbb{R}^n\setminus(\mathrm{supp} f)}\int_{\mathrm{supp} f} \frac{|f(x)|^p}{|x-y|^{n+sp}}\,\mathrm{d}x\mathrm{d}y +\int_{\mathrm{supp} f}\int_{\mathrm{supp} f}\frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}}\,\mathrm{d}x\mathrm{d}y. \end{equation*} I don't know how to continue to calculate the previous equations. So could you provide me with more details?

Answer 1

$\newcommand{\R}{\mathbb R}$Yes, this is true for any Lipschitz compactly supported function $f$.

Indeed, we have $f(x)=0$ for some real $R>0$ and all $x\in B_R^c$, where $B_R^c:=\R^n\setminus B_R$ and $B_R$ is the closed ball of radius $R$ centered at $0$, and $|f(x)-f(y)|\le L|x-y|$ for some real $L>0$ and all $x,y$. The double integral in question is \begin{equation} I:=\int_{\R^n}dx \int_{\R^n}dy\,\frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} =2I_1+I_2, \end{equation} where \begin{equation} I_1:=\int_{B_{2R}}dx \int_{B_{2R}^c}dy\,\frac{|f(x)|^p}{|x-y|^{n+sp}}, \quad I_2:=\int_{B_{2R}}dx \int_{B_{2R}}dy\,\frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}}. \end{equation} Next, \begin{equation} I_1=\int_{B_R}dx \int_{B_{2R}^c}dy\,\frac{|f(x)|^p}{|x-y|^{n+sp}} \le M^p \int_{B_R}dx \int_{B_R^c}\frac{dz}{|z|^{n+sp}}<\infty, \end{equation} where $M$ is a finite upper bound on $|f|$, and \begin{equation} I_2\le L^p\int_{B_{2R}}dx \int_{B_{2R}}\frac{dy}{|x-y|^{n-(1-s)p}} \le L^p\int_{B_{2R}}dx \int_{B_{4R}}\frac{dz}{|z|^{n-(1-s)p}}<\infty. \end{equation} Thus, $I<\infty$. $\quad\Box$