Is it possible to show $C^\alpha$ is dense in $W^{p,\alpha}$ in fractional Sobolev norm?

Question

$C_c^\alpha(\mathbb{R})$ is Holder continuous function with compact support. $W^{p,\alpha}(\mathbb{R}),0<\alpha\leq 1$ is fractional Sobolev space with norm $$\|f\|_{W^{p,\alpha}}=\|f\|_{L^p}+\bigg[\int_\mathbb{R}\int_{\mathbb{R}}\frac{|f(x)-f(y)|^p}{|x-y|^{ps+1}}dxdy\bigg]^{1/p}$$ Is it possible to show $C^\alpha$ is dense in $W^{p,\alpha}$ in the above norm?

Answer 1

Yes, this is true. In fact, $C_c^\alpha(\mathbb{R})$ contains $C_c^\infty(\mathbb{R})$, and the latter is dense in $W^{p,\alpha}(\mathbb{R})$. The proof is a little too technical to reproduce it here in detail, you can find it in Adams. Sobolev Spaces, Theorem 7.38. But the basic idea is the usual one: You convolve the function with a mollifier to make it smooth and multiply it with a smooth cut-off function to guarantee compact support. Then you have to check that everything converges in the norm of $W^{p,\alpha}$, and that is where you have to work.