I'm trying to prove that for $f\in H^{s,p}:=\{f/f=G_s*g,\,\,g\in L^p\}$ where $G_s$ is the bessel potential: $G_s(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}(1+|w|^2)^{-s/2}e^{iw\cdot x}dw$ the following holds:
$$|f(x)-f(y)|\leq ||f||_{H^{s,p}}|x-y|^{\alpha}$$
for an adequate $\alpha$. I could prove a similar result for the $H^s$ space using the Fourier Transform, but cannot prove the first one.
My attempt: given that $||f||_{H^{s,p}}=||g||_{L^p}$
$|f(x)-f(y)|\leq \int_{\mathbb{R}^n}|g(t)||G_s(x-t)-G_s(y-t)|$, and by Holder inequality,
$|f(x)-f(y)|\leq ||f||_{H^{s,p}} ||G_s(x-t)-G_s(y-t)||_{L^{p'}}$
The problem is that
$||G_s(x-t)-G_s(y-t)||_{L^{p'}}=\left(\int_{\mathbb{R}^n}\left((1+|w|^2)^{-s/2}e^{-iw\cdot t}[e^{iw\cdot x}-e^{iw\cdot y}]dw\right)^{p'}dt\right)^{1/p'}$
I thought about using Minkowski integral inequality, but this kills the $t$ inside the integral and therefore is infinite.
I would like to come up with something like
$$||G_s(x-t)-G_s(y-t)||_{L^{p'}}\leq C \left(\int_{\mathbb{R}^n} (1+|w|^2)^{-sp'/2}(e^{-iw\cdot t}[e^{iw\cdot x}-e^{iw\cdot y}])^{p'}\right)^{1/p'}$$
Am I on the right track or missing something? Any help is highly appreciated. Regards!