Show that if $f\in\mathcal {S}(\mathbb R^N)$ and $\alpha\ge 0$ then the following function belongs to } $L^p(\mathbb R^N), \quad \forall p\ge 1$: $$ G(x)=\left(\widehat f(\cdot)|\cdot|^\alpha\right)^{\widecheck{}}(x)=\int_{\mathbb R^N} \widehat f(\xi)|\xi|^\alpha e^{2\pi ix\cdot\xi}d\xi%\in L^p(\mathbb R^N), \quad \forall p\ge 1. $$
I need to prove that $F$ is in $L^1$, where $F$ is gien by:
$F(x)=\displaystyle\int_{|y|\le 1}\frac{f(x+y)-f(x)-\nabla f(x)\cdot y}{|y|^{N+\alpha}}dy+ \int_{|y|>1}\frac{f(x+y)-f(x)}{|y|^{N+\alpha}}dy$,
I have bounded as follows in the case $|y|\le 1$:
$$|f(x+y)-f(x)-\nabla f(x)\cdot y| \leq ||D^2f||_{L^{\infty}} \cdot |y|^2$$
the problem is that when I plug it into the integral I have
$$\int_{\mathbb{R}^N} ||D^2f||_{L^{\infty}} \cdot C dx$$
where $C$ is the constat of the integral of $1/|y|^{N+\alpha-2}$, and this integrand does not depend on $x$ so I cannot prove anything like this.
Any hint of how to use Taylor or something to prove this?
Thank you in advance.