fourier transform for $H_s$ function

Question

I am looking for a result that I think is a standard in sobolev spaces, here is the question:

for $ 1\le p \le2$ and $g\in H_s (R^n)$ for $ s>n(1/p-1/2)$ I want to prove thst $g$ is a Fourier transform for a function in $L^p (R^n)$

$H_s$ is a sobolev space.

I do think this is a standard result, but I couldn`t find it. Any help please

Answer 1

By Fourier inversion you have that $g = \mathscr{F} \mathscr{F}^{-1} g$. By assumption you have $g \in H_s$ so that $$\int |\mathscr{F}^{-1}g(\xi)|^2 (1 + |\xi|^2)^{s} \mathrm{d} \xi < \infty.$$

Observe that this implies $f := \mathscr{F}^{-1} g \in L^p$, for $$ \int |f(\xi)|^p \mathrm{d}\xi = \int |f(\xi)|^p \frac{(1 + |\xi|^2)^{sp/2}}{(1 + |\xi|^2)^{sp/2}} \mathrm{d}\xi $$ and hence by Holder's inequality you get $$ \leq \left(\int |f(\xi)|^2 (1 + |\xi|^2)^s \mathrm{d}\xi\right)^{p/2} \cdot \left( \int (1 + |\xi|^2)^{sp/(2-p)} \mathrm{d}\xi \right)^{1 - p/2}.$$ To conclude, by assumption you have that $2sp/(2-p) > n$ and hence $(1 + |\xi|^2)^{sp/(2-p)}$ is integrable, so $f$ is indeed in $L^p$.