My attempt:
By the Fourier inversion formula, $$u(x) = (2\pi)^{-n}\int_{\mathbb{R}^n} \hat{u}(\xi) e^{i x \cdot \xi} ~d\xi,$$ $$(2\pi)^{n}|u(x)| = |\int_{\mathbb{R}^n} \hat{u}(\xi) e^{i x \cdot \xi} ~d\xi| \leq \left( \int_{\mathbb{R}^n} (1 + |\xi|)^s |\hat{u}(\xi)|^2 ~d\xi\right)^{1/2} \left( \int_{\mathbb{R}^n} (1 + |\xi|)^{-s}e^{i x \cdot \xi} ~d\xi \right)^{1/2} < \infty$$ when $s > n/2$, so the result follows.
My question:
Is $\int_{R^n}(1+|\xi|^s)^{-2}e^{ix\xi}d\xi$ bounded when $s>n/2$?
I know $\int_{R^n}(1+|\xi|^s)^{-2}d\xi$ is clearly bounded when $s>n/2$, then when adding $e^{ix\xi}$, does it still holds? I am asking this question because I am very confused about this. Thanks very much!
I do believe so, since $|e^{ix\xi}|\le 1$, so $$\int_{\Bbb R^n}(1+|\xi|^s)^{-2}e^{ix\xi}\,d\xi\le\int_{\Bbb R^n}(1+|\xi|^s)^{-2}\,d\xi<\infty.$$