Norm inequality of Fourier transform

Question

Let $f\in L^1(R^d)$, where $$\hat{f}(\xi) = \int f(x) e^{-2\pi\xi i x}dx$$ and $$\hat{f} = f$$ How may we prove $$\|\hat{f}\|_p \leq \|f\|_1$$ holds?

I tried to use Holder's inequality, but end up in opposite direction, and the Hausdorff-Young was unable to deal with the gap between 1 and 2.

And insights are greatly appreciated!

Answer 1

We have : $\| \hat{f} \|_{\infty} \le \| f\|_1$
Thus, $$ \int_{\mathbb{R}} | \hat{f}(t)|^p dt \le \| f\|_1^{p-1} \int_{\mathbb{R}} | \hat{f}(t)|dt= \| f\|_1^p$$ Done.