estimate an fourier transform

Question

I am reading a book and don't understand an inequality. We have $\phi$ is an integrable $C^1$ function with mean value 0. Then we may write $$\hat{\phi}{(\xi)}=\int_{\mathbb{R}^n}(e^{-2\pi ix\cdot\xi}-1)\phi(x)dx$$Then how we get the estimate $$|\hat{\phi}{(\xi)}|\leq\sqrt{4\pi |\xi|}\int_{\mathbb{R}^n}|x|^{\frac{1}{2}}|\phi(x)|dx$$ I guess the author may apply the Vande Corput lemma here. But I am not sure about the application of Vande Corput lemma to multivariables and there is a function $\phi$ inside the integral. Thanks for any help!

Answer 1

Let us distinguish two cases for $y \in \Bbb{R}$.

1) We have $|y| \leq 1$. Then (since $x\mapsto e^{i x}$ is 1-Lipschitz), we have $$ |e^{2\pi iy} - 1| \leq 2\pi |y| \leq 2\pi \sqrt{|y|}. $$ Here, we used $|y| \leq \sqrt{|y|}$ because of $|y| \leq 1$.

2) We have $|y| \geq 1$. Then $$ |e^{2\pi iy } -1| \leq 2 \leq 2\pi \sqrt{|y|}. $$

Together, we see $$ |e^{2\pi iy} - 1| \leq 2\pi \sqrt{|y|} $$ for all $y \in \Bbb{R}$.

Now apply this with $y = x\cdot \xi$.

Together with the triangle inequality, this yields the desired estimate (up to slightly different constants which don't relly matter and which I leave to you to figure out).