# Reverse Poincaré inequality and Fourier Transform?

My advisor gave me the other day a draft of a paper he's been working on to study. The following is only a part of a proof which I am trying to figure out at the moment.

Now since, I'm afraid of asking something too trivial to my advisor, I thought to pose a couple of questions here, hoping you will shed some light on my confusion.

Questions:

1. What kind of Poincare inequality is that, in which the derivative lies on the left hand-side?
2. Is $$\partial_X^{-1} B$$ the inverse derivative of B or what?
3. Is there any way, one can modify the classical Poincare inequality (see Evans, PDEs, §5.8) using Fourier transform in order to obtain something similar to this?

I have been taught how to solve integral equations using Laplace Transformation and I am also familiar with Fourier analysis in general. However, I'm having a really hard time getting my head around this one.

Many thanks in advance for the understanding and the help!

• For Q2: Yes; notice that for any $B$ with vanishing zeroth-term in its Fourier expansion we have $\partial_X\partial_X^{-1}B=B$ (where $\partial_X$ is defined on the Fourier side via multiplication with $ik$). With this in hand, Q1 and Q3 follow: if you write $A=\partial_X^{-1}B$, you recover the usual Poincaré inequality for $A$. Dec 8, 2022 at 13:31
• @Jose27 Oh! Thank you very much for the comment. It definitely makes things a lot more clear! However, I would appreciate if you could elaborate more on the "$\partial_X$ is defined on the Fourier side via multiplication with ik " Dec 8, 2022 at 13:39