$\| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le C\left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f g''\|_{L^2}\right)$ holds?

Question

I want to know that the following inequality holds $$ \| f' g\|_{L^2(\mathbb R)} + \|fg'\|_{L^2(\mathbb R)}\le ^\exists C_{>0} \left( \|f g\|_{L^2} +\|f'g'\|_{L^2} + \|f'' g \|_{L^2} + \| f g''\|_{L^2}\right). $$ Here $'$ means derivatives.

Answer 1

If $f' g$ and $g\, f'$ are compact supported, the inequality holds as a consequence of the Wirtinger's inequality and the Minkowski's inequality.

I would not bet that your inequality holds without further assumptions on the decay of $f' g$ and $g\,f'$. The classical Poincaré's inequality also requires that $\Omega$ is a bounded regular domain.

To provide a counter-example, I would try something like: $$(f g')(x)=\frac{\sin\sqrt{|x|}}{\log(2+|x|) \sqrt{1+|x|}},\qquad (f' g)(x)=\frac{\cos\sqrt{|x|}}{\log(2+|x|) \sqrt{1+|x|}},$$ in order to have functions that behave like trigonometrical polynomials with a slowly increasing period. This should make the constant in the Wirtinger's inequality arbitrarily large.