Hello,

I'm looking for references on various inequalities involving the wedge product and exterior forms. The only references I could hunt down are references on Hadamard-Schwarz Inequality. The article of Iwaniec-Kauhanen-Kravetz-Scott may be quoted as an example.

More specifically, I'm interested in the validity of Cauchy-Schwarz type inequality with respect to the wedge product. As a very special case of what I have in mind, I'm interested the necessary and sufficient condition (or some reasonable sufficient condition) for the existence of $b\in\Lambda^4(\mathbb{R}^n),b\neq 0$ satisfying $$ \langle b;\omega_1\wedge\omega_2\rangle^2< \langle b;\omega_1^2\rangle \langle b;\omega_2^2\rangle,\ \langle b;\omega_1^2\rangle>0, \langle b;\omega_2^2\rangle>0, $$ where $\omega_1,\omega_2\in\Lambda^2(\mathbb{R}^n)$ is given.

Thank you.