In this talk we will try to prove the following Monotonicity inequality for a pair of operators \((L,A)\) in the space of tempered distributions \(\mathcal{S}'\), which is the dual of Schwartz space \(\mathcal{S}\)
\(\langle \phi, L\phi\rangle_p + \|A\phi\|_{HS(p)}^2 \leq C\, \|\phi\|_p^2, \ \ \ \forall \phi \in \mathcal{S} \text{and}\ p\in\mathcal{R},\)
where \(C\) is a positive constant and \(L, A\) contains forth order and second order linear differential operators respectively.
** The talk will be based on a joint work with Dr, Suprio Bhar (IIT Kanpur), online version is available on [ https://arxiv.org/abs/2312.16550 | https://arxiv.org/abs/2312.16550 ] . We will mainly concentrate in the calculations of proving above inequality.
