Let $X$ be a complex algebraic variety with a stratification $X_{\alpha}$ (all the strata are even dimensional) with $d = \dim X$. We write $i_{\alpha} : X_{\alpha} \to X$ for the inclusion.
We fix a perversity, i.e a function $p : \Bbb N \to \Bbb Z$ with $p(0) = 0$ and $p(n) - p(n-1) \in \{0,-1\}$ for all $n \geq 0$.
We define the perverse $t$-structure by $$^pD^{\leq 0} = \{A \mid \text{for all $\alpha$, }\mathscr H^k(i_{\alpha}^*A) = 0, \forall k > p(\alpha) \} $$ $$^pD^{\geq 0} = \{A \mid \text{for all $\alpha$, } \mathscr H^k(i^!_{\alpha}A) = 0, \forall k < p(\alpha)\}$$
Question : How to see that $\Bbb D (^pD^{\leq 0}) \subset \ ^qD^{\geq 0}$ ?
Here $q$ is the complementary perversity, defined by $q(n) = -n-p(n)$ and $\Bbb D$ is the Verdier duality functor.
What I tried :
So let $A \in \ ^pD^{\leq 0}$. We have $\mathscr H^i(i_{\alpha}^!(\Bbb D A)) = \mathscr H^i(\Bbb D i^*_{\alpha} A) = \mathscr H^i(\mathscr{RHom}(i^*_{\alpha}A,\omega_X))$
Now I assume that $\omega_X \cong \Bbb C_X[d]$. We get $$\mathscr H^i(\mathscr{RHom}(i^*_{\alpha}A,\omega_X)) = \mathscr H^{i+d}(\mathscr{RHom}(i^*_{\alpha}A, \Bbb C_X)) \cong \mathscr H^{-i-d}(i^*_{\alpha}A)$$ The claim follows in this case but I don't know how to deal with the general case.