I'm studying Tao's Dispersive PDE book. In section 2.6, he discusses $X^{s,b}_{\tau=h(\xi)}$ spaces, which are defined by the norm $$ \| u\|_{X^{s,b}_{\tau=h(\xi)}} = \| \langle \xi \rangle^s \langle \tau - h(\xi) \rangle^b \hat{u}(\xi,\tau) \|_{L^2_\xi L^2_\tau}.$$
Here $u: \mathbb{R}^d \times \mathbb{R} \rightarrow \mathbb{C}$, $\xi \in \mathbb{R}^d$, $\tau \in \mathbb{R}$, and $\langle y \rangle = \sqrt{1+y^2}$.
The book says it's an application of Parseval and Cauchy-Schwarz to show that the dual of $X^{s,b}_{\tau = h(\xi)}$ is $$ X^{-s,-b}_{\tau = -h(-\xi)}.$$
I can see where the negative $s$ and $b$ come from, but I don't see where the change $h \rightarrow -h(- \cdot)$ comes from. For simplicity, I assume $s=0$ and $b=1$. Then Parseval gives $$ \iint u \bar{v} \; dx dt = \iint \hat{u} \bar{\hat{v}} \; d\xi d \tau = \iint \Bigl(\langle \tau - h(\xi)\rangle\hat{u} \Bigr) \Bigl(\langle \tau - h(\xi) \rangle^{-1} \bar{\hat{v}} \Bigl) d \xi d \tau.$$ Cauchy-Schwarz bounds this by $$ \| u\|_{X^{0,1}_{\tau=h(\xi)}} \| \langle \tau - h(\xi) \rangle^{-1} \bar{\hat{v}}\|_{L^2_\xi L^2_\tau} = \| u\|_{X^{0,1}_{\tau=h(\xi)}} \| \langle \tau + h(-\xi) \rangle^{-1} \bar{\hat{v}}(-\xi, -\tau)\|_{L^2_\xi L^2_\tau}.$$
But I can't see where to go from here. Any advice would be appreciated.