key duality relation between $X^{s,\theta}$ spaces with positive, respectively negative $\theta$.

For $ 0 < \theta < \frac12$ we have the duality relation $$X^{-s,-\theta}= (X^{s,\theta} + L^2 H^{s+\theta})'
\label{dual1}$$

a\) We first show that $$X^{-s,-\theta} \subset (X^{s,\theta} + L^2 H^{s+\theta})'$$ From Corollary \[sw\] we obtain $
X^{-s,-\theta} \subset (L^2 H^{s+\theta})' $. It remains to prove the bound $$\left|\int u\cdot f\,\, dx\,dt\right|\lesssim
\|u\|_{X^{s,\theta}}\,\|f\|_{X^{-s,-\theta}}$$ We consider Littlewood-Paley decompositions of $u$ and $v$ as in Definition \[xst\], $$u =  \sum_{{\lambda}=1}^\infty  \sum_{d=1}^{{\lambda}}  S_\lambda u_{\lambda,d},
 \qquad f = f_0 + \sum_{{\lambda}=1}^\infty  \sum_{d=1}^{{\lambda}}  \tilde \Box_{g_{<\sqrt{\lambda}}}
S_\lambda f_{\lambda,d}$$ with $\tilde \Box_{g_{<\sqrt{\lambda}}}$ in divergence form, see Corollary \[self\]. The summation with respect to ${\lambda}$ is essentially diagonal therefore it follows by orthogonality. To handle the $d$ summation it suffices to obtain the off-diagonal decay $$\begin{split}
\left | \int S_{\lambda}u_{\lambda,d_1} \cdot\tilde\Box_{g_{<\sqrt{\lambda}}}
S_{\lambda}f_{\lambda,d_2} dx dt\right|  \lesssim &
\min\left\{\left(\frac{d_2}{d_1}\right)^\theta,
 \left (\frac{d_1}{d_2}\right)^{\frac12-\theta}\right\}
\\ & \|u_{\lambda,d_1}\|_{X^{s,\theta}_{\lambda,d_1}
}\,\|f_{\lambda,d_2}\|_{X^{1-s,1-\theta}_{\lambda,d_2}}
\end{split}$$ If $d_2 < d_1$ then this follows directly from and . Otherwise we integrate by parts $$\begin{split}
 \int S_{\lambda}u_{\lambda,d_1}\cdot \, &\tilde\Box_{g_{<\sqrt{\lambda}}} S_{\lambda}f_{\lambda,d_2} dx
dt = \int\tilde \Box_{g_{<\sqrt{\lambda}}}  S_{\lambda}u_{\lambda,d_1} \cdot
S_{\lambda}f_{\lambda,d_2} dx dt \\  + & \left. \int  (S_{\lambda}u_{\lambda,d_1}\cdot g^{0\alpha}_{<\sqrt{{\lambda}}}\partial_\alpha  S_{\lambda}f_{\lambda,d_2} -
 g^{0\alpha}_{<\sqrt{{\lambda}}}\partial_\alpha  S_{\lambda}u_{\lambda,d_1} \cdot S_{\lambda}f_{\lambda,d_2}) dx \right|_{-1}^1
\end{split}$$ For the first term we use and 