in the form $$S_\lambda u = f_{\lambda,1} + \sum_{d=1}^{\frac\lambda 2}
\tilde \Box_{g_{< \sqrt\lambda}} u_{\lambda,d} + \tilde \Box_{g_{<
\sqrt\lambda}} h_{\lambda,\lambda} \qquad u_{\lambda,d} =
h_{\lambda,d} -h_{\lambda,2d}
\label{py}$$ This yields for $u$ the representation $$u = \sum_{\lambda = 1}^\infty \tilde S_\lambda \left(  f_{\lambda,1} + \sum_{d=1}^{\frac\lambda 2}
\tilde \Box_{g_{< \sqrt\lambda}} u_{\lambda,d} + \tilde \Box_{g_{<
\sqrt\lambda}} h_{\lambda,\lambda}\right)
\label{pya}$$ This is very close to but not exactly the form in . However the multipliers $\tilde S_\lambda$ can be easily replaced by $S_\lambda$ by reapplying the Paley-Littlewood decomposition on the right, and then $S_\lambda$ can be commuted to the right of $\tilde \Box_{g_{<
\sqrt\lambda}}$ due to the Corollary \[tx\] and the commutator bound . Hence we have $$\| u\|_{X^{-s,-\theta} }^2 \lesssim \sum_{\lambda = 1}^\infty
\left(\lambda^{-2s} \|f_{\lambda,1} \|_{L^2}^2 +
\sum_{d=1}^{\lambda/2}
 \|  u_{\lambda,d}\|_{X_{\lambda,d}^{1-s,1-\theta}}^2 +
 \| h_{\lambda,\lambda}\|_{X_{\lambda,\lambda}^{1-s,1-\theta}}^2\right)$$ and due to it remains to bound the right hand side by $$M + \| u\|_{L^2 H^{-s-\theta}}^2$$ There is nothing to do for the $f_{\lambda,1}$ term. On the other hand we can bound $$\begin{split}
\|u_{\lambda,d}\|_{X^{1-s,1-\theta}_{\lambda,d}}^2 \lesssim &\
\lambda^{2(1-s)} d^{2(1-\theta)} \| u_{\lambda,d}\|_{L^2}^2
+ \lambda^{-2s} d^{-2\theta}  \| \tilde \Box_{g_{<
\sqrt\lambda}} u_{\lambda,d}\|_{L^2}^2 \\
= &\ \lambda^{2(1-s)} d^{2(1-\theta)} \| h_{\lambda,d} - h_{\lambda,2d}\|_{L^2}^2
+ \lambda^{-2s} d^{-2\theta}  \| f_{\lambda,d} - f_{\lambda,2d}\|_{L^2}^2
\\
\lesssim &\ \lambda^{-2s} d^{-2\theta}
 (\|f_{\lambda,d}\|_{L^2}^2+ \|f_{\lambda,2d}\|_{L^2}^2)
\\ 