Problem 16.3 Wong, An introduction to pseudo differential operators. Let $\sigma\in S^m,\, m>0$ be an elliptic symbol and let $f\in L^p,\, 1<p<\infty$. Prove that every weak solution $u$ in $L^p$ of the pseudo differential equation $T_{\sigma}u=f$ over $\mathbb{R}^n$ is in $H^{m,p}$.
Hello. I would like to have a hint to demonstrate this problem. I know that I must prove that $\mathcal{F}^{-1}((1+|\xi|^2)^{s/2}\widehat{u}(\xi))\in L^p$ but I don't see how to relate the definition of weak solution with the Sobolev space where the weak solution belongs.
Why the weak solution $u$ , i.e., $(u,T_{\sigma}^{*}\varphi)=(f,\varphi),\quad \varphi\in\mathcal{S}$ implies that $u \in H^{m,p}$?
My attempt: \begin{align} \left\|u\right\|_{H^{m,p}}^{p}&=\int |\int e^{ix\xi}(1+|\xi|^2)^{m/2}\widehat{u}(\xi)d\xi|^p dx\\ &\leq \int |\int_{|\xi|\leq R}e^{ix\xi}(1+|\xi|^2)^{m/2}\widehat{u}(\xi) d\xi|^p dx+\int |\int_{|\xi|\geq R}e^{ix\xi}\sigma(x,\xi)\widehat{u}(\xi) d\xi|^p dx \end{align} when $|\xi|\geq R$ some $R>0.$
In the above I used the ellipticity condition on the symbol. In addition to an elementary inequality for $(1+|\xi|^2)^{m/2}\leq C(1+|\xi|^2)$ some $C$
But I am not sure how to use the inequality that appears in Theorem 16.3 to relate it to the above and be able to show that $ \left\|u\right\|_{H^{m,p}}<\infty.$