My question is about a claim in Stein's book"Harmonic Analysis:Real-Variable Methods, Orthogonality, and Oscillatory Integrals", page 379. It says that $$|K_{\lambda}(\xi,\eta)|\leq A_N(1+\lambda|\xi-\eta|)^{-N}, \quad N>0.$$ Where $$K_{\lambda}(\xi,\eta)=\int_{\mathbb{R}^n} e^{i\lambda[\Phi(x,\xi)-\Phi(x,\eta)]}(\dot{D}_x)^N[\psi(x,\xi)\bar{\psi}(x,\eta)]\mathrm{d}x,$$ $\dot{D}_x$ is the transpose of the operator $$D_x=[i\lambda \Delta(x,\xi,\eta)]^{-1}\nabla_{x}^{a(x)} =[i\lambda \Delta(x,\xi,\eta)]^{-1}\langle a(x),\nabla_x\rangle,\quad a(x)\in \mathbb{R}^n.$$ And where $\Delta(x,\xi,\eta)=\nabla_x^{a(x)}[\Phi(x,\xi)-\Phi(x,\eta)]$. It's also supposed that $\psi$ is a fixed smooth function of compact support (a "cut-off" function); the phase $\Phi$ is real-valued and smooth; on the support of $\psi$, the Hessian of $\Psi$ is nonvanishing.
The symbols maybe a little complex, hope I explained them clearly. You can ask me for more details or see this Stein's book.
Now, from the book (by using a partition of unity) I can get that $$\Delta(x,\xi,\eta)\geq c|\xi-\eta|, \quad (\xi,\eta)\in\mathrm{supp}K_{\lambda}$$ and I have computed that $$\dot{D}_xg(x)=-\sum_{k=1}^{n}\frac{\partial}{\partial x_k}\left[\frac{a_k(x)g(x)}{i\lambda \Delta(x,\xi,\eta)}\right].$$ Then I tried to proof this claim by directly estimate $$\int_{\mathbb{R}^n} \left|(\dot{D}_x)^N[\psi(x,\xi)\bar{\psi}(x,\eta)]\right|\mathrm{d}x.$$ It seems a considerable way but I failed. For convenience, proofing the case $N=2$ is enough. Anyone who can help me? Thanks a lot in advance.