Pseudo-differential operators

Question

What is the meaning of the formula $\sigma (PQ)=\sum \frac{1}{\alpha!}\partial _{\xi }^{\alpha}pD_{x}^{\alpha}q\; ;\;\; \sigma (Q)=q,\;\;\; \sigma (P)=p$

if the series on right side is infinite?

Thanks a lot!

Answer 1

The equality sign is irrelevant. It's actually the asymptotic relation $$ \sigma_{AB}(x,\xi)\sim\sum_\alpha\frac{1}{\alpha!}\partial^\alpha_{\xi}\sigma_A(x,\xi)\cdot \partial^\alpha_x\sigma_B(x,\xi) $$ where the sum is taken over all possible multi-indices $\alpha\in\mathbb{Z}_+^n$ if $(x,\xi)\in\mathbb{R}^n\times\mathbb{R}^n$. By definition, a symbol $\sigma$ is asymptotic to a series $\sum_{j=1}^\infty{\sigma_j}$ where $\sigma_j\in S^{m_j}(\Omega),\;\Omega\subset\mathbb{R}^n$ and $m_j\rightarrow -\infty$ iff for each N the following relation is satisfied: $$ (\sigma-\sum_{j=1}^N\sigma_j )\in S^{\overline{m_{N+1}}}(\Omega),\;\overline{m_k}=\max[m_j\mid j\geq k]. $$

It follows easily that two symbols asymptotic to the same series may differ by not more than a symbol of the class $S^{-\infty}(\Omega)$.