What is the meaning of $\flat$ and $\sharp$ in this smoothing operator? $M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $

Question

I found this:

$$M(x,\xi) = M^\sharp(x,\xi)+M^\flat(x,\xi),$$

in this paper, where

$$M^\sharp(x,\xi)=\sum_k \Psi_0(2^{-k\delta}D)m_k(x)\;\psi_{k+1}(\xi) $$

and i'm not sure whether i understand it correctly

Is the $\sharp$ the isomorphism $T^*M \to TM$ and $\flat$ the opposite?

Also feel free to recommend any good books of Microlocal Analysis/$\Psi DO$ if you know any.