Assume $\chi_+$ is a smooth cut-off function away from origin, more precisely, it is defined as below: \begin{equation} \chi_+(\lambda)= \begin{cases} 1, & \lambda - \mu > 2\lambda_1, \\ 0, & \lambda- \mu \le \lambda_1, \end{cases} \end{equation} where $\lambda_1 >0$ is a small number and $\mu >0$ is a fixed constant, then how can we check the following equality: \begin{equation} \sup_{a \in \mathbb{R} } \| [\chi_+(\lambda^2+\mu) \cos(\lambda a)]^\vee \|_{\mathscr{M}} <\infty, \end{equation} where $\|\cdot \|_{\mathscr{M}}$ stands for the total variation norm of measures.
And my question is as follows:
How can we check the inequality above?
Why the auther tended to use $\|\cdot \|_{\mathscr M}$ rather than just the $L^1$-norm, what is the difference between them?
The question is origined from the estimates of $(233)$ in Wilhelm Schlag: Stable manifolds for an orbitally unstable NLS.