Green function for linearized Navier-Stokes around a boundary shear layer profile for long wavelengths

This paper is the continuation of a program, initiated in Grenier-Nguyen [8,9], to derive pointwise estimates on the Green function of Orr Sommerfeld equations. In this paper we focus on long wavelength perturbations, more precisely horizontal wavenumbers $\alpha$ of order ${\nu }^{1\mathrm{/}4}$, which correspond to the lower boundary of the instability area for monotonic profiles.