  This paper presents an extended version of the article [Franz, S., Kopteva,
N.: J. Differential Equations, 252 (2012)]. The main improvement compared to
the latter is in that here we additionally estimate the mixed second-order
derivative of the Green's function. The case of Neumann conditions along the
characteristic boundaries is also addressed. A singularly perturbed
convection-diffusion problem is posed in the unit square with a horizontal
convective direction. Its solutions exhibit parabolic and exponential boundary
layers. Sharp estimates of the Green's function and its first- and second-order
derivatives are derived in the $L_1$ norm. The dependence of these estimates on
the small diffusion parameter is shown explicitly. The obtained estimates will
be used in a forthcoming numerical analysis of the considered problem.
