  We study Rayleigh-Benard convection in the high-Rayleigh-number and
high-Prandtl-number regime, i.e., we consider a fluid in a container that is
exposed to strong heating of the bottom and cooling of the top plate in the
absence of inertia effects. While the dynamics in the bulk are characterized by
a chaotic convective heat flow, the boundary layers at the horizontal container
plates are essentially conducting and thus the fluid is motionless.
Consequently, the average temperature exhibits a linear profile in the boundary
layers.
  In this article, we rigorously investigate the average temperature and
oscillations in the boundary layer via local bounds on the temperature field.
Moreover, we deduce that the temperature profile is indeed essentially linear
close to the horizontal container plates. Our results are uniform in the system
parameters (e.g. the Rayleigh number) up to logarithmic correction terms. An
important tool in our analysis is a new Hardy-type estimate for the convecting
velocity field, which can be used to control the fluid motion in the layer. The
bounds on the temperature field are derived with the help of local maximal
regularity estimates for convection-diffusion equations.
