  We completely characterize the weighted Lebesgue spaces on the torus $\mathbb
T^n$ and waveguide manifold $\mathbb T^n \times \mathbb R^m$ for which the
solutions of the heat equation converge pointwise (as time tends to zero) to
the initial data. In the process, we also characterize the weighted Lebesgue
spaces for the boundedness of maximal operators on the torus and waveguide
manifold, which may be of independent interest.
