Pointwise convergence for the heat quation on tori Tⁿ and waveguide manifold Tⁿ Rᵐ

We completely characterize the weighted Lebesgue spaces on the torus Tⁿ and waveguide manifold Tⁿ Rᵐ 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 torus and waveguide manifold, which may be of independent interest.