The Heat Kernel of the Hodge Laplacian on an Invariant Subspace of 1-forms on the Hyperbolic Space ${\mathbb{H}}^n$

In this paper, we construct explicit formulas for the heat kernel of the Hodge Laplacian restricted to an invariant subspace of $1$-forms on the real hyperbolic space ${\mathbb{H}}^n$. Using warped product decompositions, the problem is reduced to component operators on specific subspaces, yielding precise closed-form expressions for both odd and even dimensions. We further establish Millson-type recurrence relations that connect heat kernels across differing dimensions. Based on these explicit representations, we derive sharp uniform estimates and long-time asymptotic behavior, extending classical scalar heat kernel bounds to $1$-forms on real hyperbolic spaces.