The complete heat-content asymptotic expansion for singular data

Let $M$ be a compact Riemannian manifold with smooth boundary $\partial M$, let $\delta$ denote the geodesic distance to the boundary, let $D$ be an operator of Laplace type with Dirichlet boundary conditions, and let $Q(\psi_1,\psi_2,D)(t)$ be the heat content for data satisfying $\delta^{\alpha_1}\psi_1$ and $\delta^{\alpha_2}\psi_2$ smooth near the boundary. The complete heat-content asymptotic expansion. If $\alpha_1+\alpha_2\notin\mathbb{Z}$ and $\alpha_1<2$, $\alpha_2<2$, then, as $t\downarrow0$,

$$Q(\psi_1,\psi_2,D)(t)\sim\sum_{n=0}^\infty t^{n}\beta_n^M+\sum_{j=0}^\infty t^{(1+j-\alpha_1-\alpha_2)/2}\beta_j^{\partial M}.$$

Here the $\beta_n^M$ are regularized integrals of local invariants over $M$, and the $\beta_j^{\partial M}$ are integrals of local invariants over $\partial M$. This describes the expected structure of the short-time heat-content expansion for singular initial temperature and specific heat; the source provides no resolution status beyond stating the result, so it is treated as open here.