  Most recently, in arXiv:1907.05360 [math.AP], we introduced the theory of
heatable currents and proved Onsager's conjecture on Riemannian manifolds with
boundary, where the weak solution has $B_{3,1}^{\frac{1}{3}}$ spatial
regularity. In this sequel, by applying techniques from geometric microlocal
analysis to construct the Hodge-Neumann heat kernel, we obtain off-diagonal
decay and local Bernstein estimates, and then use them to extend the result to
the Besov space $\widehat{B}_{3,V}^{\frac{1}{3}}$, which generalizes both the
space $\widehat{B}_{3,c(\mathbb{N})}^{1/3}$ from arXiv:1310.7947 [math.AP] and
the space $\underline{B}_{3,\text{VMO}}^{1/3}$ from arXiv:1902.07120 [math.AP]
-- the best known function space where Onsager's conjecture holds on flat
backgrounds.
