  We introduce the H\"older regularity condition for harmonic functions on
metric measure spaces and prove that under mild volume regular condition and
upper heat kernel estimate, the H\"older regularity condition, the weak
Bakry-\'Emery non-negative curvature condition, the heat kernel H\"older
continuity with or without exponential terms and the heat kernel near-diagonal
lower bound are equivalent. As applications, firstly, we prove the validity of
the so-called generalized reverse H\"older inequality on the Sierpi\'nski
carpet cable system, which was left open by Devyver, Russ, Yang (Int. Math.
Res. Not. IMRN (2023), no. 18, 15537-15583). Secondly, we prove that two-sided
heat kernel estimates alone imply gradient estimate for the heat kernel on
strongly recurrent fractal-like cable systems, which improves the main results
of the aforementioned paper. Thirdly, we obtain H\"older (Lipschitz) estimate
for heat kernel on general metric measure spaces, which extends the classical
Li-Yau gradient estimate for heat kernel on Riemannian manifolds.
