  Let $(\mathbb M, d,\mu)$ be a metric measure space with upper and lower
  densities: $$ \begin{cases} |||\mu|||_{\beta}:=\sup_{(x,r)\in \mathbb
M\times(0,\infty)} \mu(B(x,r))r^{-\beta}<\infty;\\
|||\mu|||_{\beta^{\star}}:=\inf_{(x,r)\in \mathbb M\times(0,\infty)}
\mu(B(x,r))r^{-\beta^{\star}}>0, \end{cases} $$ where $\beta, \beta^{\star}$
are two positive constants which are less than or equal to the Hausdorff
dimension of $\mathbb M$. Assume that $p_t(\cdot,\cdot)$ is a heat kernel on
$\mathbb M$ satisfying Gaussian upper estimates and $\mathcal L$ is the
generator of the semigroup associated with $p_t(\cdot,\cdot)$. In this paper,
via a method independent of Fourier transform, we establish the decay estimates
for the kernels of the fractional heat
  semigroup $\{e^{-t \mathcal{L}^{\alpha}}\}_{t>0}$ and the operators
$\{{\mathcal{L}}^{\theta/2} e^{-t \mathcal{L}^{\alpha}}\}_{t>0}$, respectively.
By these estimates, we obtain the regularity for the Cauchy problem of the
fractional dissipative
  equation associated with $\mathcal L$ on $(\mathbb M, d,\mu)$. Moreover,
based on the geometric-measure-theoretic analysis of a new $L^p$-type capacity
defined in $\mathbb{M}\times(0,\infty)$, we also characterize a nonnegative
Randon measure $\nu$ on $\mathbb M\times(0,\infty)$ such that $R_\alpha
L^p(\mathbb M)\subseteq L^q(\mathbb M\times(0,\infty),\nu)$ under
$(\alpha,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$, where $u=R_\alpha f$
is the weak solution of the fractional diffusion equation $(\partial_t+
\mathcal{L}^\alpha)u(t,x)=0$
  in $\mathbb M\times(0,\infty)$ subject to $u(0,x)=f(x)$ in $\mathbb M$.
