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Let $(\mathbb M, d,\mu)$ be a metric measure space with upper and lower  densities: $$ \begin{cases} |||\mu|||_{\beta}:=\sup_{(x,r)\in \mathbbM\times(0,\infty)} \mu(B(x,r))r^{-\beta}<\infty;\\|||\mu|||_{\beta^{\star}}:=\inf_{(x,r)\in \mathbb M\times(0,\infty)}\mu(B(x,r))r^{-\beta^{\star}}>0, \end{cases} $$ where $\beta, \beta^{\star}$are two positive constants which are less than or equal to the Hausdorffdimension of $\mathbb M$. Assume that $p_t(\cdot,\cdot)$ is a heat kernel on$\mathbb M$ satisfying Gaussian upper estimates and $\mathcal L$ is thegenerator of the semigroup associated with $p_t(\cdot,\cdot)$. In this paper,via a method independent of Fourier transform, we establish the decay estimatesfor 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 thefractional 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 capacitydefined in $\mathbb{M}\times(0,\infty)$, we also characterize a nonnegativeRandon measure $\nu$ on $\mathbb M\times(0,\infty)$ such that $R_\alphaL^p(\mathbb M)\subseteq L^q(\mathbb M\times(0,\infty),\nu)$ 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 $\mathbb M$.

Fractional heat semigroups on metric measure spaces with finite densities and applications to fractional dissipative equations