arXiv2019
Let $(\mathbb M, d,μ)$ be a metric measure space with upper and lower densities: $$ \begin{cases} |||μ|||_β:=\sup_{(x,r)\in \mathbb M\times(0,\infty)} μ(B(x,r))r^{-β}<\infty;\\ |||μ|||_{β^{\star}}:=\inf_{(x,r)\in \mathbb M\times(0,\infty)} μ(B(x,r))r^{-β^{\star}}>0, \end{cases} $$ where $β, β^{\star}$ 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 $\{e^{-t \mathcal{L}^α}\}_{t>0}$ and the operators $\{\mathcal{L}^{θ/2} e^{-t \mathcal{L}^α}\}_{t>0}$, respectively. By these estimates, we obtain the regularity for the Cauchy problem of the fractional dissipative equation associated with $\mathcal L$ on $(\mathbb M, d,μ)$. 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 $ν$ on $\mathbb M\times(0,\infty)$ such that $R_αL^p(\mathbb M)\subseteq L^q(\mathbb M\times(0,\infty),ν)$ under $(α,p,q)\in (0,1)\times(1,\infty)\times(1,\infty)$, where $u=R_αf$ is the weak solution of the fractional diffusion equation $(\partial_t+ \mathcal{L}^α)u(t,x)=0$ in $\mathbb M\times(0,\infty)$ subject to $u(0,x)=f(x)$ in $\mathbb M$.