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

Abstract

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 $\mathbb M$.