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

TL;DR

This paper develops decay estimates for fractional heat kernels on metric measure spaces with finite densities, leading to regularity results for fractional dissipative equations and a measure-theoretic characterization of solution spaces.

Contribution

It introduces a Fourier-transform-independent method to estimate fractional heat kernels and characterizes solution measures for fractional diffusion equations on metric measure spaces.

Findings

Established decay estimates for fractional heat kernels.

Proved regularity results for fractional dissipative equations.

Characterized measures related to fractional diffusion solutions.

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 $\mathbb M$.