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: $$\{\begin{array}{l}|||\mu ||{|}_{\beta }:={\sup }_{(x,r)\in \mathbb{M}\times (0,\infty )}\mu (B(x,r))r^{-\beta }<\infty ;\\ |||\mu ||{|}_{{\beta }^{\star }}:={\inf }_{(x,r)\in \mathbb{M}\times (0,\infty )}\mu (B(x,r))r^{-{\beta }^{\star }}>0,\end{array}$$ 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 $\mathbb{M}$.

48 pages