Parabolic mean value inequality and on-diagonal upper bound of the heat kernel on doubling spaces

Abstract

We prove the diagonal upper bound of heat kernels for regular Dirichlet forms on metric measure spaces with volume doubling condition. As hypotheses, we use the Faber–Krahn inequality, the generalized capacity condition and an upper bound for the integrated tail of the jump kernel. The proof goes though a parabolic mean value inequality for subcaloric functions.

Appendix

In this appendix, we collect some facts that have been used in this paper.

Proposition 8.1

([21, Proposition 15.1 in Appendix]) Let $({\mathcal {E}},{\mathcal {F}})$ be a regular Dirichlet form in $L^{2}$. Then the following statements are true.

• (i) If $u\in {\mathcal {F}}^{\prime }$ and $F:{\mathbb {R}}\mapsto \mathbb {R}$ is a Lipschitz function, then $F(u)\in {\mathcal {F}}^{\prime }$.

• (ii) If $u\in {\mathcal {F}}^{\prime }\cap L^{\infty }$ and $v\in {\mathcal {F}}\cap L^{\infty }$ then $uv\in {\mathcal {F}}\cap L^{\infty }$

• (iii) Let $\Omega$ be an open subset of M. If $u\in {\mathcal {F}}^{\prime }\cap L^{\infty }$ and $v\in {\mathcal {F}}(\Omega )\cap L^{\infty }$, then $uv\in {\mathcal {F}}(\Omega )$.

The following properties on weak differentiation of a function u were proved in [25, Lemma 5.1].

Proposition 8.2

Assume that both functions $u:I\rightarrow L^2$ and $v:I\rightarrow L^2$ are weakly differentiable at t. Then we have the following.

• (i) (Product rule) The inner product (u, v) is also differentiable at t, and

  $$\begin{aligned} (u,v)^{\prime }= (u^{\prime },v) + (u,v^{\prime }). \end{aligned}$$

• (ii) (Chain rule) Let $\Phi$ be a smooth real-valued function on ${\mathbb {R}}$ such that

  $$\begin{aligned} \Phi (0) = 0,\quad \sup _{{\mathbb {R}}}|\Phi ^{\prime }|<\infty ,\quad \sup _{ {\mathbb {R}}}|\Phi ^{\prime \prime }|<\infty . \end{aligned}$$

  Then $\Phi (u)$ is also weakly differentiable at t, and

  $$\begin{aligned} \Phi (u)^{\prime }= \Phi ^{\prime }(u)u^{\prime }. \end{aligned}$$

  (8.1)

Proposition 8.3

([21, Proposition 15.4 in Appendix]) Let $\{a_k\}_{k=0}^{\infty }$ be a sequence of non-negative numbers such that

$$\begin{aligned} a_k \le D\lambda ^ka_{k-1}^{1+\nu }\ \text { for }k=1,2,\cdots \end{aligned}$$

for some constants $D,\nu >0$ and $\lambda \ge 1$. Then for any $k\ge 0$,

$$\begin{aligned} a_k \le D^{-\frac{1}{\nu }}\Big (D^{\frac{1}{\nu }}\lambda ^{\frac{1+\nu }{\nu ^2} }a_0\Big )^{(1+\nu )^k}. \end{aligned}$$