2.1 Preliminaries

Suppose that $\mathcal {M}$ is a separable, connected, locally compact and metrisable space. Let $\mu$ be a Borel measure that is strictly positive on non-empty open sets and finite on compact sets. We consider a strongly local, closed, and regular Dirichlet form $\mathscr {E}$ on $L^{2}(\mathcal {M},\, \mu )$ with dense domain $\mathscr {D} \subset L^{2}(\mathcal {M},\, \mu )$ (see [Reference Fukushima, Oshima and Takeda15] or [Reference Gyrya and Saloff-Coste19] for an accurate definition). Suppose that $\mathscr {E}$ admits a ‘carré du champ’, which means that, for all $f,\, g \in \mathscr {D}$, the measure-valued non-negative and symmetric bilinear form $\Gamma (f,\, g)$ is absolutely continuous with respect to the measure $\mu$. In what follows, for simplicity of notation, let $\langle \nabla _{x} f,\, \nabla _{x} g\rangle$ denote the energy density ${\mathrm {d} \Gamma (f,\, g)}/{\mathrm {d} \mu }$ and $|\nabla _{x} f|$ denote the square root of ${\mathrm {d} \Gamma (f,\, f)}/{\mathrm {d} \mu }$. Assume that $(\mathcal {M},\, \mu,\, \mathscr {E})$ is endowed with the intrinsic (pseudo-)distance on $\mathcal {M}$ related to $\mathscr {E}$, which is defined by setting

\[ d(x, y)=\sup \left\{f(x)-f(y): f \in \mathscr{D}_{\text{loc}} \cap C(\mathcal{M}),\ \left|\nabla_{x} f\right| \leq 1 \text{ a.e.}\right\}, \]

where $C(\mathcal {M})$ is the space of continuous functions on $\mathcal {M}$. Suppose $d$ is indeed a distance and induces a topology equivalent to the original topology on $\mathcal {M}$. As a summary of the above situation, we will say that $(\mathcal {M},\, d,\, \mu,\, \mathscr {E})$ is a complete Dirichlet metric measure space.

Corresponding to such a Dirichlet form $\mathscr {E}$, there exists an operator denoted by $\Delta _\mathcal {M}$ (similar to the Laplacian), acting on a dense domain $\mathscr {D}(\Delta _\mathcal {M})$ in $L^2(\mathcal {M},\,\mu )$, $\mathscr {D}(\Delta _\mathcal {M})\subset W^{1,2}(\mathcal {M})$, such that for all $f\in \mathscr {D}(\Delta _\mathcal {M})$ and each $g\in W^{1,2}(\mathcal {M})$,

\[ \int_X \Delta_\mathcal{M} f(x)g(x)\mathrm{d}\mu(x)={-}\mathscr{E}(f,g), \]

where $W^{1,2}(\mathcal {M})$ is the Sobolev space equipped with the norm $([f]^{2}_{L^2}+\mathscr {E}(f,\, f))^{1/2}$ on the domain $\mathscr {D}$. The ‘Laplacian’ $\Delta _\mathcal {M}$ is the infinitesimal generator of the heat semigroup $H_t=\mathrm {e}^{t\Delta _\mathcal {M}}$, $t>0$. The heat semigroup $\{H_t\}_{t>0}$ has an integral kernel $h_t(x,\,y)$ (also called heat kernel), namely,

\[ \begin{cases} \displaystyle (\partial_t-\Delta_\mathcal{M})h_t(x,y)=0,{\quad} & \forall\,x\in \mathcal{M},\, t>0,\\ \displaystyle h_0(x,y)=\delta_{y}(x), & \forall\,x\in \mathcal{M}, \end{cases} \]

which is a non-negative measureable function on ${\mathbb {R}}_+\times \mathcal {M}\times \mathcal {M}$ such that

\[ \mathrm{e}^{t\Delta_\mathcal{M}}f(x)=\int_{\mathcal{M}} h_t( x, y) f(y) \mathrm{d} \mu(y),\quad\forall\,f\in L^2(\mathcal{M}),\, t>0; \]

see [Reference Sturm31] for more details.

To state the main result, we impose some assumptions on the underlying space and the heat kernel, define the $\sigma$-harmonic extension and introduce some function classes.

Denote by $B(x,\, r)$ the open ball with centre $x$ and radius $r$, by $V(x,\,r)$ its volume $\mu (B(x,\,r))$, and set $\lambda B(x,\, r)=B(x,\, \lambda r)$ for each $\lambda >0$. We suppose that the measure $\mu$ is doubling, i.e., there exists a constant $C_D >0$ such that

(2.1)\begin{equation} V(x,2r) \leq C_D V(x,r) < \infty,\quad \forall\,x\in \mathcal{M},\ r>0.\end{equation}

Note that the doubling property of $\mu$ implies there exists a constant $n\ge1$ such that

\[ V(x,R) \leq C_D \left(\frac{R}{r}\right)^n V(x,r), \quad \forall\,x\in \mathcal{M},\, 0< r< R<\infty, \]

and the reverse doubling property holds on a connected space (cf. [Reference Heinonen, Koskela, Shanmugalingam and Tyson21, Remark 8.1.15]), i.e., there exists a constant $0<\kappa \leq n$ such that

(2.2)\begin{equation} V(x,r) \leq C \left(\frac rR\right)^\kappa V(x,R),\quad \forall\,x\in \mathcal{M}, \, 0< r< R<\infty.\end{equation}

Obviously, the Ahlfors regular measure is doubling with $n=\kappa$.

Assume that the heat kernel $h_t(x,\,y)$ admits the so-called diagonal upper estimate

(2.3)\begin{equation} h_t(x,x)\le \frac{C}{V(x,\sqrt{t})},\quad \forall\,x\in \mathcal{M},\, t>0.\end{equation}

The conjunction of (2.1) and (2.3) is well understood: it is equivalent to a relative Faber–Krahn inequality

(2.4)\begin{equation} \lambda_1(\Omega)\ge\frac{c}{r^2}\left(\frac{V(x,r)}{\mu(\Omega)}\right)^\gamma,\quad \exists\, \gamma > 0,\end{equation}

for all balls $B(x,\, r)$ in $\mathcal {M}$ and each open set $\Omega \subset B(x,\, r)$, where $\lambda _1(\Omega )$ is the smallest Dirichlet eigenvalue of $\Delta _\mathcal {M}$ in $\Omega$; see [Reference Grigor'yan18] for more details. Under the doubling conditon (2.1), the diagonal upper estimate (2.3) self-improves into a Gaussian upper bound

(2.5)\begin{equation} h_t(x,y)\le \frac{C}{V(x,\sqrt{t})}\exp\left(-\frac{d(x,y)^2}{ct}\right),\quad \forall\,x,y\in \mathcal{M}, \, t>0;\end{equation}

see [Reference Grigor'yan17, Theorem 1.1] for the Riemannian manifold case, and [Reference Coulhon and Sikora9, Section 4.2] for the metric measure space case. Due to some technical reasons, we also assume the heat kernel $h_t(x,\,y)$ satisfies the Hölder condition of order $\theta$ ($0<\theta \le 1$)

(2.6)\begin{equation} |h_t(x,y)-h_t(x,z)| \le C\left(\displaystyle\frac{d(y,z)}{\sqrt{t}}\right)^\theta h_t(x,y),\quad \forall\,d(y,z)<\sqrt{t}. \end{equation}

On the other hand, note that the doubling property (2.1) together with the $L^2$-Poincaré inequality

(2.7)\begin{equation} \left({\unicode{x2A0F}}_{B}|f-f_B|^2\mathrm{d}\mu \right)^{1/2}\le C_Pr_B\left({\unicode{x2A0F}}_{B}|\nabla_x f|^2\mathrm{d}\mu\right)^{1/2},\quad\forall\,f\in W^{1,2}(B),\end{equation}

where $f_B$ denotes the mean (or average) of $f$ over $B$, implies that two sides Gaussian bounds for the heat kernel (also called Li-Yau's estimate), i.e., it holds

(2.8)\begin{equation} h_t(x, y) \approx \frac{1}{V(x, \sqrt{t})} \exp \left(-\frac{d(x, y)^{2}}{c t}\right),\quad \forall\,x,y\in \mathcal{M}, \, t>0;\end{equation}

see [Reference Sturm31] for example. Moreover, the two-sided estimate above is equivalent to a parabolic Harnack inequality for positive solutions to the heat equation; see [Reference Sturm31]. Therefore, under the validity of (2.1) and (2.7), the heat kernel is Hölder continuous.

2.2 Relationship between BMO and Carleson measure

The $\sigma$-harmonic extension is defined as follows.

Definition 2.1 Let $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ be a complete Dirichlet metric measure space. Given $0<\sigma <1$, for every $u\in L^1(\mathcal {M},\,{(1+d(x,\,x_0))^{-\varepsilon } V(x_0,\,1+d(x,\,x_0))^{-1}}\mathrm {d}\mu (x))$ with some $x_0\in \mathcal {M}$ and $0<\varepsilon <\min \{2\sigma,\,\kappa /2\}$, its $\sigma$-harmonic extension $U : \mathcal {M}\times {\mathbb {R}}_+ \to {\mathbb {R}}$ is the solution to

\[ \begin{cases} \displaystyle \Delta_{\mathcal{M}}U(x,t)+\frac{1-2\sigma}{t}\partial_tU(x,t)+\partial^2_tU(x,t)=0,{\quad} & \forall\,x\in \mathcal{M},\, t>0,\\ \displaystyle U(x,0)=u(x), & \forall\,x\in \mathcal{M}. \end{cases} \]

This solution is formally given by

\begin{align*} U(x,t)=P^\sigma_tu(x) & =\frac{1}{\Gamma(\sigma)}\int_0^\infty\left(\frac{t^2}{4s}\right)^\sigma \exp\left(-\frac{t^2}{4s}\right)\mathrm{e}^{s\Delta_{\mathcal{M}}}u(x)\frac{\mathrm{d} s}{s}\\ & =\frac{1}{\Gamma(\sigma)}\int_0^\infty s^\sigma \mathrm{e}^{{-}s} \mathrm{e}^{\frac{t^2}{4s}\Delta_{\mathcal{M}}}u(x)\frac{\mathrm{d} s}{s}, \end{align*}

and explicitly one has

\begin{align*} U(x,t) & =\frac{1}{\Gamma(\sigma)}\int_0^\infty\left(\frac{t^2}{4s}\right)^\sigma \exp\left(-\frac{t^2}{4s}\right)\int_\mathcal{M}h_s(x,y)u(y)\mathrm{d}\mu(y)\frac{\mathrm{d} s}{s} \\ & =\int_\mathcal{M}\left[\frac{1}{\Gamma(\sigma)}\int_0^\infty\left(\frac{t^2}{4s}\right)^\sigma \exp\left(-\frac{t^2}{4s}\right)h_s(x,y)\frac{\mathrm{d} s}{s}\right]u(y)\mathrm{d}\mu(y) \\ & =\int_\mathcal{M}p_t^\sigma(x,y)u(y)\mathrm{d}\mu(y), \end{align*}

where $P^\sigma _t$ is the $\sigma$-Poisson operator and $p^\sigma _t(x,\,y)$ is its integral kernel.

The previous definition is not explicitly stated in [Reference Stinga and Torrea30] (see also [Reference Galé, Miana and Stinga16]) but it is easy to check that the semigroup approach automatically carries over to such a geometric setting under very weak assumptions on the metric measure space; see [Reference Baudoin, Lang and Sire1] for the Dirichlet case and [Reference Brazke, Schikorra and Sire2] for the manifold case. Moreover, Brazke–Schikorra–Sire [Reference Brazke, Schikorra and Sire2] assume the boundary value $u(x)$ is a smooth function with compact support to ensure that its $\sigma$-harmonic extension $U(x,\,t)$ is vanishing at infinity

\[ \lim_{|(x,t)|\to\infty}U(x,t)=0. \]

However, via a more elaborate analysis, we can substitute the integrability of $u$ for the smoothness.

We define the following semi-norms. Denote the usual BMO norm as

\[ [u]_{{\mathrm{BMO}}}=\sup_{B}\left({\unicode{x2A0F}}_B|u-u_B|^2\mathrm{d}\mu\right)^{1/2}<\infty. \]

Furthermore, let $U(x,\,t)$ be the $\sigma$-harmonic extension of a function $u(x)$ to the upper half-space $\mathcal {M} \times {\mathbb {R}}_+$, and we introduce a notion of the Carleson measure

\[ [U]_{\mathrm{Car}}=\sup_{B}\left(\int_0^{r_B} {\unicode{x2A0F}}_{B}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}<\infty. \]

Above the two supremum are taken over all balls $B$ in $\mathcal {M}$.

The following Theorem 2.2 is the main result of this paper.

Theorem 2.2 Let $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ be a complete Dirichlet metric measure space satisfying the doubling condition (2.1). Assume that the heat kernel $h_t(x,\,y)$ admits the diagonal upper estimate (2.3) and the Hölder continuity (2.6). If $0<\sigma <1,$ then $u$ is a BMO function if and only if $U$ is a Carleson measure. Moreover, it holds

\[ [u]_{{\mathrm{BMO}}}\approx [U]_{\mathrm{Car}}. \]

Remark 2.4 To the best of our knowledge, when $(\mathcal {M},\,d,\,\mu )=({{{\mathbb {R}}}^n},\,|\cdot |,\,\mathrm {d} x)$, Theorem 2.2 is new even for the second-order elliptic operator $L=-\mathrm {div} A\nabla$, where $A=A(x)$ is an $n\times n$ matrix of real, symmetric, bounded measurable coefficients, defined on ${{{\mathbb {R}}}^n}$, and satisfies the ellipticity condition, i.e., there exist constants $0<\lambda \le \Lambda <\infty$ such that, for all $\xi \in {{{\mathbb {R}}}^n}$,

\[ \lambda |\xi|^2\le \langle A \xi, \xi\rangle\le \Lambda|\xi|^2. \]

Moreover, as we do not need any curvature condition, our result can be applied to different settings, such as Euclidean space with Muckenhoupt weight, Lie group of polynomial growth, sub-Riemannian manifold; see [Reference Coulhon, Jiang, Koskela and Sikora8, Section 7] for more details.

2.3 Two open problems

We pose two open problems related to Theorem 2.2.

Problem 1: As stated in [Reference Fefferman14], Fefferman in advance assumed the harmonic function can represented as the Poisson integral of an initial value. Later on, Fabes–Johnson–Neri [Reference Fabes, Johnson and Neri12] found that the Carleson measure condition actually characterises all harmonic functions $U(x,\,t)$ on $\mathbb {R}^{n+1}_+$ with boundary value in BMO space. They proved that a harmonic function $U$ satisfies the Carleson measure condition if and only if its trace $u$ is a BMO function.

Inspired by [Reference Fabes, Johnson and Neri12], rather than assuming $U=P^\sigma _tu$ in advance, can we seek the trace of the $\sigma$-harmonic function $U$? At this point we believe that the underlying space $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ needs to satisfy the doubling condition (2.1) and support the Poincaré inequality (2.7) (which is equivalent to Li-Yau’ estimate (2.8)); see [Reference Coulhon, Jiang, Koskela and Sikora8] and [Reference Jiang and Li22] for example.

Based on the arguments above, we pose that

The proof of the necessity relies heavily on the structure of the elliptic equation and the Poincaré inequality. The main difficulty lies in that the $\sigma$-harmonic function only satisfies a Hölder continuity condition (since $t^{1-2\sigma }$ is a Muckenhoupt $A_2$-weight) instead of smoothness or Lipschitz continuity. When $\sigma =1/2$, we can prove that such harmonic function enjoys Lipschitz regularity in the time direction; see [Reference Jiang and Li22] and [Reference Jin, Li and Shen25] for example. This together with the Carleson measure condition enables us to derive suitable control of the growth of the harmonic function. Whereas $\sigma \neq 1/2$, we do not know that how to establish $L^\infty$-estimate of $t\partial _tU(x,\,t)$. We look forward to proving the estimate

\[ |t\partial_tU(x,t)|\le C\sup_{B}\left(\int_0^{r_B}{\unicode{x2A0F}}_{B}|t\partial_tU|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}\le C[U]_{\mathrm{Car}} \]

holds for all $x\in \mathcal {M}$ and $t>0$.

Problem 2: In paper [Reference Fabes and Neri13] (see also [Reference Jiang, Xiao and Yang24]), Fabes-Neri solved the Dirichlet problem for the heat equation

\[ -\Delta V(x,t)+\partial_tV(x,t)=0 \]

on the upper half-space. They characterised the caloric function (the solution to the heat equation) by a parabolic Carleson measure condition

\[ [V]_{\mathrm{Car2}}=\sup_{B}\left(\int_0^{r^2_B}{\unicode{x2A0F}}_{B}|\sqrt{t}\nabla_{x}V|^2\mathrm{d} x\frac{\mathrm{d} t}{t}\right)^{1/2}<\infty. \]

More precisely, they showed that a caloric function $V$ satisfies the parabolic Carleson measure condition if and only if its trace $v$ is a BMO function.

In the setting of this paper, how to define the $\sigma$-heat equation? Note that a $W^{1,2}$-function $U : \mathcal {M}\times {\mathbb {R}}_+ \to {\mathbb {R}}$ is said to be $\sigma$-harmonic if it is the solution to the elliptic equation

\[ \Delta_{\mathcal{M}}U(x,t)+\frac{1-2\sigma}{t}\partial_tU(x,t)+\partial^2_tU(x,t)=0 \]

in the weak sense, namely, it holds

\[ \int_0^\infty\int_{\mathcal{M}}\langle \nabla_x U, \nabla_x \Phi\rangle t^{1-2\sigma}\mathrm{d} \mu \mathrm{d} t+\int_0^\infty\int_{\mathcal{M}}\partial_t U \partial_t\Phi t^{1-2\sigma} \mathrm{d} \mu \mathrm{d} t=0 \]

for all Lipschitz functions $\Phi$ on $\mathcal {M} \times {\mathbb {R}}_+$ with compact support, where the classical product measure $\mathrm {d}\mu \mathrm {d} t$ is replaced by the weighted measure $t^{1-2\sigma }\mathrm {d}\mu \mathrm {d} t$. Therefore the $\sigma$-caloric function (the solution to the $\sigma$-heat equation) $V : \mathcal {M}\times {\mathbb {R}}_+ \to {\mathbb {R}}$ should be also understood in the weighted product measure $t^{1-2\sigma }\mathrm {d}\mu \mathrm {d} t$, namely, it holds

\[ \int_0^\infty\int_{\mathcal{M}}\langle \nabla_x V, \nabla_x \Psi\rangle t^{1-2\sigma}\mathrm{d} \mu \mathrm{d} t-\int_0^\infty\int_{\mathcal{M}}V \partial_t\Psi t^{1-2\sigma} \mathrm{d} \mu \mathrm{d} t=0 \]

for all Lipschitz functions $\Psi$ on $\mathcal {M} \times {\mathbb {R}}_+$ with compact support.

Inspired by [Reference Fabes and Neri13], can we consider the Dirichlet problem for the $\sigma$-heat equation

\[ -\Delta_\mathcal{M}V(x,t)+\frac{1-2\sigma}{t}V(x,t)+\partial_tV(x,t)=0 \]

on the upper half-space, and study the relationship between the $\sigma$-caloric function and its trace?

Based on the above arguments, we pose a conjecture related to the $\sigma$-heat equation as follows.

Conjecture 2.6 Let $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ be a complete Dirichlet metric measure space satisfying the doubling condition (2.1) and supporting the Poincaré inequality (2.7). For every $0<\sigma <1$, a $\sigma$-caloric function $V$ satisfies the parabolic Carleson measure condition

\[ [V]_{\mathrm{Car2}}=\sup_{B}\left(\int_0^{r^2_B}{\unicode{x2A0F}}_{B}(|\sqrt{t}\nabla_{x}V|^2+|t\partial_tV|^2)\mathrm{d} \mu\frac{\mathrm{d} t}{t}\right)^{1/2}<\infty \]

if and only if its trace $u$ is a BMO function.

The parabolic Carleson measure condition in Conjecture 2.6 is different from that in [Reference Fabes and Neri13]. When $(\mathcal {M},\,d,\,\mu )=({{{\mathbb {R}}}^n},\,|\cdot |,\,\mathrm {d} x)$, these two parabolic Carleson measure conditions coincide with each other. The time derivative part $t\partial _tV$ in our parabolic Carleson measure condition is essential due to the complex structure of the parabolic equation; see [Reference Li, Ma, Shen, Wu and Zhang26] for example.

5.1 From BMO to Carleson measure

Proof. By the conservation property of the heat semigroup

\[ P_t^\sigma u_{B(x,t)}=\frac{1}{\Gamma(\sigma)}\int_0^\infty s^\sigma \mathrm{e}^{{-}s} \mathrm{e}^{\frac{t^2}{4s}\Delta_{\mathcal{M}}}u_{B(x,t)}\frac{\mathrm{d} s}{s}=u_{B(x,t)}, \]

the $\sigma$-Poisson upper bound (Proposition 3.1), and Hölder's inequality, we arrive at

\begin{align*} |U(x,t)| & \le |U(x,t)-u_{B(x,t)}|+|u_{B(x,t)}| \\ & \le C\int_{\mathcal{M}}\left(\frac{t}{t+d(x,y)}\right)^{2\sigma}\frac{|u(y)-u_{B(x,t)}|}{V(x,t+d(x,y))}\mathrm{d}\mu(y)+|u_{B(x,t)}| \\ & \le C \int_{B(x,t)} \left(\frac{t}{t+d(x,y)}\right)^{2\sigma}\frac{|u(y)-u_{B(x,t)}|}{V(x,t+d(x,y))}\mathrm{d}\mu(y) \\ & \quad + \sum_{k=1}^\infty\int_{B(x,2^kt)\setminus B(x,2^{k-1}t)} \cdots \mathrm{d}\mu(y) + |u_{B(x,t)}| \\ & \le C\sum_{k=0}^\infty \frac{1}{4^{k\sigma}}{\unicode{x2A0F}}_{B(x,2^kt)}|u(y)-u_{B(x,t)}|\mathrm{d}\mu(y)+|u_{B(x,t)}| \\ & \le C\sum_{k=0}^\infty \frac{1}{4^{k\sigma}}\sum_{i=0}^k {\unicode{x2A0F}}_{B(x,2^it)}|u(y)-u_{B(x,2^it)}|\mathrm{d}\mu(y)+|u_{B(x,t)}| \\ & \le C[u]_{{\mathrm{BMO}}}\sum_{k=0}^\infty \frac{k+1}{4^{k\sigma}}+|u_{B(x,t)}| \\ & \le C[u]_{{\mathrm{BMO}}}+|u_{B(x,t)}|<\infty, \end{align*}

which completes the proof.

Remark 5.2 From the argument used in the above lemma, we see that, if $u$ is a BMO function, then

\[ \int_{\mathcal{M}}\frac{|u(x)|}{{(1+d(x,x_0))^{\varepsilon} V(x_0,1+d(x,x_0))}}\mathrm{d}\mu(x)\le C(x_0, \epsilon)<\infty \]

for all $x_0 \in \mathcal {M}$ and $\varepsilon >0$.

Theorem 5.3 Let $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ be a complete Dirichlet metric measure space satisfying the doubling condition (2.1). Assume that the heat kernel $h_t(x,\,y)$ admits the diagonal upper estimate (2.3). For every $0<\sigma <1$, if $u$ is a BMO function, then its $\sigma$-harmonic extension $U$ is a Carleson measure. Moreover, there exists a constant $C>0$ such that

\[ [U]_{\mathrm{Car}}\le C[u]_{{\mathrm{BMO}}}. \]

Proof. We follow the argument from [Reference Brazke, Schikorra and Sire2, Proposition 4.1] without the Ahlfors regular condition and the pointwise bound on the gradient of the heat kernel.

Fixed a ball $B\subset \mathcal {M}$, we decompose

\[ \nabla_{(x,t)}U=\nabla_{(x,t)}U_1+\nabla_{(x,t)}U_2+\nabla_{(x,t)}U_3, \]

where $U_i$ is the $\sigma$-harmonic extension of $u_i$, respectively, given as

\[ \displaystyle \begin{cases} \displaystyle u_1=(u-u_B)\chi_{4B},\\ \displaystyle u_2=(u-u_B)(1-\chi_{4B}),\\ \displaystyle u_3=u_B. \end{cases} \]

It follows from the conservation property of the heat semigroup that

\[ U_3(x,t)=\frac{1}{\Gamma(\sigma)}\int_0^\infty s^\sigma \mathrm{e}^{{-}s} \mathrm{e}^{\frac{t^2}{4s}\Delta_{\mathcal{M}}}u_B(x)\frac{\mathrm{d} s}{s}=u_B, \]

and thus $\nabla _{(x,t)}U_3=0$.

In view of Proposition 4.1, we have

\begin{align*} \left(\int_0^{r_B}{\unicode{x2A0F}}_{B}|t\nabla_{(x,t)} U_1|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} & \le \left(\frac{1}{V(x_B,r_B)}\int_0^\infty\int_\mathcal{M}|t\nabla_{(x,t)} U_1|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} \\ & \le C(\sigma)\left(\frac{1}{V(x_B,r_B)}\int_{\mathcal{M}}|u_1|^2 \mathrm{d}\mu \right)^{1/2} \\ & =C(\sigma)\left(\frac{1}{V(x_B,r_B)}\int_{4B}|u-u_B|^2 \mathrm{d}\mu\right)^{1/2} \\ & \le C(\sigma)[u]_{{\mathrm{BMO}}}. \end{align*}

It remains to estimate $U_2$. In view of Lemma 3.2, we arrive at

\[ \left(\int_0^{r_B}{\unicode{x2A0F}}_{B}|t\nabla_{(x,t)} U_2|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} \le C\left(\int_0^{2r_B}{\unicode{x2A0F}}_{2B}\max_{0\le m\le2}|t^m\partial^m_tU_2|^2 \mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}. \]

Denote by $B_k$, $k\in {\mathbb {N}}$, the ball concentric around $B$ but with radius $4^k$ times the radius of $B$. Since ${\rm {\,supp\,}} u_2\subset \mathcal {M}\setminus 4B$, we deduce from the $\sigma$-Poisson upper bound (Proposition 3.1) that, for all $x\in 2B$,

\begin{align*} \max_{0\le m\le2}|t^m\partial^m_tU_2(x,t)| & \le C\int_{\mathcal{M}\setminus4B}\left(\frac{t}{t+d(x,y)}\right)^{2\sigma}\frac{|u(y)-u_B|}{V(x,t+d(x,y))}\mathrm{d}\mu(y) \\ & \le C\sum_{k=2}^\infty\int_{B_{k}\setminus B_{k-1}}\left(\frac{t}{t+d(x,y)}\right)^{2\sigma}\frac{|u(y)-u_B|}{V(x,t+d(x,y))}\mathrm{d}\mu(y)\\ & \le C\sum_{k=2}^\infty\int_{B_{k}\setminus B_{k-1}}\left(\frac{t}{4^kr_B}\right)^{2\sigma}\frac{|u-u_B|}{V(x_B,4^kr_B)}\mathrm{d}\mu \\ & \le C\sum_{k=2}^\infty\left(\frac{t}{4^kr_B}\right)^{2\sigma}{\unicode{x2A0F}}_{B_{k}}|u-u_B|\mathrm{d}\mu. \end{align*}

By the triangle inequality and a telescoping sum, it holds that

\[ {\unicode{x2A0F}}_{B_{k}}|u-u_B|\mathrm{d}\mu\le {\unicode{x2A0F}}_{B_{k}}|u-u_{B_k}|\mathrm{d}\mu+\sum_{i=1}^k|u_{B_{i-1}}-u_{B_i}|\le C\sum_{i=1}^k{\unicode{x2A0F}}_{B_i}|u-u_{B_i}|\mathrm{d}\mu. \]

Using Hölder's inequality and the definition of BMO function, we have

\[ {\unicode{x2A0F}}_{B_{k}}|u-u_B|\mathrm{d}\mu\le Ck[u]_{{\mathrm{BMO}}}. \]

Consequently one has

\[ \max_{0\le m\le2}|t^m\partial^m_tU_2(x,t)| \le C\left(\frac{t}{r_B}\right)^{2\sigma} [u]_{{\mathrm{BMO}}}\sum_{k=2}^\infty\frac{k}{4^{2k\sigma}} =C\left(\frac{t}{r_B}\right)^{2\sigma} [u]_{{\mathrm{BMO}}}. \]

This implies

\begin{align*} \left(\int_0^{r_B}{\unicode{x2A0F}}_{B}|t\nabla_{(x,t)} U_2|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} & \le C\left(\int_0^{2r_B}{\unicode{x2A0F}}_{2B}\max_{0\le m\le2}|t^m\partial^m_tU_2|^2 \mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}\\ & \le C\left(\int_0^{2r_B}{\unicode{x2A0F}}_{2B}\left(\frac{t}{r_B}\right)^{4\sigma} [u]^2_{{\mathrm{BMO}}} \mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} \\ & \le C[u]_{{\mathrm{BMO}}}, \end{align*}

which, combined with the estimates of $U_1$ and $U_3$, concludes the proof of Theorem 5.3.

5.2 From Carleson measure to BMO

Theorem 5.4 Let $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ be a complete Dirichlet metric measure space satisfying the doubling condition (2.1). Assume that the heat kernel $h_t(x,\,y)$ admits the diagonal upper estimate (2.3) and the Hölder continuity (2.6). For every $0<\sigma <1,$ if the $\sigma$-harmonic extension $U$ is a Carleson measure, then its trace $u$ is a BMO function. Moreover, there exists a constant $C>0$ such that

\[ [u]_{{\mathrm{BMO}}}\le C[U]_{\mathrm{Car}}. \]

To show Theorem 5.4, we need to introduce the definition of the $H^1$-atom. An $L^\infty$-function $a(x)$ is called a $H^1$-atom associated with a ball $B=B(x_B,\,r_B)\subset \mathcal {M}$, if it satisfies

1. (i) ${\rm {\,supp\,}} a \subset B$;

2. (ii) $[a]_{L^\infty } \leq V(x_B,\,r_B)^{-1 }$;

3. (iii) $\int _{\mathcal {M}} a \mathrm {d} \mu =0$.

The following lemma establishes the Calderón reproducing formula with respect to the $H^1$-atom.

Lemma 5.5 Let $U,\,A: \mathcal {M}\times {\mathbb {R}}_+\to {\mathbb {R}}$ be the $\sigma$-harmonic extension of $u$ and $a,$ respectively. Then it holds

\[ \int_{\mathcal{M}}ua\mathrm{d}\mu=\frac{1}{\sigma}\int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle\mathrm{d}\mu\frac{\mathrm{d} t}{t}. \]

Proof. We first claim

\[ \lim_{t\to\infty}|UA|=\lim_{t\to\infty}|Ut\partial_tA|=\lim_{t\to\infty}|At\partial_tU|=0. \]

In fact, on the one hand, it follows from the $\sigma$-Poisson formula that

\begin{align*} |U(x,t)|+|t\partial_tU(x,t)| & \le C\int_{\mathcal{M}}\left(\frac{t}{t+d(x,y)}\right)^{2\sigma}\frac{|u(y)|}{V(x,t+d(x,y))}\mathrm{d}\mu(y) \\ & \le C\int_{\mathcal{M}}\frac{t^{\min\{2\sigma,\kappa/2\}}}{(1+d(x,y))^{\min\{2\sigma,\kappa/2\}}}\frac{|u(y)|}{V(x,1+d(x,y))}\mathrm{d}\mu(y)\\ & \le C(x,\sigma,\kappa)t^{\min\{2\sigma,\kappa/2\}}. \end{align*}

On the other hand, by the $\sigma$-Poisson formula again, the definition of the $H^1$-atom, and the reverse doubling property (2.2), we arrive at

\begin{align*} |A(x,t)|+|t\partial_tA(x,t)| & \le C\int_{\mathcal{M}}\left(\frac{t}{t+d(x,y)}\right)^{2\sigma}\frac{|a(y)|}{V(x,t+d(x,y))}\mathrm{d}\mu(y) \\ & \le \frac{C}{V(x,t)}\int_{B}|a(y)|\mathrm{d}\mu(y) \\ & \le \frac{C}{V(x,1)}t^{-\kappa} \\ & \le C(x)t^{-\kappa}. \end{align*}

Therefore, one has

\[ \lim_{t\to\infty}|UA|\le C\lim_{t\to\infty}t^{\min\{2\sigma,\kappa/2\}-\kappa}=0 \]

and similarly

\[ \lim_{t\to\infty}|Ut\partial_tA|=\lim_{t\to\infty}|At\partial_tU|=0 \]

as claimed.

Below we verify the Calderón reproducing formula. Employing the integration by parts and using the above decay as $t\to \infty$, we deduce that

\begin{align*} \int_0^\infty t^{2\sigma}\partial_t(t^{1-2\sigma}\partial_t(UA))\mathrm{d} t & =\left.t^{2\sigma}t^{1-2\sigma}\partial_t(UA)\right|_{0}^\infty-\int_0^\infty 2\sigma t^{2\sigma-1}t^{1-2\sigma}\partial_t(UA)\mathrm{d} t \nonumber\\ & =\left.(Ut\partial_tA+At\partial_tU))\right|_{0}^\infty-\left.2\sigma UA\right|_{0}^\infty \nonumber\\ & ={-}\lim_{t\to0}(Ut\partial_tA+At\partial_tU)+2\sigma ua \nonumber\\ & ={-}\lim_{t\to0}t^{2\sigma}(Ut^{1-2\sigma}\partial_tA+At^{1-2\sigma}\partial_tU)+2\sigma ua \nonumber \\ & =C(\sigma)[u(-\Delta_\mathcal{M})^\sigma a+a(-\Delta_\mathcal{M})^\sigma u]\lim_{t\to0}t^{2\sigma}+2\sigma ua \nonumber\\ & =2\sigma ua, \end{align*}

where the second line from the bottom is due to the extension problem for the fractional of $-\Delta _\mathcal {M}$; see [Reference Baudoin, Lang and Sire1] and [Reference Stinga and Torrea30] for example. One may use the above identity, the integration by parts again, the vanishing properties of $Ut\partial _tA$ and $At\partial _tU$ at 0 and $\infty$, and the fact that $U$ and $A$ are $\sigma$-harmonic, to obtain

\begin{align*} 2\sigma\int_\mathcal{M}ua\mathrm{d}\mu =2\int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle \mathrm{d}\mu\frac{\mathrm{d} t}{t}, \end{align*}

see also [Reference Brazke, Schikorra and Sire2, Lemma 4.3]. This completes the proof.

Remark 5.6 It is worth noting the proof of the above lemma is analogous to that of [Reference Brazke, Schikorra and Sire2, Lemma 4.3]. However, based on some new observations, we can provide another proof. Indeed note that the structure of the elliptic equation. One obtains

\begin{align*} \int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle \mathrm{d}\mu\frac{\mathrm{d} t}{t} & =\int_0^\infty\int_\mathcal{M}\langle t^{2\sigma}\nabla_{(x,t)}U,\nabla_{(x,t)}A\rangle t^{1-2\sigma}\mathrm{d}\mu\mathrm{d} t \\ & =\int_0^\infty\int_\mathcal{M}\langle \nabla_{(x,t)}(t^{2\sigma}U),\nabla_{(x,t)}A\rangle t^{1-2\sigma} \mathrm{d}\mu\mathrm{d} t \\ & \quad- \int_0^\infty\int_\mathcal{M} U\partial_t(t^{2\sigma})\partial_t A t^{1-2\sigma}\mathrm{d}\mu\mathrm{d} t \\ & ={-}2\sigma\int_0^\infty\int_\mathcal{M} U\partial_t A \mathrm{d}\mu\mathrm{d} t, \end{align*}

and similarly

\[ \int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle \mathrm{d}\mu\frac{\mathrm{d} t}{t} ={-}2\sigma\int_0^\infty\int_\mathcal{M} A\partial_t U \mathrm{d}\mu\mathrm{d} t. \]

Therefore, we arrive at

\begin{align*} & \int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle \mathrm{d}\mu\frac{\mathrm{d} t}{t} \\ & \quad={-}\sigma\left(\int_0^\infty\int_\mathcal{M} U\partial_t A \mathrm{d}\mu\mathrm{d} t+\int_0^\infty\int_\mathcal{M} A\partial_t U \mathrm{d}\mu\mathrm{d} t\right) \\ & \quad={-}\sigma\int_\mathcal{M}\int_0^\infty \partial_t (UA)\mathrm{d} t \mathrm{d}\mu \\ & \quad = \sigma\int_\mathcal{M} ua \mathrm{d}\mu, \end{align*}

as desired.

Proof of Theorem 5.4 Step 1. We verify that, if $a(x)$ is a $H^1$-atom associated with a ball $B=B(x_B,\,r_B)\subset \mathcal {M}$, then its $\sigma$-harmonic extension $A$ satisfies

\[ \left|\int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right|\le C[U]_{{\mathrm{Car}}}. \]

One can write

\begin{align*} \left|\int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right| & \le \int_0^\infty\int_\mathcal{M}|t\nabla_{(x,t)}U||t\nabla_{(x,t)}A|\mathrm{d}\mu\frac{\mathrm{d} t}{t} \\ & =\left\{\int_{T(4B)}+\sum_{k=3}^\infty\int_{T(2^kB)\setminus T(2^{k-1}B)}\right\}\cdots \mathrm{d}\mu\frac{\mathrm{d} t}{t} \\ & =\sum_{k=2}^\infty I_k. \end{align*}

For the term $I_2$, it follows from Hölder's inequality, the square function estimate (Proposition 4.1) and the definition of the $H^1$-atom that

\begin{align*} & \int_{T(4B)}|t\nabla_{(x,t)}U||t\nabla_{(x,t)}A|\mathrm{d}\mu\frac{\mathrm{d} t}{t} \\ & \quad \le \left(\int_{T(4B)}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}\left(\int_{T(4B)}|t\nabla_{(x,t)}A|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} \\ & \quad\le CV(x_B,4r_B)^{1/2} [U]_{\mathrm{Car}}\left(\int_{B}|a|^2\mathrm{d}\mu\right)^{1/2} \\ & \quad\le CV(x_B,r_B) [U]_{\mathrm{Car}}[a]_{L^\infty}\\ & \quad\le C [U]_{\mathrm{Car}}. \end{align*}

To estimate the term $I_k$, one may invoke the space–time conversion technology (see the proof of Lemma 3.2) to obtain

(5.1)\begin{align} & \int_{T(2^kB)\setminus T(2^{k-1}B)}|t\nabla_x A|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t} \nonumber\\ & \quad\le C\int_{T(2^{k+1}B)\setminus T(2^{k-2}B)}\max_{0\le m\le2}|t^m\partial^m_tA|^2 \mathrm{d}\mu\frac{\mathrm{d} t}{t}. \end{align}

Below we claim that, for all $(x,\,t)\in T(2^{k+1}B)\setminus T(2^{k-2}B)$ with $k\ge 3$,

\[ \max_{0\le m\le2}|t^m\partial^m_tA(x,t)|\le C2^{{-}k\theta}\left(\frac{t}{2^kr_B}\right)^{2\sigma}\frac{1}{V(x_B,2^kr_B)}, \]

where $0<\theta \le 1$ as in (2.6). Indeed, from the $\sigma$-Poisson formula, it holds

\begin{align*} \max_{0\le m\le2}|t^m\partial^m_tA(x,t)| & \le C\int_0^\infty\left(\frac{t^2}{s}\right)^\sigma \exp\left(-\frac{t^2}{5s}\right)\left|\mathrm{e}^{s\Delta_\mathcal{M}}a(x)\right|\frac{\mathrm{d} s}{s} \\ & = C\int_0^\infty\left(\frac{t^2}{s}\right)^\sigma \exp\left(-\frac{t^2}{5s}\right)\left|\int_Bh_s(x,y)a(y)\mathrm{d}\mu(y)\right|\frac{\mathrm{d} s}{s}. \end{align*}

If $d(y,\,x_B)<\sqrt {s}$, then by the cancellation condition of $a(y)$

\begin{align*} \left|\int_Bh_s(x,y)a(y)\mathrm{d}\mu(y)\right| & =\left|\int_B[h_s(x,y)-h_s(x,x_B)]a(y)\mathrm{d}\mu(y)\right|\\ & \le C\int_B\left(\frac{d(y,x_B)}{\sqrt{s}}\right)^\theta h_s(x,y) |a(y)|\mathrm{d}\mu(y), \end{align*}

where the last inequality is due to the Hölder continuity of the heat kernel (2.6). Otherwise $d(y,\,x_B)\ge \sqrt {s}$, there holds from the Gaussian upper bound (2.5) that

\[ \left|\int_B h_s(x,y)a(y)\mathrm{d}\mu(y)\right| \le C\int_B\left(\frac{d(y,x_B)}{\sqrt{s}}\right)^\theta h_s(x,y) |a(y)|\mathrm{d}\mu(y). \]

Therefore an argument similar to the one used in Proposition 3.1 shows that

\begin{align*} & \max_{0\le m\le2}|t^m\partial^m_tA(x,t)| \\ & \quad \le C\int_0^\infty\left(\frac{t^2}{s}\right)^\sigma \exp\left(-\frac{t^2}{5s}\right)\left|\int_Bh_s(x,y)a(y)\mathrm{d}\mu(y)\right|\frac{\mathrm{d} s}{s} \\ & \quad \le C \int_0^\infty\left(\frac{t^2}{s}\right)^\sigma \left(\frac{d(y,x_B)}{\sqrt{s}}\right)^\theta \exp\left(-\frac{t^2}{5s}\right) \int_B h_s(x,y)|a(y)| \mathrm{d}\mu(y)\frac{\mathrm{d} s}{s} \\ & \quad \le C\int_B\left(\frac{d(y,x_B)}{t+d(x,y)}\right)^\theta\left(\frac{t}{t+d(x,y)}\right)^{2\sigma} \frac{1}{V(x,t+d(x,y))} |a(y)|\mathrm{d}\mu(y), \end{align*}

which, together with

\[ t+d(x,y)\ge c2^kr_B \]

provided $y\in B$ and $(x,\,t)\in T(2^{k+1}B)\setminus T(2^{k-2}B)$ with $k\ge 3$, implies

\begin{align*} \max_{0\le m\le2}|t^m\partial^m_tA(x,t)| & \le C\int_B\left(\frac{d(y,x_B)}{2^kr_B}\right)^\theta\left(\frac{t}{2^kr_B}\right)^{2\sigma} \frac{1}{V(x,2^kr_B)} |a(y)|\mathrm{d}\mu(y) \\ & \le C2^{{-}k\theta}\left(\frac{t}{2^kr_B}\right)^{2\sigma}\frac{1}{V(x_B,2^kr_B)}\int_B|a(y)|\mathrm{d}\mu(y)\\ & \le C2^{{-}k\theta}\left(\frac{t}{2^kr_B}\right)^{2\sigma}\frac{1}{V(x_B,2^kr_B)} \end{align*}

as claimed. By substituting this estimate into (5.1), we obtain

\begin{align*} & \int_{T(2^kB)\setminus T(2^{k-1}B)}|t\nabla_{(x,t)} A|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t} \\ & \quad \le C\int_{T(2^{k+1}B)\setminus T(2^{k-2}B)}\max_{0\le m\le2}|t^m\partial^m_tA|^2 \mathrm{d}\mu\frac{\mathrm{d} t}{t} \\ & \quad\le C\int_0^{2^{k+1}r_B}\int_{2^{k+1}B}2^{{-}2k\theta}\left(\frac{t}{2^kr_B}\right)^{4\sigma}\frac{1}{V(x_B,2^kr_B)^2} \mathrm{d}\mu\frac{\mathrm{d} t}{t}\\ & \quad\le C2^{{-}2k\theta}\frac{1}{V(x_B,2^kr_B)}. \end{align*}

This combined with Hölder's inequality yields

\begin{align*} I_k & \le \int_{T(2^kB)\setminus T(2^{k-1}B)}|t\nabla_{(x,t)}U||t\nabla_{(x,t)}A|\mathrm{d}\mu\frac{\mathrm{d} t}{t} \\ & \le \left(\int_{T(2^kB)\setminus T(2^{k-1}B)}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} \\ & \quad\times \left(\int_{T(2^kB)\setminus T(2^{k-1}B)}|t\nabla_{(x,t)}A|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} \\ & \le C2^{{-}k\theta} \left(\int_0^{2^{k}r_B}{\unicode{x2A0F}}_{2^{k}B}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} \\ & \le C2^{{-}k\theta} [U]_{\mathrm{Car}}. \end{align*}

Summing over $k$, we arrive at

\[ \left|\int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A\rangle\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right|\le C[U]_{{\mathrm{Car}}}. \]

Step 2. Let $f$ be a finite linear combinations of the $H^1$-atoms, namely,

\[ f=\sum_{j=1}^N\lambda_ja_j.\]

From the Calderón reproducing formula (Lemma 5.5) and the conclusion of Step 1, it follows that

\begin{align*} \left|\int_\mathcal{M}uf\mathrm{d}\mu\right| & =\left|\int_\mathcal{M}u\sum_{j=1}^N\lambda_ja_j\mathrm{d}\mu\right| \\ & \le \sum_{j=1}^N|\lambda_j|\left|\int_\mathcal{M}ua_j\mathrm{d}\mu\right| \\ & = \sum_{j=1}^N|\lambda_j|\frac{1}{\sigma}\left|\int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A_j\rangle\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right| \\ & \le C[U]_{{\mathrm{Car}}}\sum_{j=1}^N|\lambda_j|. \end{align*}

By taking the infimum over all admissible decompositions of $f$ on the both sides of the above inequality, and invoking the fact that the vector space of all finite linear combinations of the $H^1$-atoms is dense in the Hardy space $H^1=H^1(\mathcal {M})$, we conclude that, for any $g \in H^1(\mathcal{M})$, 

\[ \left|\int_\mathcal{M}ug\mathrm{d}\mu\right|\le C[U]_{{\mathrm{Car}}}[g]_{H^1}. \]

From this and the $H^1$-${\mathrm {BMO}}$ dual theorem of Fefferman–Stein (see [Reference Coifman and Weiss7] for instance), it follows that

\[ [u]_{{\mathrm{BMO}}}=\sup_{[g]_{H^1}\le 1}\left|\int_\mathcal{M}ug\mathrm{d}\mu\right|\le C\sup_{[g]_{H^1}\le 1} [U]_{{\mathrm{Car}}}[g]_{H^1}\le C [U]_{{\mathrm{Car}}}, \]

which completes the proof.

6.1 From CMO to Carleson measure

Theorem 6.3 Let $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ be a complete Dirichlet metric measure space satisfying the doubling condition (2.1). Assume that the heat kernel $h_t(x,\,y)$ admits the diagonal upper estimate (2.3). For every $0<\sigma <1$, if $u$ is a CMO function, then its $\sigma$-harmonic extension $U$ is a vanishing Carleson measure. Moreover, there exists a constant $C>0$ such that

\[ [U]_{\mathrm{Car}}\le C[u]_{{\mathrm{BMO}}}.\]

Proof. By Theorem 5.3, we know

\[ [U]_{\mathrm{Car}}\le C[u]_{{\mathrm{BMO}}}. \]

It remains to verify that $U$ satisfies the three limiting conditions. We only consider the last one since the proofs of the other two cases are similar. Fixed a ball $B\subset \mathcal {M}$, by the proof of Theorem 5.3, we see that

\begin{align*} \left(\int_0^{r_B}{\unicode{x2A0F}}_{B}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2} & \le C\sum_{k=2}^\infty 4^{{-}2k\sigma}\sum_{i=1}^k\left({\unicode{x2A0F}}_{4^iB}|u-u_{4^iB}|^2\mathrm{d}\mu\right)^{1/2}\\ & =C\sum_{k=2}^\infty 4^{{-}2k\sigma}\sum_{i=1}^k\eta_i(u,B). \end{align*}

For any fixed integer $i$, it follows from $\gamma _2(u)=\gamma _3(u)=0$ that

\begin{align*} \sup_{B: B\subset B(x_0,a)^\complement}\eta_i(u,B) & = \sup_{B: B\subset B(x_0,a)^\complement}\left({\unicode{x2A0F}}_{4^iB}|u-u_{4^iB}|^2\mathrm{d}\mu\right)^{1/2} \\ & \le \sup_{\substack{B: B\subset B(x_0,a)^\complement \\ 2^ir_B< a/2}}\left({\unicode{x2A0F}}_{4^iB}|u-u_{4^iB}|^2\mathrm{d}\mu\right)^{1/2} \\ & \quad +\sup_{\substack{B: B\subset B(x_0,a)^\complement \\ 2^ir_B\ge a/2}}\left({\unicode{x2A0F}}_{4^iB}|u-u_{4^iB}|^2\mathrm{d}\mu\right)^{1/2}\\ & \le \sup_{B: B\subset B(x_0,a/2)^\complement} \left({\unicode{x2A0F}}_{B}|u-u_{B}|^2\mathrm{d}\mu\right)^{1/2} \\ & \quad+\sup_{B: r_B\ge a/2}\left({\unicode{x2A0F}}_{B}|u-u_{B}|^2\mathrm{d}\mu\right)^{1/2}\to0 \end{align*}

as $a\to \infty$, and from the definition of BMO that

\[ \eta_i(u,B)=\left({\unicode{x2A0F}}_{4^iB}|u-u_{4^iB}|^2\mathrm{d}\mu\right)^{1/2} \le [u]_{{\mathrm{BMO}}}. \]

Therefore, we have

\begin{align*} \beta_3(U) & =\lim_{a\to\infty}\sup_{B: B\subset B(x_0,a)^\complement}\left(\int_0^{r_B}{\unicode{x2A0F}}_{B}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}\\ & \le C\lim_{a\to\infty}\sup_{B: B\subset B(x_0,a)^\complement}\sum_{k=2}^\infty 4^{{-}2k\sigma}\sum_{i=1}^k\eta_i(u,B)\\ & = C\lim_{a\to\infty}\sup_{B: B\subset B(x_0,a)^\complement}\left\{\sum_{k=2}^{K}+\sum_{k=K+1}^\infty\right\}4^{{-}2k\sigma}\sum_{i=1}^k\eta_i(u,B)\\ & \le C\sum_{k=2}^{K}4^{{-}2k\sigma}\sum_{i=1}^k\lim_{a\to\infty}\sup_{B: B\subset B(x_0,a)^\complement}\eta_i(u,B)+C\sum_{k=K+1}^{\infty}{ 4^{{-}k\sigma}}[u]_{{\mathrm{BMO}}} \\ & \le C4^{{-}K\sigma}[u]_{{\mathrm{BMO}}}, \end{align*}

where the constant $C$ is independent of $K$. As the positive integer $K$ is arbitrary, we derive that the last limiting condition $\beta _3(U)=0$ holds by letting $K\to \infty$. The proof is completed.

6.2 From Carleson measure to CMO

Theorem 6.4 Let $(\mathcal {M},\,d,\,\mu,\,\mathscr {E})$ be a complete Dirichlet metric measure space satisfying the doubling condition (2.1). Assume that the heat kernel $h_t(x,\,y)$ admits the diagonal upper estimate (2.3) and the Hölder continuity (2.6). For every $0<\sigma <1$, if the $\sigma$-harmonic extension $U$ is a vanishing Carleson measure, then its trace $u$ is a CMO function. Moreover, there exists a constant $C>0$ such that

\[ [u]_{{\mathrm{BMO}}}\le C[U]_{\mathrm{Car}}. \]

Proof. By Theorem 5.4, we know

\[ [u]_{{\mathrm{BMO}}}\le C[U]_{\mathrm{Car}}. \]

It remains to verify that $u$ satisfies the three limiting conditions. Let $f$ be a finite linear combinations of $H^1$-atoms, namely,

\[ f=\sum_{j=1}^N\lambda_ja_j. \]

From the Calderón reproducing formula (Lemma 5.5), and the conclusion of Step 1 in Theorem 5.4, it follows that

\begin{align*} \left|\int_\mathcal{M}uf\mathrm{d}\mu\right| & =\left|\int_\mathcal{M}u\sum_{j=1}^N\lambda_ja_j\mathrm{d}\mu\right| \\ & \le \sum_{j=1}^N|\lambda_j|\left|\int_\mathcal{M}ua_j\mathrm{d}\mu\right| \\ & = \sum_{j=1}^N|\lambda_j|\frac{1}{\sigma}\left|\int_0^\infty\int_\mathcal{M}\langle t\nabla_{(x,t)}U,t\nabla_{(x,t)}A_j\rangle\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right| \\ & \le C\sum_{k=2}^\infty 2^{{-}k\theta} \left(\int_0^{2^{k}r_B}{\unicode{x2A0F}}_{2^{k}B}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}\sum_{j=1}^N|\lambda_j|. \end{align*}

By taking the infimum over all admissible decompositions of $f$ on the both sides of the above inequality, and invoking the fact that the vector space of all finite linear combinations of the $H^1$-atoms is dense in the Hardy space $H^1(\mathcal{M})$, we conclude that, for any $g \in H^1(\mathcal{M})$, 

\[ \left|\int_\mathcal{M}ug\mathrm{d}\mu\right|\le C\sum_{k=2}^\infty 2^{{-}k\theta} \left(\int_0^{2^{k}r_B}{\unicode{x2A0F}}_{2^{k}B}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}[g]_{H^1}. \]

This, combined with the $H^1$-${\mathrm {BMO}}$ dual theorem of Fefferman–Stein (see [Reference Coifman and Weiss7] for instance), tells us that

\begin{align*} \left({\unicode{x2A0F}}_B|u-u_B|^2\mathrm{d}\mu\right)^{1/2} & \le \sup_{[g]_{H^1}\le 1}\left|\int_\mathcal{M}ug\mathrm{d}\mu\right| \\ & \le C\sum_{k=2}^\infty 2^{{-}k\theta} \left(\int_0^{2^{k}r_B}{\unicode{x2A0F}}_{2^{k}B}|t\nabla_{(x,t)}U|^2\mathrm{d}\mu\frac{\mathrm{d} t}{t}\right)^{1/2}. \end{align*}

Therefore an argument similar to the one used in Theorem 6.3 shows that the limiting conditions

\[ \gamma_1(u)=\gamma_2(u)=\gamma_3(u)=0 \]

hold, which completes the proof.