Stability Results for Symmetric Jump Processes on Metric Measure Spaces with Atoms

Abstract

Consider a symmetric Markovian jump process {X_t} on a metric measure space (M, d, μ). Chen, Kumagai, and Wang recently showed that two-sided heat kernel estimates and the parabolic Harnack inequality are both stable under bounded perturbations of the jumping measure, assuming (M, d, μ) satisfies the volume-doubling and reverse-volume-doubling conditions. These results do not apply if (M, d, μ) is a graph (or more generally, if M contains any atoms x such that μ(x) > 0) because it is impossible for reverse-volume-doubling to hold on a space with atoms. We generalize the results of Chen, Kumagai, and Wang to a larger class of metric measure spaces, including all infinite graphs with volume-doubling. Our main tool is the construction of an “auxiliary space” that smooths out the atoms. We show that many properties transfer from (M, d, μ) to the auxiliary space, and vice versa, including heat kernel estimates, the parabolic Harnack inequality, and their stable characterizations.

Appendix A: Domains of the Regular Dirichlet Forms

In this appendix we prove Proposition 8.3 and Lemma 8.4. In some cases, we include more detailed version of these proofs in [20, Appendix A].

We use the notation ∥f∥_p for the L^p-norm of a function f, regardless of whether f is a function on M or $\widehat {M}$. It follows from Eq. 3.13 that for any $g \in \mathcal {F}$,

$$ \|g \circ \pi\|_{2} = \|g\|_{2}, \qquad \widehat{\mathcal{E}}(g \circ \pi) = \mathcal{E}(g), \qquad\text{and} \qquad \widehat{\mathcal{E}}_{1}(g \circ \pi) = \mathcal{E}_{1}(g). $$

(A.1)

We use Eq. A.1 to easily prove Proposition 8.3(a), but first let us briefly recall how $\widehat {\mathcal {F}}$ was defined. In Section 3.3, we defined the bilinear form $\widehat {\mathcal {E}}$ (given by Eq. 3.28) on a subset $\widehat {\mathcal {F}}_{\max \limits }$ of $L^{2}(\widehat {M}, \widehat {\mu })$. We defined the $\widehat {\mathcal {E}}_{1}$-norm on $\widehat {\mathcal {F}}_{\max \limits }$. by Eq. 3.29. We then constructed a set $\widehat {\mathcal {D}} \subseteq \widehat {\mathcal {F}}_{\max \limits }$ in Definition 3.4, defined $\widehat {\mathcal {E}}$ as the $\widehat {\mathcal {E}}_{1}$-closure of $\widehat {\mathcal {D}}$, and used Lemma 2.2 to show that $(\widehat {\mathcal {E}}, \widehat {\mathcal {F}})$ was a regular Dirichlet form.

Proof of Proposition 8.3(a)

Recall that $\mathcal {F} \cap C_{c}(M)$ is a core of $(\mathcal {E}, \mathcal {F}$), and is therefore dense in $\mathcal {F}$ under the $\mathcal {E}_{1}$-norm. Let {g_n} be a sequence in $\mathcal {F} \cap C_{c}(M)$ such that $\|g_{n} - g\|_{\mathcal {E}_{1}} \to 0$. By Definition 3.4, $g_{n} \circ \pi \in \widehat {\mathcal {D}}$ for all n. By Eq. A.1,

$$ \|(g_{n} \circ \pi) - (g \circ \pi)\|_{\widehat{\mathcal{E}}_{1}} = \|(g_{n} - g) \circ \pi\|_{\widehat{\mathcal{E}}_{1}} = \|g_{n} - g\|_{\mathcal{E}_{1}} \to 0. $$

(A.2)

Recall that $\widehat {\mathcal {F}}$ is defined as the $\widehat {\mathcal {E}}_{1}$-closure of $\widehat {\mathcal {D}}$. Since $g_{n} \circ \pi \in \widehat {\mathcal {D}}$ for all n, Eq. A.2 means that $g \circ \pi \in \widehat {\mathcal {F}}$. □

Now let us prove Lemma 8.4. Note that if ξ and η are random variables with finite second moment, the reverse triangle inequality gives

$$ \left| \|\xi\|_{2} - \|\eta\|_{2} \right| \leq \|\xi-\eta\|_{2}. $$

(A.3)

Proof of Lemma 8.4

If

$$ \begin{array}{@{}rcl@{}} \alpha_{1} &:=& {\int}_{E_{1}} {\int}_{E_{2}} (f_{\text{mean}}(x)-f_{\text{mean}}(y))^{2} g(x, y) \mu(dy) \mu(dx),\\ \alpha_{2} &:=& {\int}_{E_{1}} {\int}_{E_{2}} (f_{\text{rms}}(x)-f_{\text{rms}}(y))^{2} g(x, y) \mu(dy) \mu(dx),\\ \text{and} \qquad \beta &:=&{\int}_{\pi^{-1}(E_{1})} {\int}_{\pi^{-1}(E_{2})} (f(z)-f(z^{\prime}))^{2} g(\pi(z), \pi(z^{\prime})) \widehat{\mu}(dz^{\prime}) \widehat{\mu}(dz), \end{array} $$

then α₁ ≤ β by Jensen’s inequality and α₂ ≤ β by Eq. A.3. □

We still need to prove that for a function $f \in \widehat {\mathcal {F}}$, both f_mean and f_rms belong to $\mathcal {F}$. For the proof of $f_{\text {mean}} \in \mathcal {F}$, we will construct a sequence $\{f_{n}\} \subseteq \widehat {\mathcal {D}}$ such that f_n → f in the $\widehat {\mathcal {E}}_{1}$-norm, and show that $\{ (f_{n})_{\text {mean}} \} \subseteq \mathcal {F}$ and (f_n)_mean → f_mean in the $\mathcal {E}_{1}$-norm. By comparing $\mathcal {E}((f_{\text {mean}})_{n} - f_{\text {mean}})$ to $\widehat {\mathcal {E}}_{1}(f_{n}-f)$, we conclude that $f_{\text {mean}} \in \mathcal {F}$.

Remark A.1

Suppose $f \in L^{2}(\widehat {M}, \widehat {\mu })$. By Jensen’s inequality, ∥f_mean∥₂ ≤∥f∥₂. By Lemma 8.4, $\mathcal {E}(f_{\text {mean}}) \leq \mathcal {E}(f)$. Consequently, $\mathcal {E}_{1}(f_{\text {mean}}) \leq \widehat {\mathcal {E}}(f)$.

Proof of Proposition 8.3(b)

Recall that $\widehat {\mathcal {F}}$ is defined as the $\widehat {\mathcal {E}}_{1}$-closure of $\widehat {\mathcal {D}}$. Since $f \in \widehat {\mathcal {F}}$, there exists a sequence $\{f_{n}\} \subseteq \widehat {\mathcal {D}}$ such that f_n → f in the $\widehat {\mathcal {E}}_{1}$-norm.

For all n, by the definition of $\widehat {\mathcal {D}}$, f_n has a representation of the form

$$ f_{n} = (g_{n} \circ \pi) + \sum\limits_{j=1}^{N_{n}} H_{{x^{n}_{j}}, {h^{n}_{j}}} $$

(A.4)

where $g_{n} \in \mathcal {F} \cap C_{c}(M)$, N_n is a non-negative integer, and ${x^{n}_{j}} \in M_{A}$ and ${h^{n}_{j}} \in \tilde {\mathcal {D}}$ for all 1 ≤ j ≤ N_n.

By taking averages over each W_x, Eq. A.4 gives

$$ (f_{n})_{\text{mean}} = g_{n} + \sum\limits_{j=1}^{N_{n}} \left( {\int}_{W} {h^{n}_{j}} d\nu\right) \delta_{x_{j}}. $$

(A.5)

For all n, g_n belongs to $\mathcal {F}$ by construction, and each $\delta _{{x^{n}_{j}}}$ belongs to $\mathcal {F} \cap C_{c}(M)$ by Lemma 2.4(a). Thus, by Eq. A.5, $(f_{n})_{\text {mean}} \in \mathcal {F}$.

By Remark A.1,

$$ \|(f_{n})_{\text{mean}} - f_{\text{mean}}\|_{\mathcal{E}_{1}} = \|(f_{n}-f)_{\text{mean}}\|_{\mathcal{E}_{1}} \leq \|f_{n}-f\|_{\widehat{\mathcal{E}}_{1}} \to 0. $$

(A.6)

Since $(f_{n})_{\text {mean}} \in \mathcal {F}$ for all n and $\mathcal {F}$ is complete with respect to the $\mathcal {E}_{1}$-norm, Eq. A.6 implies $f_{\text {mean}} \in \mathcal {F}$. □

All that remains now is to prove Proposition 8.3(c). This is by far the hardest result in this section to prove. Note that in the proof of Proposition 8.3(b), we used the fact that (f_n − f)_mean = (f_n)_mean − f_mean. Unfortunately, the same distributive property does not hold for root-mean-square averages, so we can not use the same argument to show that $f_{\text {rms}} \in \mathcal {F}$. Instead, we approximate f_rms by a series $\{g_{n}\} \subseteq \mathcal {F}$ such that g_n = f_mean except for on a finite set I_n, on which g_n = f_rms, where the increasing union $\bigcup _{n} I_{n}$ is all of M_A. We show that $\widehat {\mathcal {E}}_{1}(f_{\text {rms}} - g_{n}) \to 0$ (at least along a subsequence), but the argument is calculation-heavy and difficult to summarize, because so many terms arise from the quantity $\widehat {\mathcal {E}}(f_{\text {rms}}-g_{n})$.

We start with the following lemma. The reader should think of u in this lemma as f − (f_mean ∘ π), where f is a function in $\widehat {\mathcal {F}}$, and f_rms is the function we are trying to show belongs to $\mathcal {F}$. By Remark 8.2(a), M_A is countable.

Lemma A.2

Suppose $(M, d, \mu , \mathcal {E}, \mathcal {F})$ satisfies Assumption 1.19, $(\mathcal {E}, \mathcal {F})$ admits a jump kernel, and M = M_A ∪ M_C. Suppose $u \in \widehat {\mathcal {F}}$ and u_mean is identically 0. (In other words, ${\int \limits }_{W_{x}} u d\widehat {\mu } = 0$ for all x ∈ M_A, and u(x) = 0 for all x ∈ M_C.) Let {x_j} be an enumeration of M_A. (Since M_A is countable, such an enumeration exists.) For all n, let

$$ I_{n} = \{x_{j} : 1\leq j \leq n\},\qquad J_{n} = \{x_{j} : j>n \}, $$

and

$$ u_{n} = u \cdot 1_{(\pi^{-1}(I_{n}))}. $$

(A.7)

Then there exists a sunsequence $\{u_{n_{k}}\}$ such that $\widehat {\mathcal {E}}_{1}(u_{n_{k}}-u) \to 0$.

Proof

Clearly, $\|u-u_{n}\|_{2}^{2} = {\sum }_{j>n} \left \|u \cdot 1_{W_{x_{j}}}\right \|_{2}^{2} \to 0$ (since this is the tail of ${\sum }_{j} \left \|u \cdot 1_{W_{x_{j}}}\right \|_{2}^{2} = \|u\|_{2}^{2}$, which is finite. Therefore, u_n → u in L²-norm. All that remains to show is that $\widehat {\mathcal {E}}(u_{n_{k}} - u) \to 0$ for some subsequence {n_k}.

Recall that $\widehat {\mathcal {F}}$ is the $\widehat {\mathcal {E}}_{1}$-closure of $\widehat {\mathcal {D}}$. Since $u \in \widehat {\mathcal {F}}$, there exists a sequence $\{v_{k}\} \subseteq \widehat {\mathcal {D}}$ such that $\widehat {\mathcal {E}}_{1}(v_{k}-u) \to 0$. For all k, since $v_{k} \in \widehat {\mathcal {D}}$, we can represent v_k as

$$ v_{k} = (g_{k} \circ \pi) + \sum\limits_{j=1}^{n_{k}} H_{x_{j}, {h^{k}_{j}}} $$

where g_k belongs to $\mathcal {F} \cap C_{c}(M)$, n_k is a non-negative integer, and ${h^{k}_{j}} \in \tilde {\mathcal {D}}$ for all j ≤ n. We will show that $\widehat {\mathcal {E}}(u-u_{n_{k}}) \leq \widehat {\mathcal {E}}(u-v_{k})$.

For all $z \in \widehat {M}$,

$$ (u-v_{k})(z) = \left\{ \begin{matrix} u(z) - g_{k}(x_{j}) - H_{x_{j}, {h^{k}_{j}}}(z) &:& \text{if} z \in W_{x_{j}} \text{for some} j \leq n_{k}\\ \\ u(z)-g_{k}(x_{j}) &:& \text{if} z \in W_{x_{j}} \text{for some} j > n_{k}\\ \\ -g_{k}(z) &:& \text{if} z\in M_{C} \end{matrix}\right. $$

(A.8)

and

$$ (u-u_{n_{k}})(z) = \left\{ \begin{matrix} u(z) &:& \text{if} z \in \pi^{-1}(J_{n_{k}})\\ \\ 0 &:& \text{otherwise}. \end{matrix}\right. $$

(A.9)

For all x ∈ M_A and $C \in \mathbb {R}$, since the mean of u on W_x is 0,

$$ {\int}_{W_{x}} (u-C)^{2} d\widehat{\mu} \geq {\int}_{W_{x}} u^{2} d\widehat{\mu}. $$

(A.10)

Similarly, if x and y are distinct elements of M_A, then

$$ {\int}_{W_{x} \times W_{y}} (u(z)-u(z^{\prime})-C)^{2} d\widehat{\mu} d\widehat{\mu} \geq {\int}_{W_{x} \times W_{y}} (u(z)-u(z^{\prime}))^{2} d\widehat{\mu} d\widehat{\mu} \qquad\text{for all} C \in \mathbb{R}. $$

(A.11)

For the sake of brevity, for any $S \subseteq \widehat {M} \times \widehat {M}$, let

$$ \alpha(S) := {\int}_{(z, z^{\prime}) \in E} \left( (u-u_{n_{k}})(z) - (u-u_{n_{k}})(z^{\prime})\right)^{2} \widehat{J}(dz, dz^{\prime}) $$

and

$$ \beta(S) := {\int}_{(z, z^{\prime}) \in E} \left( (u-v_{k})(z) - (u-v_{k})(z^{\prime})\right)^{2} \widehat{J}(dz, dz^{\prime}) $$

We would like to show that $\alpha (\widehat {M}\times \widehat {M}) = \widehat {\mathcal {E}}(u_{n_{k}}-u)$ is less than or equal to $\beta (\widehat {M}\times \widehat {M}) = \widehat {\mathcal {E}}(u-v_{k})$. We will do so by breaking $\widehat {M} \times \widehat {M}$ into various sets S, and showing that α(S) ≤ β(S) for each one. In each case, we use Eqs. A.8 and A.9 to express α(S) and β(S), and use Eqs. A.10 and A.11 to conclude that β(S) is the larger one.

Note that $\widehat {M} = M_{C} \cup \pi ^{-1}(I_{n_{k}}) \cup \pi ^{-1}(J_{n_{k}})$. By Eq. A.9, whenever z and $z^{\prime }$ both belong to $M_{C} \cup \pi ^{-1}(I_{n_{k}})$, we have $(u-u_{n_{k}})(z)-(u-u_{n_{k}})(z^{\prime }) = 0$. Therefore,

$$ \alpha \left( \left( M_{C} \cup \pi^{-1}(I_{n_{k}}) \right) \times \left( M_{C} \cup \pi^{-1}(I_{n_{k}}) \right) \right) = 0 \leq \beta \left( \left( M_{C} \cup \pi^{-1}(I_{n_{k}}) \right) \times \left( M_{C} \cup \pi^{-1}(I_{n_{k}}) \right) \right). $$

(A.12)

If $x \in J_{n_{k}}$ and $z^{\prime } \in M_{C} \cup \pi ^{-1}(I_{n})$, then

$$ \begin{array}{@{}rcl@{}} &&\quad{\int}_{W_{x}} ((u-v_{k})(z) - (u-v_{k})(z^{\prime}))^{2} \widehat{\mu}(dz)\\ &=& {\int}_{W_{x}} (u(z)-g_{k}(x)-(u-v_{k})(z^{\prime}))^{2} \widehat{\mu}(dz) \qquad\text{(by Eq.~A.8)}\\ &\geq& {\int}_{W_{x}} u^{2}(z) \widehat{\mu}(dz) \qquad\text{(by Eq.~A.10, with} C = g_{k}(x) + (u-v_{k})(z^{\prime}))\\ &=& {\int}_{W_{x}} ((u-u_{n_{k}})(z) - (u-u_{n_{k}})(z^{\prime}))^{2} \widehat{\mu}(dz) \qquad\text{(by Eq.~A.9)}. \end{array} $$

Therefore,

$$ \begin{array}{@{}rcl@{}} {}\beta(\pi^{-1}(J_{n_{k}}) \times (M_{C} \cup \pi^{-1}(I_{n}))) &=& \sum\limits_{x \in J_{n_{k}}} {\int}_{z^{\prime} \in M_{C} \cup \pi^{-1}(I_{n_{k}})} J(x, \pi(z^{\prime})) {\int}_{W_{x}} ((u-v_{k})(z)\\ &&- (u-v_{k})(z^{\prime}))^{2} \widehat{\mu}(dz) \widehat{\mu}(dz^{\prime})\\ &\geq& \sum\limits_{x \in J_{n_{k}}} {\int}_{z^{\prime} \in M_{C} \cup \pi^{-1}(I_{n_{k}})} J(x, \pi(z^{\prime})) {\int}_{W_{x}} ((u-u_{n_{k}})(z) \\ &&- (u-u_{n_{k}})(z^{\prime}))^{2} \widehat{\mu}(dz) \widehat{\mu}(dz^{\prime}) \\ &=& \alpha(\pi^{-1}(J_{n_{k}}) \times (M_{C} \cup \pi^{-1}(I_{n}))). \end{array} $$

(A.13)

If x and y are distinct elements of $J_{n_{k}}$, then

$$ \begin{array}{@{}rcl@{}} \beta(W_{x} \times W_{y}) &=& J(x, y) {\int}_{z \in W_{x}} {\int}_{z^{\prime} \in W_{y}} \left( u(z) - g_{k}(x) - u(z^{\prime}) + g_{k}(y)\right)^{2} \widehat{\mu}(dz^{\prime}) \widehat{\mu}(dz) \qquad\text{(by Eq.~A.8)}\\ &\geq& J(x, y) {\int}_{W_{x} \times W_{y}} (u(z)-u(z^{\prime}))^{2} \widehat{\mu}(dz^{\prime}) \widehat{\mu}(dz) \qquad\text{(by Eq.~A.11 with $C=g_{k}(x)-g_{k}(y)$)}\\ &=& \alpha(W_{x} \times W_{y}) \qquad\text{(by Eq.~A.9)}. \end{array} $$

(A.14)

If $x \in J_{n_{k}}$, then by Eqs. A.8 and A.9,

$$ \alpha(W_{x} \times W_{x}) = {\int}_{W_{x} \times W_{x}} (u(z)-u(z^{\prime}))^{2} \widehat{J}(dz, dz^{\prime}) = \beta(W_{x} \times W_{x}). $$

(A.15)

By Eqs. A.12–A.15, we see that $\alpha (\widehat {M} \times \widehat {M}) \leq \beta (\widehat {M} \times \widehat {M})$, or $\widehat {\mathcal {E}}(u-u_{n_{k}}) \leq \widehat {\mathcal {E}}(u-v_{k})$. Since $\widehat {\mathcal {E}}(u-v_{k}) \to 0$, this means that $\widehat {\mathcal {E}}(u-u_{n_{k}}) \to 0$ as $k \to \infty$. □

Note that the key idea in our proof of Lemma A.2 (using Eqs. A.10 and A.11 to compare energies) would not work if it weren’t for the assumption that u_mean is identically 0. Let us now prove that $f_{\text {rms}} \in \mathcal {F}$ for all $f \in \widehat {\mathcal {F}}$. Recall that our strategy is to approximate f_rms by a series $\{g_{n}\} \subseteq \mathcal {F}$ such that g_n = f_mean except for on a finite set I_n, on which g_n = f_rms (where the increasing union $\bigcup _{n} I_{n}$ is all of M_A), and show that $\widehat {\mathcal {E}}_{1}(f_{\text {rms}} - g_{n}) \to 0$ along a subsequence. We decompose the quantity $\widehat {\mathcal {E}}_{1}(f_{\text {rms}} - g_{n})$ into a sum of several terms, each of which we show goes to 0 (along a subsequence). One of these terms is bounded above by $\widehat {\mathcal {E}}(u-u_{n_{k}})$, where u = f − (f_mean ∘ π). We handle this term using Lemma A.2.

Proof of Proposition 8.3(c)

Let us assume without loss of generality that f ≥ 0. We can do this because $|f| \in \widehat {\mathcal {F}}$ and |f|_rms = f_rms, so proving the result for |f| proves it for f.

As in Lemma A.2, let {x_j} be an enumeration of M_A. Let u := f − (f_mean ∘ π). Note that $u \in \widehat {\mathcal {F}}$ and u_mean is identically 0, just like in the setting of Lemma A.2. Let {u_n} be as in Eq. A.7.

Even though we have not yet shown that $f_{\text {rms}} \in \mathcal {F}$, we do know that by Lemma 8.4 that $\mathcal {E}(f_{\text {rms}}) \leq \widehat {\mathcal {E}}(f) < \infty$, and that $\|f_{\text {rms}}\|_{2}^{2} = \|f\|_{2}^{2} < \infty$, so $f_{\text {rms}} \in \mathcal {F}_{\max \limits }$.

Let h := f_rms − f_mean. By Proposition 8.3(b), $f_{\text {mean}} \in \mathcal {F} \subseteq \mathcal {F}_{\max \limits }$. Since $\mathcal {F}_{\max \limits }$ is a vector space, $h \in \mathcal {F}_{\max \limits }$. Thus,

$$ \begin{array}{@{}rcl@{}} \infty > \mathcal{E}(h) &=& {\int}_{M\times M \setminus \text{diag}_{M}} (h(x)-h(y))^{2} J(dx, dy)\\ &=& {\int}_{M_{A} \times M_{A} \setminus \text{diag}_{M_{A}}} (h(x)-h(y))^{2} J(dx, dy) \\ &&+ 2 {\int}_{x \in M_{A}} {\int}_{y \in M_{C}} h^{2}(x) J(dx, dy) \qquad\text{(since \textit{h} is 0 on $M_{C}$)}. \end{array} $$

In particular, we have both

$$ {\int}_{M_{A} \times M_{A} \setminus \text{diag}_{M_{A}}} (h(x)-h(y))^{2} J(dx, dy) < \infty $$

(A.16)

and

$$ {\int}_{x \in M_{A}} {\int}_{y \in M_{C}} h^{2}(x) J(dx, dy) < \infty. $$

(A.17)

For all n, let

$$ g_{n} := f_{\text{mean}} + {\sum}_{j \leq n} h(x_{j}) \delta_{x_{j}} $$

so that

$$ f_{\text{rms}} - g_{n} = {\sum}_{j>n} h(x_{j}) \delta_{x_{j}}. $$

(A.18)

By Proposition 8.3(b) and Lemma 2.4(a), $g_{n} \in \mathcal {F}$. We will show that some subsequence of {g_n} converges to f_rms under the $\mathcal {E}_{1}$-norm.

First,

$$ \begin{array}{@{}rcl@{}} \|f_{\text{rms}}-g_{n}\|_{2}^{2} &=& {\sum}_{j>n} \mu(x_j) h^2(x_j) \\ &=& {\sum}_{j>n} \mu(x_j) \left[ f_{\text{rms}}(x_j) - f_{\text{mean}}(x_j) \right]^2\\ &\leq& {\sum}_{j>n}\mu(x_j) f_{\text{rms}}^2(x_j) \to 0 \end{array} $$

(A.19)

since ${\sum }_{j>n}\mu (x_{j}) f_{\text {rms}}^{2}(x_{j})$ is the tail of ${\sum }_{j} \mu (x_{j}) f_{\text {rms}}(x_{j}) = {\int \limits }_{M_{A}} f_{\text {rms}}^{2} d\mu = {\int \limits }_{\pi ^{-1}(M_{A})} f^{2} d\widehat {\mu }$, which is finite.

By Eq. A.18,

$$ \mathcal{E}(f_{\text{rms}}-g_{n}) = {\int}_{J_{n} \times J_{n} \setminus \text{diag}_{J_{n}}} (h(x)-h(y))^{2} J(dx, dy) + 2 {\int}_{x \in J_{n}} {\int}_{y \in M\setminus J_{n}} h^{2}(x) J(dx, dy). $$

(A.20)

Note that the decreasing intersection $\bigcap _{n \in \mathbb {N}} J_{n} \times J_{n} \setminus \text {diag}_{J_{n}}$ is empty. Therefore, by Eq. A.16 and continuity from above,

$$ \lim_{n\to\infty} {\int}_{J_{n}\times J_{n} \setminus \text{diag}_{J_{n}}} (h(x)-h(y))^{2} J(dx, dy) = 0. $$

(A.21)

For all x ∈ M_A, f ξ is uniformly distributed on W_x, then by Eq. A.3,

$$ h(x) = f_{\text{rms}}(x) - f_{\text{mean}}(x) = \|f(\xi)\|_{2} - \|f_{\text{mean}}(x)\|_{2} \leq \|f(\xi)-f_{\text{mean}}(x)\|_{2} = u_{\text{rms}}(x). $$

Therefore,

$$ \begin{array}{@{}rcl@{}} {}{\int}_{x \in J_{n}} {\int}_{y \in M\setminus J_{n}} h^{2}(x) J(dx,dy) &\leq& {\int}_{x \in J_{n}} {\int}_{y \in M\setminus J_{n}} u_{\text{rms}}^{2}(x) J(dx,dy)\\ &\leq& {\int}_{z \in \widehat{J_{n}}} {\int}_{z^{\prime} \in \widehat{M}\setminus \widehat{J_{n}}} u^{2}(z) \widehat{J}(dz, dz^{\prime}) \qquad\text{(by Lemma 8.4)}\\ &=& {\int}_{z \in \widehat{J_{n}}} {\int}_{z^{\prime} \in \widehat{M}\setminus \widehat{J_{n}}} \left( (u-u_{n})(z) - (u-u_{n})(z^{\prime})\right) \widehat{J}(dz, dz^{\prime})\\ &\leq& \widehat{\mathcal{E}}(u-u_{n}). \end{array} $$

(A.22)

Let {n_k} be the subsequence given by Lemma A.2 such that $\widehat {\mathcal {E}}_{1}(u-u_{n_{k}}) \to 0$. By Eqs. A.20, A.21, and A.22,

$$ \widehat{\mathcal{E}}(f_{\text{rms}} - g_{n_{k}}) \to 0. $$

(A.23)

By Eqs. A.19 and A.23,

$$ \widehat{\mathcal{E}}_{1}(f_{\text{rms}} - g_{n_{k}}) = \|f_{\text{rms}} - g_{n}\|_{2}^{2} + \widehat{\mathcal{E}}(f_{\text{rms}} - g_{n_{k}}) \to 0. $$

(A.24)

Since $\widehat {\mathcal {F}}$ is closed and $g_{n_{k}} \in \widehat {\mathcal {F}}$ for all k, Eq. A.24 implies that $f_{\text {rms}} \in \mathcal {F}$. □