Abstract
Quasisymmetric maps are well-studied homeomorphisms between metric spaces preserving annuli, and the Ahlfors regular conformal dimension \(\dim _\textrm{ARC}(X,d)\) of a metric space (X, d) is the infimum over the Hausdorff dimensions of the Ahlfors regular images of the space by quasisymmetric transformations. For a given regular Dirichlet form with the heat kernel, the spectral dimension \(d_s\) is an exponent that indicates the short-time asymptotic behavior of the on-diagonal part of the heat kernel. In this paper, we consider the Dirichlet form induced by a resistance form on a set X and the associated resistance metric R. We prove \(\dim _\textrm{ARC}(X,R)\le \overline{d_s}<2\) for \(\overline{d_s}\), a variation of \(d_s\) defined through the on-diagonal asymptotics of the heat kernel. We also give an example of a resistance form whose spectral dimension \(d_s\) satisfies the opposite inequality \(d_s<\dim _\textrm{ARC}(X,R)<2.\)
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Acknowledgements
This work has been done as the author’s Ph.D. thesis at Kyoto University. I would like to thank Naotaka Kajino for his kind grammatical corrections to the introduction and technical comments. In particular, Appendix A was motivated by his remark. I also thank my supervisor Takashi Kumagai for his comments on the introduction. This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP20J23120.
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This work was supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP20J23120.
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Kôhei Sasaya is was a paid fellow of Japan Society for the Promotion of Science. He was also received travel support from Professor Takashi Kumagai from Kyoto University, Assistant Professor Xiaodan Zhou from Okinawa Institute of Science and Technology Graduate University and Associate Professor Satoshi Ishiwata from Yamagata University for his talks.
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: Kôhei Sasaya is was a Ph.D. student at Kyoto University. His supervisor was Professor Takashi Kumagai until March 2022 and is Associate Professor Naotaka Kajino from April 2022. He usually has seminars with Doctor Ryosuke Shimizu, Professors Naotaka Kajino and Jun Kigami, and a student of Kigami
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A Equivalence of local properties
A Equivalence of local properties
Let us recall that X is a set, \((\mathcal {E,F})\) is a resistance form on X and R is the resistance metric associated with \((\mathcal {E,F}).\) In this appendix we discuss the relation between the local property of \((\mathcal {E,F})\) and that of the Dirichlet form induced by \((\mathcal {E,F}).\) We also recall that the local property of a Dirichlet form is defined as follows.
Definition A.1
(Local) Let \((Y,\rho )\) be a locally compact separable metric space and \(\nu \) be a Radon measure on Y with full support. A Dirichlet form (E, D) on \(L^2(Y,\nu )\) is called local if \(E(u,v)=0\) whenever \(u,v\in D\) have disjoint compact supports, where the support \(\textrm{supp}(u)\) of \(u\in L^2(Y,\nu )\) is defined as the support of the measure \(ud\nu \) on \((Y,\rho ).\)
By [20, Theorem 9.4], if \((\mathcal {E,F})\) is a regular resistance form satisfying (ACC), then for each Radon measure \(\mu \) on X with full support, \((\mathcal {E}_\mu ,\mathcal {D}_\mu ),\) defined in the same way as Lemma 1.8, is a regular Dirichlet form on \(L^2(X,\mu ).\) Here we remark that \(\textrm{supp}(u)=\overline{\{x\in X\mid u(x)\ne 0\}}\) because \(\mathcal {F}\subset C(X,R).\) Therefore by the definition of \(\mathcal {D}_\mu \), \((\mathcal {E}_\mu ,\mathcal {D}_\mu )\) is a local Dirichlet form (over (X, R)) if \((\mathcal {E,F})\) is a local resistance form. In this appendix, we prove that, under Assumption 1.6, the converse direction is also true. Indeed, the following holds.
Proposition A.2
Assume that R is complete and doubling. Then for any \(u\in \mathcal {F},\) there exists \(\{u_n\}_{n\ge 0}\subset \mathcal {F}\cap C_0(X,R)\) such that \(\textrm{supp}(u_n)\subset \textrm{supp}(u)\) for any n and \(\lim _{n\rightarrow \infty }\mathcal {E}(u-u_n,u-u_n)=0.\)
Corollary A.3
We make the Assumption 1.6 and let \(\mu \) be a Radon measure on X with full support. Then the following conditions are equivalent.
-
1.
\(\mathcal {E}(u,v)=0\) if \(u,v\in \mathcal {F}\) and \(\textrm{supp}(u)\cap \textrm{supp}(v)=\emptyset .\)
-
2.
\((\mathcal {E,F})\) is a local resistance form.
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3.
\((\mathcal {E}_\mu ,\mathcal {D}_\mu )\) is a local Dirichlet form.
Proof
\(1\Rightarrow 2\Rightarrow 3\) is obvious. \(3\Rightarrow 1\) follows from Theorem 3.5 and Proposition A.2.\(\square \)
In the remainder of this appendix, we assume that R is complete and doubling, and prove Proposition A.2. In the same way as the proof of Lemma 2.13, the following inequality holds without the uniform perfectness condition.
Lemma A.4
\(\mathcal {R}(\overline{B(x,r)}, B(x,2r)^c)\gtrsim r\) for any \(x\in X\) and \(r>0.\)
The following Proposition A.5, Corollary A.6 and Lemma A.7 were proved in [14] for a general resistance form whose associated resistance metric is not necessarily doubling. Here we give proofs for the same reason as we did for Lemma 2.11.
Proposition A.5
(cf. [14, Theorem 2.38 (2)]) Let \(u\in \mathcal {F}\) and \(\{u_n\}_{n\ge 0}\subset \mathcal {F}.\) Then \(\lim _{n\rightarrow \infty }\mathcal {E}(u-u_n,u-u_n)=0\) if and only if \(\limsup _{n\rightarrow \infty }\mathcal {E}(u_n,u_n)\le \mathcal {E}(u,u)\) and \(\lim _{n\rightarrow \infty }(u-u_n)(x)\) exists in \(\mathbb {R}\) and is constant on X.
Proof
The necessity is clear by the triangle inequality of \(\mathcal {E}^{1/2}\) and Eq. 1.4. For the sufficiency, let \(\{V_m\}_{m\ge 0}\) be a spread sequence of (X, R) then
which proves \(\lim _{n\rightarrow \infty }\mathcal {E}(u_n,u_n)=\mathcal {E}(u,u).\) Let \(u^*_n:=(u+u_n)/2,\) then by the triangle inequality of \(\mathcal {E}^{1/2},\) \(\{u^*_n\}_{n\ge 0}\) also satisfies the same condition as \(\{u_n\}_{n\ge 0}.\) Therefore \(\lim _{n\rightarrow \infty }\mathcal {E}(u^*_n,u^*_n)=\mathcal {E}(u,u)\) and
Corollary A.6
(cf. [14, Corollary 2.39 (4)]) \(\lim _{n\rightarrow \infty }\mathcal {E}(u-\hat{u}_n,u-\hat{u}_n)=0\) for any \(u\in \mathcal {F},\) where \(\hat{u}_n=(u\wedge n)\vee (-n).\)
Proof
It immediately follows from Proposition A.5 and Eq. 1.5.\(\square \)
Lemma A.7
(cf. [14, Corollary 2.39 (3)]) Let \(u,v\in \mathcal {F}\) be bounded. Then \(uv\in \mathcal {F}\) and \(\mathcal {E}(uv,uv)^{1/2}\le \Vert u\Vert _{\infty }\mathcal {E}(v,v)^{1/2}+\Vert v\Vert _{\infty }\mathcal {E}(u,u)^{1/2},\) where \(\Vert u\Vert _{\infty }=\sup _{x\in X}|u(x)|.\)
Proof
This follows from Proposition 2.10 with easy calculation.\(\square \)
Proof of Propositions
A.2 Since Corollary A.6 holds, we only need to show the case that \(u\in \mathcal {F}\) is bounded and \(\textrm{diam}(X,R)=\infty \). Fix some \(x\in X\) and let \(f_n\) be the optimal function for \(\mathcal {R}(\overline{B(x,2^n)}, B(x,2^{n+1})^c)\) for each \(n\ge 0.\) Let \(u_n=f_nu,\) then \(u_n\in C_0(X,R)\) and \(\textrm{supp}(u_n)\subset \textrm{supp}(u).\) Moreover, \(u_n\in \mathcal {F}\) and
because of Lemmas A.4 and A.7. Therefore \(\lim _{n\rightarrow \infty }\mathcal {E}(u-u_n,u-u_n)=0\) by Proposition A.5, which completes the proof.\(\square \)
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Sasaya, K. Some Inequalities Between Ahlfors Regular Conformal Dimension And Spectral Dimensions For Resistance Forms. Potential Anal 61, 347–377 (2024). https://doi.org/10.1007/s11118-023-10112-6
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DOI: https://doi.org/10.1007/s11118-023-10112-6