Uniqueness of the canonical $p$-energy on Sierpiński carpets and the conformal-dimension continuity threshold for generalized Sierpiński carpets

On Euclidean space, nonlinear potential theory rests on the $(1,p)$-Sobolev space and the energy $\int|\nabla f|^p\,dx$. Shimizu (Trans. Amer. Math. Soc. 377 (2024), 951–1032) constructed the analogue on generalized Sierpiński carpets — a $p$-energy $\mathcal{E}_p$ with domain $\mathcal{F}_p$, energy measures, and Besov/Hölder regularity — but only for $p$ above the carpet's Ahlfors regular conformal dimension $\dim_{\mathrm{ARC}}$, and posed: (Conjecture 1.1) $\dim_{\mathrm{ARC}}(K,d)=\inf\{p:\mathcal{F}_p\subset C(K)\}$; uniqueness of the $p$-energy up to constants; the construction of a two-variable form $\mathcal{E}_p(f;g)$ and a $p$-Laplacian; and $\beta_p>p$ beyond planar carpets. Subsequent work has resolved parts of this program: the two-variable form/$p$-harmonic program and $\beta_p>p$ for all generalized Sierpiński carpets are resolved by Kajino–Shimizu, and Conjecture 1.1 is affirmed for the planar standard carpet by Cao–Chen–Kumagai combined with Murugan–Shimizu, who construct the $p$-energy there for all $p\in(1,\infty)$. The surviving open core is precise: (A) the threshold identity $\inf\{p:\mathcal{F}_p\subset C(K)\}=\dim_{\mathrm{ARC}}$ for the concretely defined Sobolev scale $\mathcal{F}_p$ on generalized Sierpiński carpets beyond the planar standard carpet — notably non-planar carpets such as the Menger sponge, where even $p$-conductive homogeneity below $\dim_{\mathrm{ARC}}$ is unknown — and (B) uniqueness, up to constants, of the self-similar $p$-energy satisfying the canonical axioms, known only for $p=2$ (Barlow–Bass–Kumagai–Teplyaev); on the planar standard carpet the corresponding statement for a related axiom class is Murugan–Shimizu's Conjecture 12.3.

Background

On $\mathbb{R}^n$, nonlinear potential theory is built on the $(1,p)$-Sobolev space $W^{1,p}$ — the space of functions whose weak first derivative lies in $L^p$ — and the $p$-energy $\int|\nabla f|^p\,dx$, whose Euler–Lagrange equation is the $p$-Laplace equation. On fractals there is no gradient and no useful abundance of rectifiable curves, so both objects must be constructed as scaling limits of discrete energies. The standard test bed is the Sierpiński carpet: the compact set obtained from the unit square by dividing it into a $3\times3$ grid of congruent subsquares, deleting the open middle one, and iterating. More generally, a generalized Sierpiński carpet (GSC) is a self-similar set $K\subset[-1,1]^D$ built from a subdivision into $a^D$ congruent cubes ($a\ge3$) keeping a symmetric, connected, non-diagonal pattern $S$ of subcubes that touches the outer boundary; the standard carpet is $GSC(2,3,\{0,1,2\}^2\setminus\{(1,1)\})$ and the Menger sponge is a three-dimensional GSC. GSCs are infinitely ramified — no finite set of points disconnects them — which makes their analysis substantially harder than on post-critically finite (p.c.f.) fractals, i.e. self-similar sets such as the Sierpiński gasket whose finitely many constituent pieces intersect in only finitely many points.

For $p=2$ the theory exists: Barlow–Bass constructed the Brownian motion on the standard carpet as a limit of reflected Brownian motions in approximating domains, and Kusuoka–Zhou constructed the associated regular Dirichlet form (a symmetric, closed, Markovian bilinear form on $L^2(K,\mu)$ — a '2-energy') as the renormalized scaling limit of discrete Dirichlet energies on graph approximations $G_n$. The domain consists of Hölder-continuous functions, and Barlow–Bass–Kumagai–Teplyaev proved that the carpet Dirichlet form is unique up to constant multiples — the $p=2$ prototype of the uniqueness question studied here. On p.c.f. fractals, $p$-energies for $p\neq2$ were first constructed by Herman–Peirone–Strichartz (Sierpiński gasket), with a $p$-Laplacian proposed by Strichartz–Wong, and in full generality by Cao–Gu–Qiu; Kigami developed the parallel framework of $p$-conductive homogeneity (a scale-uniform comparison between combinatorial $p$-modulus and capacity on approximating graphs) covering large classes of infinitely ramified sets.

Shimizu's construction extends the Kusuoka–Zhou scheme to all $p>1$: define discrete $p$-energies on $G_n$, renormalize by the constant $\rho_p$ (the exponential rate across scales of the combinatorial $p$-modulus of the curve families crossing the construction's annuli — the $p$-modulus of a curve family being the least total mass of a nonnegative weight function that every curve in the family carries with total weight at least $1$, a quantitative measure of how hard the family is to connect; its two-sided bounds go back to Bourdon–Kleiner), and set $\mathcal{F}_p=\{f\in L^p(K,\mu):\sup_n\rho_p^n\mathcal{E}_p^{G_n}(M_nf)<\infty\}$. The condition $\rho_p>1$ is equivalent to $p>\dim_{\mathrm{ARC}}(K,d)$, where $\dim_{\mathrm{ARC}}$ is the Ahlfors regular conformal dimension: the infimum of exponents $\alpha$ such that $K$ carries an $\alpha$-Ahlfors-regular measure (a measure with $\mu(B(x,r))\asymp r^\alpha$) for some metric quasisymmetric to $d$ — quasisymmetric maps being the metric-space analogue of conformal maps, i.e. homeomorphisms distorting relative distances in a uniformly controlled way. For $p>\dim_{\mathrm{ARC}}$, Shimizu proves that $\mathcal{F}_p$ is a reflexive separable Banach space embedded in a Hölder space $C^{0,(\beta_p-\alpha)/p}$ (here $\alpha=\log N_*/\log a$ is the Hausdorff dimension and $\beta_p=\log(N_*\rho_p)/\log a$ generalizes the walk dimension, the escape-rate exponent of the diffusion), realizes a Besov-type space $B^{\beta_p/p}_{p,p}$ (a smoothness scale on metric measure spaces, defined by the finiteness of double integrals of $|f(x)-f(y)|^p$ against powers of the distance), and carries a canonical $p$-energy $\mathcal{E}_p$ satisfying: (1) vanishing exactly on constants; (2) regularity, i.e. $\mathcal{F}_p\cap C(K)$ dense in $C(K)$; (3) the Markov (Lipschitz-contraction) property; (4) invariance under the carpet's isometry group; (5) exact self-similarity $\mathcal{E}_p(f)=\rho_p\sum_i\mathcal{E}_p(f\circ F_i)$; (6) strong locality (additivity on disjoint supports). At $p=2$ this recovers the unique carpet Dirichlet form. The paper also constructs $\mathcal{E}_p$-energy measures $\mu^p_{\langle f\rangle}$ — the nonlinear analogue of $|\nabla f|^p\,dx$, satisfying the chain rule $d\mu^p_{\langle\Phi\circ f\rangle}=|\Phi'\circ f|^p\,d\mu^p_{\langle f\rangle}$; at $p=2$ such measures are singular to the Hausdorff measure by Hino's theorem — proves $\beta_p>p$ for planar GSCs, and states the open problems treated below. The exact value — and even the attainment — of $\dim_{\mathrm{ARC}}$ of the standard carpet is a famous open problem in conformal geometry, connected to quasisymmetric uniformization questions (whether the carpet's conformal gauge contains a Loewner-type metric, i.e. one supporting a modulus lower bound for curve families), which is what makes the conjectural identification of $\dim_{\mathrm{ARC}}$ with an analytic threshold significant.

Problem Statement

Let $K=GSC(D,a,S)$ be a generalized Sierpiński carpet with the Euclidean metric $d$ normalized on $[-1,1]^D$ and the self-similar measure $\mu$, and let $\dim_{\mathrm{ARC}}(K,d)$ denote its Ahlfors regular conformal dimension. For $p>\dim_{\mathrm{ARC}}(K,d)$, let $(\mathcal{E}_p,\mathcal{F}_p)$ be the canonical $p$-energy constructed by Shimizu, satisfying the axioms: (1) $\mathcal{E}_p(f)=0$ exactly on constants; (2) regularity ($\mathcal{F}_p\cap C(K)$ dense in $C(K)$); (3) Markov property; (4) invariance under the carpet's isometry group; (5) exact self-similarity with scaling $\rho_p$; (6) strong locality. Two linked conjectures — together asserting that every generalized Sierpiński carpet carries one and only one canonical $p$-energy theory whose continuity threshold is exactly its conformal dimension — remain open in general.

Part A (continuity threshold; Shimizu's Conjecture 1.1 beyond the planar standard carpet). Prove or refute, for every generalized Sierpiński carpet,

$$\dim_{\mathrm{ARC}}(K,d)=\inf\{p\in(1,\infty):\ \mathcal{F}_p\subset C(K)\},$$

where, for each $p\in(1,\infty)$, $\mathcal{F}_p$ is the concretely defined $(1,p)$-Sobolev space $\mathcal{F}_p=\{f\in L^p(K,\mu):\sup_n\rho_p^n\mathcal{E}_p^{G_n}(M_nf)<\infty\}$, with $\rho_p$ the combinatorial $p$-modulus growth rate and $\mathcal{E}_p^{G_n}$, $M_n$ the discrete $p$-energies and averaging operators on the graph approximations $G_n$ described in the Background — this definition makes sense for every $p>1$, so the infimum is taken over a fixed family of spaces and not over those $p$ for which an axiom-satisfying energy happens to exist; for $p>\dim_{\mathrm{ARC}}(K,d)$, $\mathcal{F}_p$ is the domain of the canonical $p$-energy above, and it coincides with Kigami's space $\mathcal{W}^p$ whenever $p$-conductive homogeneity holds. The embedding $\mathcal{F}_p\subset C(K)$ for $p>\dim_{\mathrm{ARC}}$ is known, and the identity is now a theorem for the planar standard carpet; the open cases are the remaining GSCs. Concretely, for a non-planar GSC such as the Menger sponge — and for non-standard planar GSCs — the open cases concern (i) the structure of the space $\mathcal{F}_p$ for $p\le\dim_{\mathrm{ARC}}(K,d)$ — in particular whether $\mathcal{F}_p\cap C(K)$ is dense in $C(K)$, the ingredient Shimizu identifies as missing — and (ii) whether $\mathcal{F}_p$ then necessarily contains discontinuous, even unbounded, functions — so that the continuity threshold equals $\dim_{\mathrm{ARC}}(K,d)$ — or whether some GSC fails this.

Part B (uniqueness of the canonical $p$-energy). Prove or refute: on the Sierpiński carpet — and more generally on any generalized Sierpiński carpet — for each $p\in(1,\infty)$, any two $p$-energies on $L^p(K,\mu)$ satisfying the canonical axioms (1)–(6) are constant multiples of each other — in particular, each is a constant multiple of the constructed energy wherever a construction is known. This is known for $p=2$ by the Barlow–Bass–Kumagai–Teplyaev uniqueness theorem; for $p\neq2$ it is open even on the planar standard carpet, where Murugan–Shimizu's Conjecture 12.3 formulates the uniqueness for a related but distinct axiom class (their Assumption 10.22 for self-similar $p$-energies, which imposes unit contractivity, a uniformly convex Banach structure and a spectral gap in place of the full Markov property and strong locality of (1)–(6)) and expects the uniqueness to hold for a wide class of generalized Sierpiński carpets.

The two parts are the uniqueness and the sharp-regularity halves of the same canonical-object statement.

References

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