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Energy inequalities for cutoff functions of $p$-energies on metric measures spaces

Meng Yang

TL;DR

This work develops a comprehensive framework to deduce the cutoff Sobolev inequality for $p$-energies on general metric measure spaces from geometric and functional hypotheses, including volume doubling, LLC, LSC, EHI, PI, and capacity bounds. A central achievement is a Wolff potential estimate for nonnegative superharmonic functions, which bridges potential theory with the intrinsic CS inequality and enables Nash–Moser–De Giorgi-type arguments to yield elliptic Harnack and related regularity. The authors show that under VD, LSC, EHI, PI, and cap, CS holds; with fast or RSVR volume growth, LLC can be derived or CS can hold under weaker hypotheses, and in the $p=2$ case, two-sided heat kernel bounds follow, addressing capacity conjectures. A notable corollary establishes CS and energy-measure singularity on the Sierpiński carpet for all $p>1$, resolving a standing problem in that setting. Altogether, the results provide a robust, scalable pathway to regularity, capacity, and heat-kernel behavior on fractals and general metric measure spaces, with concrete implications for energy measures and potential theory.

Abstract

For $p>1$, and for a $p$-energy on a metric measure space, we provide various geometric and functional conditions for the validity of the cutoff Sobolev inequality. In particular, we employ a technique of Trudinger and Wang [Amer. J. Math. 124 (2002), no. 2, 369--410] to derive a Wolff potential estimate for superharmonic functions, and a method of Holopainen [Contemp. Math. 338 (2003), 219--239] to prove the elliptic Harnack inequality for harmonic functions. As applications, we make progress toward the capacity conjecture of Grigor'yan, Hu, and Lau [Springer Proc. Math. Stat. 88 (2014), 147--207], and we prove that the $p$-energy measure is singular with respect to the Hausdorff measure on the Sierpiński carpet for all $p>1$, resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608].

Energy inequalities for cutoff functions of $p$-energies on metric measures spaces

TL;DR

This work develops a comprehensive framework to deduce the cutoff Sobolev inequality for -energies on general metric measure spaces from geometric and functional hypotheses, including volume doubling, LLC, LSC, EHI, PI, and capacity bounds. A central achievement is a Wolff potential estimate for nonnegative superharmonic functions, which bridges potential theory with the intrinsic CS inequality and enables Nash–Moser–De Giorgi-type arguments to yield elliptic Harnack and related regularity. The authors show that under VD, LSC, EHI, PI, and cap, CS holds; with fast or RSVR volume growth, LLC can be derived or CS can hold under weaker hypotheses, and in the case, two-sided heat kernel bounds follow, addressing capacity conjectures. A notable corollary establishes CS and energy-measure singularity on the Sierpiński carpet for all , resolving a standing problem in that setting. Altogether, the results provide a robust, scalable pathway to regularity, capacity, and heat-kernel behavior on fractals and general metric measure spaces, with concrete implications for energy measures and potential theory.

Abstract

For , and for a -energy on a metric measure space, we provide various geometric and functional conditions for the validity of the cutoff Sobolev inequality. In particular, we employ a technique of Trudinger and Wang [Amer. J. Math. 124 (2002), no. 2, 369--410] to derive a Wolff potential estimate for superharmonic functions, and a method of Holopainen [Contemp. Math. 338 (2003), 219--239] to prove the elliptic Harnack inequality for harmonic functions. As applications, we make progress toward the capacity conjecture of Grigor'yan, Hu, and Lau [Springer Proc. Math. Stat. 88 (2014), 147--207], and we prove that the -energy measure is singular with respect to the Hausdorff measure on the Sierpiński carpet for all , resolving a problem posed by Murugan and Shimizu [Comm. Pure Appl. Math. 78 (2025), no. 9, 1523--1608].

Paper Structure

This paper contains 7 sections, 52 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. Statement of main results
  3. Proof of Theorem \ref{['thm_main']}
  4. Proof of Theorem \ref{['thm_main_conv']}
  5. Proof of Theorem \ref{['thm_main_FVR']}
  6. Proofs of Theorem \ref{['thm_main_RSVR']} and Theorem \ref{['thm_cap_conj']}
  7. Proof of Proposition \ref{['prop_RBA']}

Key Result

Theorem 2.1

Assume eq_VD, eq_LSC, eq_EHI, eq_PI, eq_ucap. If either holds, then eq_CS holds.

Theorems & Definitions (99)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Theorem 2.6
  • Remark 2.7
  • Theorem 2.8
  • Remark 2.9
  • Theorem 2.10
  • ...and 89 more