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Reverse inequalities for quasi-Riesz transform on the Vicsek cable system

Baptiste Devyver, Emmanuel Russ

Abstract

This work is devoted to the study of so-called ``reverse Riesz'' inequalities and suitable variants in the context of some fractal-like cable systems. It was already proved by L. Chen, T. Coulhon, J. Feneuil and the second author that, in the Vicsek cable system, the inequality $\left\Vert Δ^{1/2}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$ is false for all $p\in [1,2)$. Following a recent joint paper by the two authors and M. Yang, we examine the validity of ``reverse quasi-Riesz'' inequalities, of the form $\left\Vert Δ^γe^{-Δ}f\right\Vert_p\lesssim \left\Vert \nabla f\right\Vert_p$, in the (unbounded) Vicsek cable system, for $p\in (1,+\infty)$ and $γ>0$. These reverse inequalities are strongly related to the problem of $L^p$ boundedness of the operators $\nabla e^{-Δ}Δ^{-\varepsilon}$, the so-called ``quasi-Riesz transforms'' (at infinity), introduced by L. Chen in her PhD thesis. Our main result is an almost complete characterization of the sets of $γ\in (0,1)$ and $p\in (1,+\infty)$ such that the reverse quasi-Riesz inequality holds in the Vicsek cable system. It remains an open question to investigate reverse quasi-Riesz inequalities for other cable systems, or for manifolds built out of these.

Reverse inequalities for quasi-Riesz transform on the Vicsek cable system

Abstract

This work is devoted to the study of so-called ``reverse Riesz'' inequalities and suitable variants in the context of some fractal-like cable systems. It was already proved by L. Chen, T. Coulhon, J. Feneuil and the second author that, in the Vicsek cable system, the inequality is false for all . Following a recent joint paper by the two authors and M. Yang, we examine the validity of ``reverse quasi-Riesz'' inequalities, of the form , in the (unbounded) Vicsek cable system, for and . These reverse inequalities are strongly related to the problem of boundedness of the operators , the so-called ``quasi-Riesz transforms'' (at infinity), introduced by L. Chen in her PhD thesis. Our main result is an almost complete characterization of the sets of and such that the reverse quasi-Riesz inequality holds in the Vicsek cable system. It remains an open question to investigate reverse quasi-Riesz inequalities for other cable systems, or for manifolds built out of these.
Paper Structure (13 sections, 19 theorems, 160 equations, 4 figures)

This paper contains 13 sections, 19 theorems, 160 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The setting
  3. The Vicsek cable system
  4. Statements of our results
  5. $L^p$ reverse quasi-Riesz inequalities in the Vicsek cable system
  6. Some elementary facts about "quasi-Riesz" inequalities
  7. More on the Vicsek cable system
  8. A Poincaré inequality for skeletons
  9. Extension of Poincaré inequalities to more general subsets of $X$
  10. A partition of unity associated with a covering by balls
  11. A Calderón-Zygmund decomposition
  12. Reverse quasi-Riesz inequalities: the positive result
  13. Reverse quasi-Riesz inequalities: negative results

Key Result

Lemma 1.2

Let $(X,d,m,\mathcal{E},\mathcal{F})$ be a MMD space with a "carré du champ". Let $\varepsilon\in (0,1)$ and $p\in (1,+\infty)$ be such that the quasi-Riesz transform $\nabla e^{-\Delta}\Delta^{-\varepsilon}$ is bounded on $L^p$. Then, for every $0<s\leq \varepsilon$, the quasi-Riesz transform $\nab

Figures (4)

  • Figure 1: $V^{(0)}$, $V^{(1)}$ and $V^{(2)}$ for $N=2$
  • Figure 2: The set $A$ with its soul $\Gamma$ (in red)
  • Figure 3: The construction of the soul $\Gamma_i$ (in red) in $B_i$
  • Figure 4: The sets $U_{4k+j}$

Theorems & Definitions (42)

  • Definition 1.1
  • Lemma 1.2
  • Corollary 1.3
  • Lemma 1.4
  • Corollary 1.5
  • Lemma 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • Proposition 1.11
  • ...and 32 more