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The metric for matrix degenerate Kato square root operators

Gianmarco Brocchi, Andreas Rosén

Abstract

We prove a Kato square root estimate with anisotropically degenerate matrix coefficients. We do so by doing the harmonic analysis using an auxiliary Riemannian metric adapted to the operator. We also derive $L^2$-solvability estimates for boundary value problems for divergence form elliptic equations with matrix degenerate coefficients. Main tools are chain rules and Piola transformations for fields in matrix weighted $L^2$ spaces, under $W^{1,1}$ homeomorphism.

The metric for matrix degenerate Kato square root operators

Abstract

We prove a Kato square root estimate with anisotropically degenerate matrix coefficients. We do so by doing the harmonic analysis using an auxiliary Riemannian metric adapted to the operator. We also derive -solvability estimates for boundary value problems for divergence form elliptic equations with matrix degenerate coefficients. Main tools are chain rules and Piola transformations for fields in matrix weighted spaces, under homeomorphism.
Paper Structure (9 sections, 17 theorems, 152 equations, 7 figures)

This paper contains 9 sections, 17 theorems, 152 equations, 7 figures.

Table of Contents

  1. Bisectorial operators
  2. Bounded holomorphic functional calculus
  3. Quadratic estimates
  4. Weights
  5. Two scalar weights in one dimension
  6. The (μ,W) manifold $M$
  7. Matrix degenerate Boundary Value Problems
  8. Sobolev pullbacks and Piola transformations
  9. Approximation in weighted Sobolev spaces

Key Result

Lemma 1.1

Let $\mu,w$ be two weights that are smooth on an interval $I \subset \mathbb{R}$. Let $\rho \colon I \to \mathbb{R}$ be such that $\rho'(x) = \sqrt{\mu(x)/w(x)}$. Set $M \coloneqq \rho(I) \subset \mathbb{R}$. Let $\nu(\rho(x)) \coloneqq \sqrt{\mu(x) w(x)}$ and Then the map $\mathsf{P}$ defined in eq:rubber_band_pullback is an isometry between the Hilbert spaces $\mathcalboondox{H} = L^2(I,\mu) \o

Figures (7)

  • Figure 1: Geodesic disks in the metric of \ref{['ex:ellipses']} are ellipses whose principal axes are the eigenvectors of the matrix $A(x)$. These ellipses shrink anisotropically towards the origin.
  • Figure 2: Geodesic disks in the metric of \ref{['ex:example2']} for $a = 1$ are ellipses with increasing eccentricity.
  • Figure 3: We will use a unitary map $\mathsf{P}$ and its inverse, introduced in \ref{['sec:1dim']} and defined in \ref{['eq:definition_rubber_band_higher_dim']}.
  • Figure 4: Completeness of the $y$-axes. In Case 1, $\rho(x) = \sqrt{x}$ on $\mathbb{R}_+$ can be extended to an odd bijection $\mathbb{R} \to \mathbb{R}$. In Case 2, $\rho(x) = -1/x$ is not surjective onto $\mathbb{R}$. In Case 3, $\rho(x) = \ln(x)$ is a bijection from $\mathbb{R}_+$ to $\mathbb{R}$.
  • Figure 5: The Riemannian manifold $M$ with a chart $\varphi$ from its smooth atlas. A function $f$ on $M$ is smooth if $f \circ \varphi$ is smooth. But $f \circ \rho$ is not in general smooth since the map $\varphi^{-1} \circ \rho$ is only in $W^{1,1}$.
  • ...and 2 more figures

Theorems & Definitions (46)

  • Definition 1: Bisectorial operator
  • Definition 2
  • Definition 3: Muckenhoupt $A_2^R$ weights
  • Definition 4: Local Muckenhoupt weights
  • Example 5
  • Definition 6
  • Lemma 1.1
  • proof
  • Theorem 1.2
  • proof
  • ...and 36 more