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The Kato problem and extensions for degenerate elliptic operators of higher order in weighted spaces

Guoming Zhang

Abstract

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a Muckenhoupt $A_{2}$-weight; this generalizes the work of [Cruz-Uribe, Martell and Rios 2018]. As an application, the unweighted $L^{p}$-Dirichlet, regularity and Neumann boundary value problems associated to such an operator are solved when $p$ is sufficiently close to $2.$

The Kato problem and extensions for degenerate elliptic operators of higher order in weighted spaces

Abstract

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order (), whose coefficients are measurable, complex-valued and satisfy the Grding inequality with respect to a Muckenhoupt -weight; this generalizes the work of [Cruz-Uribe, Martell and Rios 2018]. As an application, the unweighted -Dirichlet, regularity and Neumann boundary value problems associated to such an operator are solved when is sufficiently close to

Paper Structure

This paper contains 23 sections, 46 theorems, 359 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weights and the associated Sobolev spaces.
  4. Generalized Poincaré-Sobolev inequalities.
  5. Off-diagonal estimates in higher-order setting.
  6. Theorems on weighted boundedness of sublinear operators.
  7. Off-diagonal estimates for the semigroup of $e^{-t\mathcal{L}_{w}}$
  8. $H^{\infty}$ Functional calculi in $L^{2}(w)$.
  9. Off-diagonal estimates in $L^{2}(w)$
  10. Off-diagonal estimates in $L^{p}(w)$
  11. The weighted $L^{p}$ functional calculus for $\mathcal{L}_{w}$
  12. Reverse inequalities for square roots in weighted spaces
  13. The higher-order weighted Calder$\acute{o}$n-Zygmund decomposition
  14. The weighted conservation property in higher-order case
  15. Proof of the reverse inequalities in $L^{p}(w)$ and $L^{p}(vdw)$
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $\mathcal{L}_{w}$ be given by eq: z00 with $\{a_{\alpha, \beta}(x)\}_{|\alpha|=|\beta|=m}\in \mathcal{E}(w, c_{1}, c_{2}).$ Then, if $w\in A_{1}\cap \hbox{RH}_{1+\frac{n}{2m}},$ we have for every $f\in H^{m}({{{\Bbb R}}^n})$ that where the implicit constants depend only on $n, m, c_{1}, c_{2}$ and the $A_{1}$ and $\hbox{RH}_{1+\frac{n}{2m}}$ constants of $w$ (see Section 2.1 for the rigorous

Theorems & Definitions (58)

  • Theorem 1.1
  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • ...and 48 more