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The Kato problem for weighted elliptic and parabolic operators of higher order

Guoming Zhang

Abstract

We solve the Kato square root problem for parabolic operators of arbitrary order $2m$ whose coefficients are allowed to depend on both space and time in a merely measurable way and possess boundedness and ellipticity controlled by a Muckenhoupt $A_{2}-$weight. Notably, the proof applies to the weighted Kato problem within an elliptic framework.

The Kato problem for weighted elliptic and parabolic operators of higher order

Abstract

We solve the Kato square root problem for parabolic operators of arbitrary order whose coefficients are allowed to depend on both space and time in a merely measurable way and possess boundedness and ellipticity controlled by a Muckenhoupt weight. Notably, the proof applies to the weighted Kato problem within an elliptic framework.

Paper Structure

This paper contains 7 sections, 20 theorems, 258 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The higher order parabolic operator and the central estimate to \ref{['eq: t1']}
  4. Proof of \ref{['eq: a2.54']}: Part I
  5. Proof of \ref{['eq: a2.54']}: Part II
  6. Proof of \ref{['eq: a2.54']}: Part III
  7. Appendix

Key Result

Theorem 1.1

Any operator $\mathcal{H}$ given in eq: c001-eq: a2.29 can be defined as a maximal accretive operator in $L^{2}_{\upsilon}$ via an accretive sesquilinear form with domain $\mathbf{E}_{\upsilon}.$ Moreover, the domain of its unique maximal accretive square root $\sqrt{\mathcal{H}}$ coincides with $\m

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 4.1
  • Lemma 4.2
  • Lemma 5.1
  • Lemma 5.2
  • Definition 5.3
  • ...and 15 more