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The Kato square root problem for parabolic operators with an anti-symmetric part in BMO

Alireza Ataei, Kaj Nyström

TL;DR

This work resolves the parabolic Kato square root problem for operators $\mathcal{H}=\partial_t-\operatorname{div}_x(A(x,t)\nabla_x)$ with $A=S+D$, where $S$ is coercive and $D$ is a real anti-symmetric part in $\text{BMO}$ in $x$. The authors develop a time–space decoupling strategy that leverages hidden coercivity via the Hilbert transform in time, coupled with off-diagonal estimates and a Carleson-measure argument for the principal-part operator $\mathcal{U}_\lambda A$, to establish maximal accretivity and a square-root bound. The main achievement is proving that the domain of $\sqrt{\mathcal{H}}$ equals the energy space $\mathsf{E}(\mathbb{R}^{n+1})$ and that $\|\sqrt{\mathcal{H}}u\|_2$ is comparable to $\|\nabla_x u\|_2+\|D_t^{1/2}u\|_2$, with the same conclusions for $\mathcal{H}^*$. These results extend the parabolic Kato theory to operators with unbounded coefficients and a BMO anti-symmetric part, providing a robust analytic framework toward $L^p$ solvability for boundary-value problems in the upper half-space inspired by Dirichlet/Neumann problems for such operators.

Abstract

We solve the Kato square root problem for parabolic operators whose coefficients can be written as the sum of a complex part, which is coercive, and a real anti-symmetric part, which is in BMO. In particular, we allow for certain unbounded coefficients.

The Kato square root problem for parabolic operators with an anti-symmetric part in BMO

TL;DR

This work resolves the parabolic Kato square root problem for operators with , where is coercive and is a real anti-symmetric part in in . The authors develop a time–space decoupling strategy that leverages hidden coercivity via the Hilbert transform in time, coupled with off-diagonal estimates and a Carleson-measure argument for the principal-part operator , to establish maximal accretivity and a square-root bound. The main achievement is proving that the domain of equals the energy space and that is comparable to , with the same conclusions for . These results extend the parabolic Kato theory to operators with unbounded coefficients and a BMO anti-symmetric part, providing a robust analytic framework toward solvability for boundary-value problems in the upper half-space inspired by Dirichlet/Neumann problems for such operators.

Abstract

We solve the Kato square root problem for parabolic operators whose coefficients can be written as the sum of a complex part, which is coercive, and a real anti-symmetric part, which is in BMO. In particular, we allow for certain unbounded coefficients.
Paper Structure (23 sections, 20 theorems, 226 equations)

This paper contains 23 sections, 20 theorems, 226 equations.

Table of Contents

  1. Introduction and the main result
  2. Statement of our main result
  3. The Proof of Theorem \ref{['thm:Kato']}
  4. Organization of the Paper
  5. Preliminaries
  6. Notation
  7. Convention concerning constants
  8. The energy space
  9. The space $\textup{BMO}$
  10. Assumptions on the coefficients
  11. Hardy spaces
  12. Compensated compactness
  13. Maximal functions and Littlewood-Paley theory
  14. The sesquilinear form and maximal accretivity
  15. Maximal accretivity
  16. ...and 8 more sections

Key Result

Theorem 1.1

Assume that $A=S+D$ satisfies ellip and bmocoeff. Then, the part of $\mathcal{H}$ in $\operatorname{L}^2({\mathbb{R}^{n+1}})$, with maximal domain $\mathop{\mathrm{\mathsf{D}}}\nolimits(\mathcal{H})=\{u \in \mathsf{E}({\mathbb{R}^{n+1}}): \mathcal{H} u \in \operatorname{L}^2({\mathbb{R}^{n+1}})\}$, holds with implicit constants that only depend on the dimension, the boundedness and coercivity par

Theorems & Definitions (39)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • ...and 29 more