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Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials

Haruya Mizutani, Xiaohua Yao

Abstract

Let \( H = (-Δ)^m + V \) be a higher-order elliptic operator on \( L^2(\mathbb{R}^n) \), where \( V \) is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with \( H \). In particular, we first establish sharp global Kato smoothing estimates for \( e^{itH} \), based on uniform resolvent estimates of Kato-Yajima type for the absolutely continuous part of \( H \). As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig-Ruiz-Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.

Global Kato smoothing and Strichartz estimates for higher-order Schrödinger operators with rough decay potentials

Abstract

Let \( H = (-Δ)^m + V \) be a higher-order elliptic operator on \( L^2(\mathbb{R}^n) \), where is a general bounded decaying potential. This paper focuses on the global Kato smoothing and Strichartz estimates for solutions to Schrödinger-type equation associated with . In particular, we first establish sharp global Kato smoothing estimates for , based on uniform resolvent estimates of Kato-Yajima type for the absolutely continuous part of . As a consequence, we also obtain optimal local decay estimates. Using these local decay estimates, we then prove the full set of Strichartz estimates, including the endpoint case. Notably, we derive Strichartz estimates with sharp smoothing effects for higher-order cases with rough potentials, which are applicable to the study of nonlinear higher-order Schrödinger equations. Finally, we introduce new uniform Sobolev estimates of the Kenig-Ruiz-Sogge type, incorporating an additional derivative term, which are crucial for establishing the sharp Kato smoothing estimates.

Paper Structure

This paper contains 15 sections, 14 theorems, 119 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Further remarks
  4. Organization of the paper
  5. Notations
  6. Free resolvent estimates for $H_0=(-\Delta)^m$
  7. The limiting absorption principle for $R_0(z)$
  8. The compactness of ${\langle}x{\rangle}^{-s} R_0(z){\langle} x{\rangle}^{-s}$
  9. Kato smoothing estimates of $e^{itH}$
  10. The spectrum of $H_0+V$
  11. Uniform resolvent estimates and Kato smoothing estimates
  12. Strichartz estimates of $e^{itH}$
  13. The proofs of Theorems \ref{['LIP-free case']} and \ref{['Sobolev estimates']}
  14. The proof of Theorem \ref{['LIP-free case']}
  15. The proof of Theorem \ref{['Sobolev estimates']}

Key Result

Theorem 1.1

Let $m\in {\mathbb{N}}$, $m\ge2$, $n>2m$, $H=(-\Delta)^m+V$ and $|V(x)|\le C {\langle}x{\rangle}^{-s}$ for $s>2m$. Assume that $H$ has no positive eigenvalues and no zero resonance/eigenvalue (see Definition resonace in Section local uniform resolvent estimates). Let $P_{ac}(H)$ denote the projecti where $D=-(i\partial_{x_1},i\partial_{x_2},\cdots,i\partial_{x_n})$, $|D|=\sqrt{-\Delta}$. In parti

Theorems & Definitions (29)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 19 more