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$L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field

Jialu Wang, Fang Zhang, Junyong Zhang, Jiqiang Zheng

Abstract

In this paper, we study the $L^{p}$-estimates for the solution to the $2\mathrm{D}$-wave equation with a scaling-critical magnetic potential. Inspired by the work of \cite{FZZ}, we show that the operators $(I+\mathcal{L}_{\mathbf{A}})^{-γ}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}$ is bounded in $L^{p}(\mathbb{R}^{2})$ for $1<p<+\infty$ when $γ>|1/p-1/2|$ and $t>0$, where $\mathcal{L}_{\mathbf{A}}$ is a magnetic Schrödinger operator. In particular, we derive the $L^{p}$-bounds for the sine wave propagator $\sin(t\sqrt{\mathcal{L}_{\mathbf{A}}})\mathcal{L}^{-\frac12}_{\mathbf{A}}$. The key ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.

$L^p$-estimates for the 2D wave equation in the scaling-critical magnetic field

Abstract

In this paper, we study the -estimates for the solution to the -wave equation with a scaling-critical magnetic potential. Inspired by the work of \cite{FZZ}, we show that the operators is bounded in for when and , where is a magnetic Schrödinger operator. In particular, we derive the -bounds for the sine wave propagator . The key ingredients are the construction of the kernel function and the proof of the pointwise estimate for an analytic operator family .

Paper Structure

This paper contains 12 sections, 10 theorems, 160 equations, 1 figure.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Notation
  4. Functional calculus associated with the operator $\mathcal{L}_{\mathbf{A}}$
  5. Some results about spectral multipliers
  6. The proof of Theorem \ref{['operator-family']}
  7. The kernel of analytic operator family $f_{w,t}(\mathcal{L}_{\mathbf{A}})$.
  8. Pointwise estimate for $G(r_{1}, r_{2}, \theta_{1}, \theta_{2})$ and $D(r_{1}, r_{2}, \theta_{1}, \theta_{2})$
  9. The proof of Theorem \ref{['operator-family']}
  10. The proof of Theorem \ref{['Lpestimate']}
  11. The proof of Theorem \ref{['sinLA']}
  12. The Bessel function

Key Result

Theorem 1.1

Let $\mathcal{L}_{{\bf A}}$ be given magnetic Schrödinger operator in LA and let $\gamma>0$ and $1< p<+\infty$ satisfy $|\frac{1}{p}-\frac{1}{2}|<\gamma$. Then, there exists a constant $C(p, \gamma)>0$ such that for $t>0$ and for all $f\in L^{p}(\mathbb{R}^{2})$

Figures (1)

  • Figure 1: Here $A=(\frac{1}{2},0)$, $B=(1,\frac{1}{4})$, $C=(0,\frac{1}{4})$, $D=(1,\frac{1}{2})$, $E=(0,\frac{1}{2})$, respectively. The line $AB: l=\frac{1}{2}(\frac{1}{p}-\frac{1}{2})$. The line $AC: l=\frac{1}{2}(\frac{1}{2}-\frac{1}{p})$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Proposition 3.1: Kernel of the operator $f_{w,t}(\mathcal{L}_{\mathbf{A}})$
  • proof
  • Proposition 3.2: Pointwise estimate for the kernel of $f_{w,t}(\mathcal{L}_{\mathbf{A},a})$
  • ...and 6 more