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

2 ⁢ D {2\mathrm{D}}

-wave equation with a scaling-critical magnetic potential. Inspired by the work of [L. Fanelli, J. Zhang and J. Zheng, Dispersive estimates for 2D-wave equations with critical potentials, Adv. Math. 400 2022, Article ID 108333], we show that the operators

( I + ℒ 𝐀 ) - γ ⁢ e i ⁢ t ⁢ ℒ 𝐀 {(I+\mathcal{L}_{\mathbf{A}})^{-\gamma}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}}

is bounded in

L p ⁢ ( ℝ 2 ) {L^{p}(\mathbb{R}^{2})}

for

1 < p < + ∞ {1<p<+\infty}

when

γ > | 1 p - 1 2 | {\gamma>|\frac{1}{p}-\frac{1}{2}|}

and

t > 0 {t>0}

, where

ℒ 𝐀 {\mathcal{L}_{\mathbf{A}}}

is a magnetic Schrödinger operator. In particular, we derive the

L p {L^{p}}

-bounds for the sine wave propagator

sin ⁡ ( t ⁢ ℒ 𝐀 ) ⁢ ℒ 𝐀 - 1 2 {\sin(t\sqrt{\mathcal{L}_{\mathbf{A}}})\mathcal{L}^{-\frac{1}{2}}_{{\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 ⁢ ( ℒ 𝐀 ) {f_{w,t}(\mathcal{L}_{\mathbf{A}})}

.