<i>L</i> ^<i>p</i> -estimates for the 2D wave equation in the scaling-critical magnetic field

Abstract In this paper, we study the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {L^{p}} -estimates for the solution to the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>2</m:mn> <m:mo>⁢</m:mo> <m:mi mathvariant="normal">D</m:mi> </m:mrow> </m:math> {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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>I</m:mi> <m:mo>+</m:mo> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>𝐀</m:mi> </m:msub> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mrow> <m:mo>-</m:mo> <m:mi>γ</m:mi> </m:mrow> </m:msup> <m:mo>⁢</m:mo> <m:msup> <m:mi>e</m:mi> <m:mrow> <m:mi>i</m:mi> <m:mo>⁢</m:mo> <m:mi>t</m:mi> <m:mo>⁢</m:mo> <m:msqrt> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>𝐀</m:mi> </m:msub> </m:msqrt> </m:mrow> </m:msup> </m:mrow> </m:math> {(I+\mathcal{L}_{\mathbf{A}})^{-\gamma}e^{it\sqrt{\mathcal{L}_{\mathbf{A}}}}} is bounded in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mn>2</m:mn> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {L^{p}(\mathbb{R}^{2})} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mn>1</m:mn> <m:mo>&lt;</m:mo> <m:mi>p</m:mi> <m:mo>&lt;</m:mo> <m:mrow> <m:mo>+</m:mo> <m:mi mathvariant="normal">∞</m:mi> </m:mrow> </m:mrow> </m:math> {1&lt;p&lt;+\infty} when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>γ</m:mi> <m:mo>&gt;</m:mo> <m:mrow> <m:mo stretchy="false">|</m:mo> <m:mrow> <m:mfrac> <m:mn>1</m:mn> <m:mi>p</m:mi> </m:mfrac> <m:mo>-</m:mo> <m:mfrac> <m:mn>1</m:mn> <m:mn>2</m:mn> </m:mfrac> </m:mrow> <m:mo stretchy="false">|</m:mo> </m:mrow> </m:mrow> </m:math> {\gamma&gt;|\frac{1}{p}-\frac{1}{2}|} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>t</m:mi> <m:mo>&gt;</m:mo> <m:mn>0</m:mn> </m:mrow> </m:math> {t&gt;0} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="script">ℒ</m:mi> <m:mi>𝐀</m:mi> </m:msub> </m:math> {\mathcal{L}_{\mathbf{A}}} is a magnetic Schrödinger operator. In particular, we derive the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {L^{p}}