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

In this paper, we study the L^p-estimates for the solution to the 2D-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)^−γe^it√LA is bounded in L^p(ℝ²) for 1 < p < +∞ when γ > |1/p − 1/2| and t > 0, where L_A is a magnetic Schrödinger operator. In particular, we derive the L^p-bounds for the sine wave propagator sin(t√L_A)L_A^{− 1/2}. 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).