L^p-estimates for the Wave Equation with Critical Magnetic Potential in Higher Dimensions

Abstract

In this paper, we study the L^p-estimates for solution to the wave equation with a scaling-critical magnetic Schrödinger operator $\mathscr{L}_{A}$ in Euclidean space ℝ^N with N ≥ 3. Inspired by the work of [Math. Z., 2012, 272(12): 551–575] and [Forum Math., 2026, doi:https://doi.org/10.1515/forum-2025-0323], we address the challenge of determining the explicit kernel function associated with the analytic operator family by employing the Hadamard parametrix, thereby establishing its L^p regularity. As applications, we derive the L^p estimates involving oscillatory multiplier operators $(I+\mathscr{L}_{A})^{-\gamma/2}e^{it\sqrt{\mathscr{L}}_{A}}$ and the sine wave propagator $\sin(t\sqrt{\mathscr{L}}_{A}){\mathscr{L}}_{A}^{-1/2}$ by leveraging the asymptotic behavior of Bessel functions and applying spectral multiplier theorem. Furthermore, the estimates hold precisely for the range of p satisfying $|\frac{1}{p}-\frac{1}{2}|<\frac{\gamma}{N-1}(\gamma>0)$.

Keywords

L^p-estimates / scaling-critical magnetic field / wave equation / 35Q40 / 42B37

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