Multiplication operators on a class of Hardy spaces for Fourier integral operators

We show boundedness of multiplication operators $M_g$ on Hardy spaces for Fourier integral operators $H^p_{FIO,a}(\mathbb{R}^d)$, which are adapted to structured Lipschitz coefficients. The boundedness is described in terms of regularity of $g$ in adapted Besov and BMO spaces. This improves a recent result on multiplication operators on $H^p_{FIO,a}(\mathbb{R}^d)$ where more regularity on $g$ was required