Oscillatory multiplier estimates near $L^1$

Fourier multiplier operators are among the most important operators in analysis due to their wide range of applications. In this talk, we will consider a class of oscillatory Fourier multipliers of the form
$$e^{it|\xi|}(1+|\xi|^2)^{-b/2}, \qquad t>0,\; b\in \mathbb{R},$$
and study their boundedness properties on $\beta$-dimensional spaces of measures lying between the Hardy space $H^1(\mathbb{R}^n)$ and the space of finite Radon measures. Our results refine the classical estimates of Miyachi and Peral in the Hardy space setting. We will also discuss an application of these estimates to solutions of the wave equation with measure data. This talk is based on joint work with Daniel Spector.