Georgia Institute of Technology College of Sciences

It is well known that the wave operators cos(t (−∆)) and sin(t (−∆)) are not bounded on Lp(Rn), for n≥2 and 1≤p≤∞, unless p=2 or t=0. In fact, for 1 < p < ∞ these operators are bounded from W2s(p),p  to Lp(Rn) for s(p) := (n−1)/2 | 1/p − 1/2 |, and this exponent cannot be improved. This phenomenon  is symptomatic of the behavior of Fourier integral operators, a class of oscillatory operators which includes wave propagators, on Lp(Rn).

In this talk, I will introduce a class of Hardy spaces HFIOp (Rn), for p ∈ [1,∞],on which Fourier integral operators of order zero are bounded. These spaces also satisfy Sobolev embeddings which allow one to recover the optimal boundedness results for Fourier integral operators on Lp(Rn).

However, beyond merely recovering existing results, the invariance of these spaces under Fourier integral operators allows for iterative constructions that are not possible when working directly on Lp(Rn). In particular, we shall indicate how one can use this invariance to obtain the optimal fixed-time Lp regularity for wave equations with rough coefficients. We shall also mention the connection of these spaces to the phenomenon of local smoothing.

This talk is based on joint work with Andrew Hassell and Pierre Portal (Aus- tralian National University), and Zhijie Fan, Naijia Liu and Liang Song (Sun Yat- Sen University).