Global and local regularity of Fourier integral operators on weighted and unweighted spaces

Source abstract

We investigate the global continuity on L p L^p spaces with p ∈ [ 1 , ∞ ] p\in [1,\infty ] of Fourier integral operators with smooth and rough amplitudes and/or phase functions subject to certain necessary non-degeneracy conditions. In this context we also prove the optimal global L 2 L^2 boundedness result for Fourier integral operators with non-degenerate phase functions and the most general smooth Hörmander class amplitudes i.e. those in S ϱ , δ m S^{m} _{\varrho , \delta } with ϱ , δ ∈ [ 0 , 1 ] \varrho , \delta \in [0,1] . We also prove the very first results concerning the continuity of smooth and rough Fourier integral operators on weighted L p L^{p} spaces, L w p L_{w}^p with 1 > p > ∞ 1> p > \infty and w ∈ A p , w\in A_{p}, (i.e. the Muckenhoupt weights) for operators with rough and smooth amplitudes and phase functions satisfying a suitable rank condition. These results are shown to be optimal for operators with amplitudes in classical Hörmander classes and can also be given a geometrically invariant formulation. The weighted results are in turn applied to prove, for the first time, weighted and unweighted estimates for the commutators of Fourier integral operators with functions of bounded mean oscillation BMO, estimates on weighted Triebel-Lizorkin spaces, and finally global unweighted and local weighted estimates for the solutions of the Cauchy problem for m m -th and second order hyperbolic partial differential equations on R n . \mathbf {R}^n . The global estimates in this context, when the Sobolev spaces are L 2 L^2 based, are the best possible.