Global and local regularity of Fourier integral operators on weighted and unweighted spaces (Q2925676)

In this interesting paper, the authors study systematically the global \(L^p\) boundedness of Fourier integral operators with amplitudes in Hörmander classes \(S^m_{\varrho, \delta}\) with \(\varrho\) and \(\delta\) in \([0, 1]\), that is, in the range of all possible parameters \(\varrho\) and \(\delta\). Both operators with smooth or rough phases, and smooth or rough amplitudes are considered, cf. Chapter 2. Moreover, they study global and local weighted norm inequalities for Fourier integral operators with weights in the Muckenhoupt, \(A_p\). The proof of a sharp weighted \(L^p\) boundedness theorem for Fourier integral operators is based on global unweighted \(L^p\) results, cf. Chapter 3. The weighted results in turn are used to establish the validity of certain vector-valued inequalities, the boundedness of certain Fourier integral operators in weighted Triebel-Lizorkin spaces, and the boundedness and weighted boundedness of commutators of Fourier integral operators with functions of bounded mean oscillation BMO, cf. Chapter 4. Finally, global unweighted and local weighted estimates for the solutions of the Cauchy problem for \(m\)-th and second-order hyperbolic partial differential equations on \({\mathbb R}^n\) are proved.