We establish new local and global estimates for evolutionary partial differential equations in classical Banach and quasi-Banach spaces that appear most frequently in the theory of partial differential equations. More specifically, we obtain optimal (local in time) estimates for the solution to the Cauchy problem for variable-coefficient evolutionary partial differential equations. The estimates are achieved by introducing the notions of Schrödinger and general oscillatory integral operators with inhomogeneous phase functions and prove sharp local and global regularity results for these in Besov–Lipschitz and Triebel–Lizorkin spaces.

We establish the global regularity of multilinear Fourier integral operators that are associated to nonlinear wave equations on products of $L^p$ spaces by proving endpoint boundedness on suitable product spaces containing combinations of the local Hardy space, the local BMO and the $L^2$ spaces.

In this paper we prove weighted norm inequalities with weights in the ${{A}_{p}}$ classes, for pseudodifferential operators with symbols in the class $S_{\rho ,\delta }^{n(\rho -1)}$ that fall outside the scope of Calderón–Zygmund theory. This is accomplished by controlling the sharp function of the pseudodifferential operator by Hardy–Littlewood type maximal functions. Our weighted norm inequalities also yield ${{L}^{p}}$ boundedness of commutators of functions of bounded mean oscillation with a wide class of operators in $\text{OPS}_{\rho ,\delta }^{m}$ .