In this paper we obtain new estimates for bilinear pseudodiferential operators with symbol in the class , when both arguments belong to Triebel-Lizorkin spaces of the type . The inequalities are obtained as a consequence of a refnement of the classical Sobolev embedding ↪bmo(ℝn), where we replace bmo(ℝn) by an appropriate subspace which contains L^∞(ℝn). As an application, we study the product of functions on when 1 < p < ∞, where those spaces fail to be multiplicative algebras.

We establish sharp global regularity of a class of multilinear oscillatory integral operators that are associated to nonlinear dispersive equations with both Banach and quasi-Banach target spaces. As a consequence we also prove the (local in time) continuous dependence on the initial data for solutions of a large class of coupled systems of dispersive partial differential equations.

In this paper we establish some endpoint estimates for bilinear pseudodifferential operators with symbol in the class BS, involving the space of functions with local bounded mean oscillation bmo(Rⁿ). As a consequence we also obtain an endpoint estimate of Kato-Ponce type.

This article is devoted to a study of the Hardy space H^log(R^d) introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space H¹ and a function in BMO to distributions that belong to H^log based on dyadic paraproducts. We also point out analogues of classical results of Hardy–Littlewood, Zygmund, and Stein for H^log and related Musielak–Orlicz spaces.

We prove weighted norm inequalities with Muckenhoupt’s A_p-weights, for a wide class of oscillatory integral operators. As a consequence, one also obtains the boundedness of commutators of the aforementioned operators with functions in BMO.

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.

We establish optimal local and global Besov-Lipschitz and Triebel-Lizorkin estimates for the solutions to linear hyperbolic partial differential equations. These estimates are based on local and global estimates for Fourier integral operators that span all possible scales (and in particular both Banach and quasi-Banach scales) of Besov-Lipschitz spaces B-p,q(s)(R-n) and certain Banach and quasi-Banach scales of Triebel-Lizorkin spaces F-p,q(s)(R-n).

In this paper we establish mapping properties of bilinear Coifman-Meyer multipliers acting on the product spaces H-1 (R-n) x bmo(R-n) and L-p (R-n) x bmo(R-n), with 1 < p < infinity. As application of these results, we obtain some related Kato-Poncetype inequalities involving the endpoint space bmo(R-n), and we also study the pointwise product of a function in bmo(R-n) with functions in H-1 (R-n), h(1) (R-n) and L-p(R-n), with 1 < p < infinity.

Extending work of Pichorides and Zygmund to the d-dimensional setting, we show that the supremum of L-p-norms of the Littlewood-Paley square function over the unit ball of the analytic Hardy spaces H-A(p) (T-d) blows up like (p-1)(-d) as p -> 1(+). Furthermore, we obtain an Llog(d) L-estimate for square functions on H-A(1) (T-d). Euclidean variants of Pichorides' theorem are also obtained.

In this paper we give an elementary proof for transference of local to global maximal estimates for dispersive PDEs. This is done by transferring local L-2 estimates for certain oscillatory integrals with rough phase functions, to the corresponding global estimates. The elementary feature of our approach is that it entirely avoids the use of the wave packet techniques which are quite common in this context, and instead is based on scalings and classical oscillatory integral estimates.