Estimates for rough Fourier integral and pseudodifferential operators and applications to the boundedness of multilinear operators

Abstract

We study the boundedness of rough Fourier integral and pseudodifferential operators, defined by general rough Hörmander class amplitudes, on Banach and quasi-Banach L p spaces. Thereafter we apply the aforementioned boundedness in order to improve on some of the existing boundedness results for Hörmander class bilinear pseudodifferential operators and certain classes of bilinear (as well as multilinear) Fourier integral operators. For these classes of amplitudes, the boundedness of the aforementioned Fourier integral operators turn out to be sharp. Furthermore we also obtain results for rough multilinear operators.

On the boundedness of certain bilinear oscillatory integral operators

“…Of direct relevance to this paper is another result in [16] which proved a corresponding boundedness result for oscillatory integral operators. This result, again contained in Theorem 5.12, states that in the case that σ verifies estimate (2) with m < 0, and if the phases ϕ j ∈ C ∞ (R n × R n ) (are not assumed to be homogeneous), satisfy (6) and the condition (8) |∂ α x ∂ β ξ ϕ j (x, ξ)| C α,β for all α and β with 2 |α| + |β| for both j = 1, 2, then T ϕ 1 ,ϕ 2 σ is bounded from L 2 × L 2 → L 1 .…”

“…In [16] the boundedness of linear oscillatory integral operators with amplitudes in the class L p S m ρ (n, 1) is proved. This amplitude class was first introduced by N. Michalowski, D. Rule and W. Staubach in [15].…”

“…They also assume the phase functions ϕ j (x, ξ) ∈ C ∞ (R n × (R n \ {0})) are homogeneous of degree one in the frequency variable ξ and verify the non-degeneracy condition for j = 1, 2. They then deduce that the bilinear operator (1) This result for Fourier integral operators was extended to the full Banach range of exponents 1 q 1 , q 2 ∞ and the assumption of compact support in x was eliminated by S. Rodríguez-López and W. Staubach in [16]. This result, contained in Theorem 5.12 of [16], concentrates on amplitudes which were only measurable in x, replacing (2) with…”

“…Indeed, the global regularity of bilinear Fourier integral operators with amplitudes that are neither compactly supported nor smooth in the first variable is studied in [16] in part by proving the boundedness of linear operators with amplitudes in the aforementioned classes.…”

“…The main goal of this paper is to confirm that this is the case. Most of the methods we employ are different to those used in [11] and [16], essentially due to the fact we must take advantage of more subtle cancellation properties at the end-point m = 0.…”