2020/12/21 by James Wright, Wright, James
Mathematics · #42B20 #Advanced Differential Equations and Dynamical Systems #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2012.11256
openalex publication_date 2020/12/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we develop a theory for oscillatory integrals with complex phases. When f:\mathbb Cn → \mathbb C, we evaluate this phase function on the basic character \rm e(z) := e2πi x e2πi y of \mathbb C ≃ \mathbb R2 (here z = x+iy ∈ \mathbb C or z = (x,y) ∈ \mathbb R2) and consider oscillatory integrals of the form I = ∫_\mathbb Cn \rm e(f(\underlinez)) ϕ(\underlinez) d\underlinez where ϕ∈ C∞c(\mathbb Cn). Unfortunately basic scale-invariant bounds for the oscillatory integrals I do not hold in the generality that they do in the real setting. Our main effort is to develop a perspective and arguments to locate scale-invariant bounds in (necessarily) less generality than we are accustomed to in the real setting.