L^p-L^q boundedness of integral operators with oscillatory kernels: Linear versus quadratic phases

2015/07/13 by Abdelhakim, Ahmed A.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1507.03346

Abstract

Let T^j,k_N:L^p(B) → L^q([0,1]) be the oscillatory integral operators defined by T^j,k_Nf(s):=∫_B f(x) e^\imath N|x|^js^k dx, (j,k)∈\1,2\², where B is the unit ball in ℝⁿ and N >>1. We compare the asymptotic behaviour as N→ +∞ of the operator norms ∥ T^j,k_N ∥_ L^p(B)→ L^q([0,1]) for all p, q∈ [1,+∞]. We prove that, except for the dimension n=1, this asymptotic behaviour depends on the linearity or quadraticity of the phase in s only. We are led to this problem by an observation on inhomogeneous Strichartz estimates for the Schrödinger equation.

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