Sharp L^p decay estimates for degenerate and singular oscillatory integral operators

2021/01/27 by Xu, Shaozhen
#Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.2101.11341

Abstract

We consider the following model of degenerate and singular oscillatory integral operators: Tf(x)=∫_ℝ e^iλS(x,y)K(x,y)ψ(x,y)f(y)dy, where the phase functions are homogeneous polynomials of degree n and the singular kernel K(x,y) satisfies suitable conditions related to a real parameter μ. We show that the sharp decay estimates on L² spaces, obtained in \citeliu1999model, can be preserved on more general L^p spaces with an additional condition imposed on the singular kernel. In fact, we obtain that ‖Tf‖_L^p≤ C_{E,S,ψ,μ,n,p}λ^-(1-μ)/(n)‖f‖_L^p, (n-2μ)/(n-1-μ)≤ p ≤(n-2μ)/(1-μ). The case without the additional condition is also discussed.

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