Rough oscillatory singular integrals on $\mathbb {R}^{n}$
Tom 221 / 2014
Studia Mathematica 221 (2014), 249-267
MSC: Primary 42B20; Secondary 26D05.
DOI: 10.4064/sm221-3-4
Streszczenie
We establish sharp bounds for oscillatory singular integrals with an arbitrary real polynomial phase $P$. The kernels are allowed to be rough both on the unit sphere and in the radial direction. We show that the bounds grow no faster than $\log\deg(P) $, which is optimal and was first obtained by Papadimitrakis and Parissis (2010) for kernels without any radial roughness. Among key ingredients of our methods are an $L^1 \to L^2$ estimate and extrapolation.
