1Department of Mathematics and Physics Qatar University Doha, Qatar 2Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010, U.S.A. 3Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260, U.S.A.
Studia Mathematica, Tome 221 (2014) no. 3, pp. 249-267
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.
Keywords:
establish sharp bounds oscillatory singular integrals arbitrary real polynomial phase kernels allowed rough unit sphere radial direction bounds grow faster log deg which optimal first obtained papadimitrakis parissis kernels without radial roughness among key ingredients methods estimate extrapolation
Affiliations des auteurs :
Hussain Mohammad Al-Qassem
1
;
Leslie Cheng
2
;
Yibiao Pan
3
1
Department of Mathematics and Physics Qatar University Doha, Qatar
2
Department of Mathematics Bryn Mawr College Bryn Mawr, PA 19010, U.S.A.
3
Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260, U.S.A.
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author = {Hussain Mohammad Al-Qassem and Leslie Cheng and Yibiao Pan},
title = {Rough oscillatory singular integrals on $\mathbb {R}^{n}$},
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Hussain Mohammad Al-Qassem; Leslie Cheng; Yibiao Pan. Rough oscillatory singular integrals on $\mathbb {R}^{n}$. Studia Mathematica, Tome 221 (2014) no. 3, pp. 249-267. doi: 10.4064/sm221-3-4
