On rough oscillatory singular integral operators

Abstract Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>P</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:mrow> <m:mo>→</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> </m:math> {P:\mathbb{R}^{n}\times\mathbb{R}^{n}\mathbb{\rightarrow R}} be a polynomial mapping where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} is the n -dimensional euclidean space (with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>n</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> {n\geq 2} ). Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>h</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">∞</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mo>→</m:mo> <m:mi>ℝ</m:mi> </m:mrow> </m:mrow> </m:math> {h:(0,\infty)\rightarrow\mathbb{R}} be a measurable function, and let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:msup> <m:mi>𝕊</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:mrow> </m:math> {\Omega\in L^{1}(\mathbb{S}^{n-1})} be a homogeneous function of degree zero on <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>n</m:mi> </m:msup> </m:math> {\mathbb{R}^{n}} with mean value zero over the unit sphere <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>𝕊</m:mi> <m:mrow> <m:mi>n</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {\mathbb{S}^{n-1}} . The aim of this paper is to investigate the long-standing problem concerning the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:mi>p</m:mi> </m:msup> </m:math> {L^{p}} boundedness of the oscillatory singular integral operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msub> <m:mi>T</m:mi> <m:mrow> <m:mi>P</m:mi> <m:mo>,</m:mo> <m:mi mathvariant="normal">Ω</m:mi> <m:mo>,</m:mo> <m:mi>h</m:mi> </m:mrow> </m:msub> <m:mo>⁢</m:mo> <m:mi>f</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mtext>p.v.</m:mtext> <m:mo>⁢</m:mo> <m:mrow> <m:msub>