A class of oscillatory singular integrals on triebel-lizorkin spaces

Abstract

The boundedness on Triebel-Lizorkin spaces of oscillatory singular integral operator T in the form e(superscript i|x|(superscript α))Ω(x)|x|(superscript -n) is studied, where α∈R, α≠0, 1 and Ω∈L^1(S(superscript n-1)) is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω(x')∈Llog(superscript +)L(S(superscript n-1)), the F(superscript a,q subscript p)(R(superscript n)) boundedness of the above operator is obtained. Meanwhile, when Ω(x) satisfies L^1-Dini condition, the above operator T is bounded on F(superscript 0,1 subscript 1)(R(superscript n)).