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 ^{i|x| a}Ω(x)|x|⁻ⁿ is studied, where a∈)R, a≠0,1 and Ω≠L ¹(Sⁿ⁻¹) is homogeneous of degree zero and satisfies certain cancellation condition. When kernel Ω(x′≠)Llog⁺ L(Sⁿ⁻¹), the $\dot F_p^{\alpha ,q} (R^n )$ boundedness of the above operator is obtained. Meanwhile, when Ω(x) satisfies L ¹-Dini condition, the above operator T is bounded on $\dot F_1^{0,1} (R^n )$.