A rough hypersingular integral operator with an oscillating factor on function space

Abstract

The singular integral operator

$$T_{\alpha ,\beta } f(x) = p.v.\int_{R^n } {\frac{{e^{i\left| y \right|^{ - \beta } } \Omega (y')}}{{\left| y \right|^{n + \alpha } }}} f(x - y)dy,$$

defined for all test functions f is studied, where Ω(y′) is a distribution on the unit sphere S ⁿ⁻¹ satisfying certain cancellation condition. It is proved that T _α,β is a bounded operator from the Triebel-Lizorkin space F ^s,q _np to the Triebel-Lizorkin space F ^s+γ,q _p , provided that Ω(y′) is a distribution in the Hardy space H ^r (S ^{n − 1}) with r = (n − 1)/(n − 1 + γ).