We consider a convolution operator Tf = p.v. Ω ⁎ f with $\Omega \left(x\right)=K\left(x\right)e^{ih\left(x\right)}$, where K(x) is an (n,β) kernel near the origin and an (α,β), α ≥ n, kernel away from the origin; h(x) is a real-valued $C^{\infty }$ function on ${ℝ}^n\setminus 0$. We give a criterion for such an operator to be bounded from the space $H_0^p\left({ℝ}^n\right)$ into itself.

Let $Tf\left(x\right)=p.v.{ʃ}_{ℝ¹}{e}^{iP(x-y)}f\left(y\right)/(x-y)dy$, where P is a real polynomial on ℝ. It is proved that T is bounded on the weighted H¹(wdx) space with w ∈ A₁.