Certain oscillating operators on Herz-type Hardy spaces

Let 0 < p ≤ 1 < q < ∞, and ω₁, ω₂ ∈ A₁ (Muckenhoupt-class). We study an oscillating multiplier operator T_γ,β and obtain that it is bounded on the homogeneous weighted Herz-type Hardy spaces \(H\dot{K}_{q}^{\alpha,p}(\mathbb{R}^{n};\omega _{1},\omega _{2})\) $$H{\dot{K}}_q^{\alpha ,p}({\mathbb{R}}^n;{\omega }_1,{\omega }_2)$$ when \(\gamma=\frac{n\beta}{2}, \alpha =n(1-1/q)\) $$\gamma =\frac{n\beta }{2},\alpha =n(1-1/q)$$ . Also, for the unweighted case, we obtain the \(H\dot{K}_{q}^{\alpha,p}(\mathbb{R}^{n})\) $$H{\dot{K}}_q^{\alpha ,p}({\mathbb{R}}^n)$$ boundedness of T_γ,β under certain conditions on γ. These results are substantial improvements and extensions of the main results in the papers by Li and Lu and by Cao and Sun. As an application, we prove the \(H\dot{K}_{q}^{\alpha,p}(\mathbb{R}^{n})\) $$H{\dot{K}}_q^{\alpha ,p}({\mathbb{R}}^n)$$ boundedness of the spherical average S _t ^δ uniformly on t > 0.