Oscillatory hyper-Hilbert transform on Wiener amalgam spaces

Abstract

We study the boundedness of the oscillatory integral $$T_{\alpha ,\beta }f\left(x,y\right)=\underset{Q^2}{\int }f\left(x-{\gamma }_1\left(t\right),y-{\gamma }_2\left(s\right)\right)e^{-2\pi i{t}^{-{\beta }_1}{s}^{-{\beta }_2}}{t}^{{-\alpha }_1-1}{s}^{-{\alpha }_2-1}\mathrm{d}t\mathrm{d}s$$ on Wiener amalgam spaces, where $Q^2=\left[0,1\right]\times \left[0,1\right]$ is the unit square in two dimensions, $\left(x,y\right)\in {\mathbb{R}}^n\times {\mathbb{R}}^m,{\gamma }_1\left(t\right)=\left(t^{p_1},t^{p_2},\dots ,t^{p_n}\right),{\gamma }_2\left(s\right)=\left(s^{q_1},s^{q_2},\dots ,s^{q_m}\right)$ are homogeneous curves on ${\mathbb{R}}^n$ and ${\mathbb{R}}^m$.