Oscillatory hyper-Hilbert transform on Wiener amalgam spaces

Abstract

We study the boundedness of the oscillatory integral Tα,βf(x,y)=∫Q2f(x−γ1(t),y−γ2(s))e−2πit−β1s−β2t−α1−1s−α2−1dtds{T}_{\alpha ,\beta }f\left(x,y)=\mathop{\int }\limits_{{Q}^{2}}f\left(x-{\gamma }_{1}\left(t),y-{\gamma }_{2}\left(s)){e}^{-2\pi i{t}^{-{\beta }_{1}}{s}^{-{\beta }_{2}}}{t}^{{-\alpha }_{1}-1}{s}^{-{\alpha }_{2}-1}{\rm{d}}t{\rm{d}}s on Wiener amalgam spaces, where Q2=[0,1]×[0,1]{Q}^{2}=\left[0,1]\times \left[0,1] is the unit square in two dimensions, (x,y)∈Rn×Rm,γ1(t)=(tp1,tp2,…,tpn),γ2(s)=(sq1,sq2,…,sqm)\left(x,y)\in {{\mathbb{R}}}^{n}\times {{\mathbb{R}}}^{m},{\gamma }_{1}\left(t)=\left({t}^{{p}_{1}},{t}^{{p}_{2}},\ldots ,{t}^{{p}_{n}}),{\gamma }_{2}\left(s)=\left({s}^{{q}_{1}},{s}^{{q}_{2}},\ldots ,{s}^{{q}_{m}}) are homogeneous curves on Rn{{\mathbb{R}}}^{n} and Rm{{\mathbb{R}}}^{m}.

Keywords

• oscillatory integral
• wiener amalgam spaces
• curves
• product domain
• 42b20
• 42b25