Oscillatory hyper-Hilbert transform on Wiener amalgam spaces

Abstract

Abstract We study the boundedness of the oscillatory integral T α , β f ( x , y ) = ∫ Q 2 f ( x − γ 1 ( t ) , y − γ 2 ( s ) ) e − 2 π i t − β 1 s − β 2 t − α 1 − 1 s − α 2 − 1 d t d s {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 Q 2 = [ 0 , 1 ] × [ 0 , 1 ] {Q}^{2}=\left[0,1]\times \left[0,1] is the unit square in two dimensions, ( x , y ) ∈ R n × R m , γ 1 ( t ) = ( t p 1 , t p 2 , … , t p n ) , γ 2 ( s ) = ( s q 1 , s q 2 , … , s q m ) \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 R n {{\mathbb{R}}}^{n} and R m {{\mathbb{R}}}^{m} .