Mapping properties of certain oscillatory integrals on modulation spaces

Suppose β₁ > α₁ ≥ 0, β₂ > α₂ ≥ 0 and (k, j) ∈ R². In this paper, we mainly investigate the mapping properties of the operator

$${T_{\alpha ,\beta }}f\left( {x,y,z} \right) = \int_{{Q^2}} {f\left( {x - t,y - s,z - {t^k}{s^j}} \right){e^{ - 2\pi i{t^{ - {\beta _1}}}{s^{ - {\beta _2}}}}}{t^{ - 1 - {\alpha _1}}}{s^{ - 1 - {\alpha _2}}}dtds} $$

on modulation spaces, where Q² = 0, 1] × 0, 1] is the unit square in two dimensions.