Estimate for generalized unimodular multipliers on modulation spaces

In this paper, we consider the operator eitϕ(h(D))eitϕ(h(D)) where hh defined on RnRn is a C∞(Rn∖{0})C∞(Rn∖{0}) positive homogeneous function with degree λ>0λ>0 and ϕ:R→R is a smooth function satisfying the following.(A1) There exists a constant m1>0m1>0 such that for all μ∈N0:=N∪{0}μ∈N0:=N∪{0}|ϕ(μ)(r)|≲rm1−μ,r≥1.(A2) There exists a constant m2>0m2>0 such that for all μ∈N0μ∈N0|ϕ(μ)(r)|≲rm2−μ,0