Kloosterman sums over smooth numbers

We give nontrivial upper bounds in various ranges for Kloosterman sums of the form∑′n∈S(x,y)exp⁡(2πian‾/m), where m,a are integers with m≥2, (a,m)=1, and S(x,y) is the set of y-smooth numbers up to x. We also obtain a better bound on average over m as∑m∼Mmax(a,m)=1⁡|∑′n∈S(x,y)exp⁡(2πian‾/m)|, where m∼M means M