I am wondering whether the following uniform upper bound holds:

$|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le Cab^2,$

where $0<a<b<1$, $N>N_0(a,b)\gg1$, and $C$ is a constant independent of $N,a,b$.

Remark: (1) A trivial upper bound is 1; (2) If we write $\frac1t\sin(Nb^2t)=b^2\int_0^N \cos(ub^2t)du$, then we get an easy upper bound $Nab^2$; (3) This uniform upper bound can be verified by Matlab but I have no idea how to prove it; (4) Equivalently, $\limsup_{N\to\infty}\frac1{ab^2}|\int_a^{2a}\frac1t\sin(N b^2t)\exp(iNbt^2)dt|\le C.$

Any comments are welcome!:)