How to estimate the exponential sum $\sum_{|m|\le N} \frac{e^{im^{3/2}}}{m^{1/4}}$

Question

I am interested in upper-bounding tightly the following sum $$\mathcal{S}(N)=\sum_{|m|\le N} \frac{e^{im^{3/2}}}{m^{1/4}}$$ better than the naive triangle inequality estimate $$\mathcal{S}(N)\lesssim N^{3/4}.$$ This sum originates from trying to estimate the following exponential sum, for irrational $\alpha$, $$\mathcal{A}(N)=\sum_{|n|\le N} \exp 2\pi i \alpha n^3,$$ related to the possible equidistribution of the sequence $\alpha \,n^3 \operatorname{mod} 1$. The sum $\mathcal{S}$ originates from using $\sum_{|n|\le N}=\sum_{n\in\mathbb{Z}}\eta(n/N)$ for some suitable cut-off function $\eta$ followed by Poisson summing and a suitable application of the stationary phase principle.