Fourier-Mukai transforms of slope stable torsion-free sheaves on a product elliptic threefold

On the product elliptic threefold $X=C\times S$ where $C$ is an elliptic curve and $S$ is a K3 surface of Picard rank 1, we define a notion of limit tilt stability, which satisfies the Harder-Narasimhan property. We show that under the Fourier-Mukai transform $\mathrm{\Phi }$ on $D^b(X)$ induced by the classical Fourier-Mukai transform on $D^b(C)$, a slope stable torsion-free sheaf satisfying a vanishing condition in codimension 2 (e.g. a reflexive sheaf) is taken to a limit tilt stable object. We also show that a limit tilt semistable object on $X$ is taken by $\mathrm{\Phi }$ to a slope semistable sheaf, up to modification by the transform of a codimension 2 sheaf.