Preservation of semistability under Fourier-Mukai transforms

For a trivial elliptic fibration $X=C\times S$ with $C$ an elliptic curve and $S$ a projective K3 surface of Picard rank $1$, we study how various notions of stability behave under the Fourier-Mukai autoequivalence $\mathrm{\Phi }$ on $D^b(X)$, where $\mathrm{\Phi }$ is induced by the classical Fourier-Mukai autoequivalence on $D^b(C)$. We show that, under some restrictions on Chern classes, Gieseker semistability on coherent sheaves is preserved under $\mathrm{\Phi }$ when the polarisation is fiber-like'. Moreover, for more general choices of Chern classes, Gieseker semistability under a fiber-like’ polarisation corresponds to a notion of ${\mu }_{\ast }$-semistability defined by a `slope-like’ function ${\mu }_{\ast }$.