Different behaviour of the spin structure functions $g_1(x)$ and $h_1(x)$ at $x\to 0$

We consider low-$x$ behaviour of the spin structure functions $g_1(x)$ and $h_1(x)$ in the unitarized chiral quark model which combines ideas on the constituent quark structure of hadrons with a geometrical scattering picture and unitarity. A nondiffractive singular low-$x$ dependence of $g_1^p(x)$ and $g_1^n(x)$ indicated by the recent SMC experimental data is described. A diffractive type smooth behaviour of $h_1(x)$ is predicted at small $x$. The expectations for the double-spin asymmetries in the low-mass Drell-Yan production at RHIC in the central region are discussed alongside.