  Solitons in the fractional space, supported by lattice potentials, have
recently attracted much interest. We consider the limit of deep one- and
two-dimensional (1D and 2D) lattices in this system, featuring finite bandgaps
separated by nearly flat Bloch bands. Such spectra are also a subject of great
interest in current studies. The existence, shapes, and stability of various
localized modes, including fundamental gap and vortex solitons, are
investigated by means of numerical methods; some results are also obtained with
the help of analytical approximations. In particular, the 1D and 2D gap
solitons, belonging to the first and second finite bandgaps, are tightly
confined around a single cell of the deep lattice. Vortex gap solitons are
constructed as four-peak \textquotedblleft squares" and \textquotedblleft
rhombuses" with imprinted winding number $S=1$. Stability of the solitons is
explored by means of the linearization and verified by direct simulations.
