An accelerating gradient larger than 1 GeV/m, can potentially be produced by the Cherenkov Smith-Purcell radiative mechanism in a dielectric capillary with a variable inner radius. When an electron bunch propagates through the capillary, its self-field interacts with the dielectric material and produces wakefields that propagate inside the material at the Cherenkov angle. The wakefields are slowed down below the speed of light, as the relative dielectric permittivity of the material is larger than 1. The radiation is then reflected from the capillary's metallic boundary and diffracted back into the vacuum region, producing high accelerating fields on the capillary axis with a distinct frequency signature. In presence of a periodic boundary the Smith-Purcell radiation imposes frequency dispersion.