Space charge effects and mitigation in the CERN PS Booster, in view of the upgrade

The CERN PS Booster (PSB) is presently running with a space-charge tune spread larger than 0.5 at injection. Since the High Luminosity LHC (HL-LHC) will require beams with twice the intensity and brightness of today, the LHC Injector Upgrade (LIU) Project is putting in place an upgrade program for all the injector chain and, in particular, it relies on the important assumption that the PS Booster can successfully produce these beams after the implementation of the 160 MeV H$^{-}$ injection from Linac4. This contribution describes the studies (measurements and simulations) that have been carried out to confirm that the PSB can indeed perform as needed in terms of beam brightness for the future HL-LHC runs. The importance of the mitigation measures already in place, such as the correction of the half-integer line, and the effects of non-linear resonances on the beam are also discussed