NCT Controller reported Aircraft X pilot misunderstood Tower instructions led to early level off below the MVA resulted in a CFTT event.

Narrative

Aircraft X was released from LVK on the LVK3 departure. Aircraft are required to climb to a minimum of 4000. MVA in the area is 3800. Radio communications are difficult until the aircraft reaches 1500. The first few transmissions to the aircraft were not understood by the Pilot. It took a few attempts before the Pilot could understand my instructions. I instructed the Pilot to ident and say altitude leaving for radar identification. The Pilot reported they were level at 1500. I verified that they had leveled off below the MVA and issued a low altitude alert with an immediate climb to 4000. The Pilot stated that the Tower had assigned 1500. Although the Pilot idented and altitude was verified I failed to say 'radar contact'. During this time I was busy with an OAK arrival rush in low IMC; a departure from HWD; and other south bay airport departures. LVK Tower reported that they instructed Aircraft X to contact departure leaving 1500. I was instructed to issue a warning to the Pilot. Aircraft X was switched to the next sector without further incident. Later it was discovered the Tower instructed the Pilot to contact departure leaving 1500 but the Pilot read back maintain 1500. Don't issued freq. change associated with an altitude leaving. This can easily be confused by the Pilot and read back error hard to catch. Switch departures to NCT when radio reception is likely.