A TRACON Controller reported a corporate jet descended below their assigned altitude below the Minimum Vectoring Altitude and through a satellite airport Class D airspace.

Narrative

Aircraft X was inbound from the North requesting to land runway XX. I gave Aircraft X a vector; 'Turn Left Heading 180 Descend and Maintain 8000 feet vector Visual Approach RWY XX.' A few minutes later I observed the pilot at 7700 feet. By the time I issued a Low Altitude alert to the pilot and climbed him to 8000 feet. the aircraft was at 7300 feet and descending. The aircraft went into ZZZ1 class D airspace. ZZZ1 called on the shout line at that time but was inaudible to me as I was fixing the altitude. I called back to ZZZ1 letting them know the pilot was climbing back up and they 'approved' the transition. Aircraft X was below the 7500 feet MVA but near or on final of RWY XY for ZZZ1 so it appeared to be in a safe proximity. Pilot was soon after given a Visual approach to ZZZ RWY XX and landed without issue. Catching the aircraft sooner would have stopped any need for a Low Altitude alert. I distinctly remember seeing the aircraft at 8000 feet. and on the next scan pass; they were at 7700 feet.