Walk to the right and use the elevator there. Jump over the elevator below for now and duck under the two balls to the platform below. Walk left and hit the flashing brick above the elevator and then head to the right and use the elevator there. Head to the top, face left and fire at the flashing brick far to the right. After that, go under the ball on your left and enter the door. When you reappear, jump over the ball to the left and use the elevator. After that, walk to the right, hit the flashing brick in the distance, and then head back down elevator. Now, wait for the ball to go up and then fall into the hole and use the elevator at the far right.
You're back at the beginning, but now the path is clear. Use the elevator farther ahead and then another one at the top. Climb the ascending platforms to the right and enter the first door at the top. When you reappear, jump up the ascending platforms and then head over the hole to the door on the left. Now, get onto the elevator but don't take it up. Instead, jump onto the stack of platforms to the right and they will disappear. Take the elevator you land on to the top and hit the flashing brick you see ahead. Head back down and take the elevator right beside you, and then the next one further to the right.
Enter the door. Jump onto the elevator beside you and take it up. Now carefully make your way through the three balls bouncing around ahead. The best way to do this is to wait for all three to line up and then walk pass them cleanly. It's the middle ball that's the most trouble, since it has the least distance to travel. You can't wait too long though, or the elevator will go back down again and you'll have to make your way back up. When you make it, jump across the hole to the right when the ball isn't around. Stand on the right edge of the elevator before you jump or you might not make it. Now finally jump up the ascending platforms ahead to finish the stage. Avoid the balls that materialize while you're jumping. They are fairly easy to dodge but don't get hit.