Flight Instructor reported a near miss during a training flight from a non-towered airport. The instructor pilot did not see the other aircraft in time to maneuver but the other pilot took evasive action to avoid a collision.

Narrative

Took off of Runway 19 at 0W3 with a busy pattern. The skydive aircraft had just called jumpers away just as we were beginning our takeoff roll. We made our call that we were departing runway 19 for closed traffic. As we call turning a normal crosswind leg of the traffic pattern; we hear the skydive aircraft call a 'high left crosswind runway 19; I have you in sight can I overtake you'. To which I respond that I do not have the aircraft in sight. I know that the aircraft is above us and descending on us however; I know he is behind us and I was unable to see him. As we are turning left downwind; he states that he is in front of us already. I know this to be false; however just as I am processing this information; I see him descend from above at an alarming rate and coming within a wingspan of the aircraft. He overtook me from the left and cut me off from above. Had I inadvertently gone to either side he would have collided with me. I could not see him until he was already below me. The behavior of this particular pilot has been noted by numerous people to be a threat to aviation safety completely disregarding basic traffic pattern procedures and flat out lying about his position on the common frequency in order to cut in front of other aircraft in an extremely reckless manner. He was descending at such a rate that I was unable to take any evasive action because I did not see him; with this being said he did have to take evasive action to avoid colliding with me on his descent.