The strategic air campaign over Germany by 8th US Air Force and "Bomber Harris" for the RAF has elicited decidedly mixed opinions. Depending on who you listen to it was a noble effort or it was a waste.
At the same time, it is true that, after Dieppe, the alternatives of a true second front in continental Europe were limited, so that bombing was probably one of the few way to strike at Germany.
However, we also know that the Luftwaffe gradually lost its air superiority on the Eastern Front which made Germany's situation there ever more untenable.
Is there any study that quantifies how much of a Westward shift of German fighters happened from early Barbarossa to say late 43, once Allied bombing really got underway? Were the Allied raids a major cause to the gradual weakening of the East Front Luftwaffe?
Please, I am not asking about the losses suffered by Luftwaffe transports during Stalingrad. This is about losing control of the skies, i.e. weakening ME109/FW190 availability over Russia because of transfer to protect Germany.
Edit: it's not specifically about moving planes around, East to West. Allocation of replacement planes and pilots, even fuel allocation, if shortages resulted in less Luftwaffe activity in the East. In a roundabout way, even prioritization of design and production of bomber-destroying aircraft, like the ME163, would have affected the East front.
Basically, how many Luftwaffe resources, defined broadly, do we see used to protect Germany, as opposed to used elsewhere, the most important other location being beating the Red Army on the ground and in the air?