WARNING: The following contains spoilers for X-Men/Fantastic Four #4 by Chip Zdarsky, Terry Dodson, Rachel Dodson, Ranson Getty, Laura Martin, and VC's Joe Caramagna, on sale now.
Doctor Doom has come up against most of the heroes in the Marvel Universe at one point or another and has developed workarounds for most of them in battle. But in the process of his most recent analysis, he may have also uncovered the key to a long-standing question for the mutants on Krakoa.
Doom has been analyzing Shadowcat's powers and has even figured out a way to counter them -- which means he could also figure out what about her makes it impossible for her to come to Krakoa.
Doctor Doom is no stranger to Shadowcat and her powers. During the events of the Mutant Massacre, Shadowcat tried to phase an unsuspecting Rogue out of the way of an incoming energy spear thrown by Harpoon. The energy-based weapon affected her in a unique way, however, leaving Kitty trapped in her phased form. The longer she was in this form, the less chance she had of ever being able to fully reconstruct her physical form. It was only Psylocke's telepathy that kept Shadowcat in one piece.
When Reed Richards declined to help due to a recent series of blows to his confidence, the X-Men were forced to turn to Doctor Doom. The villainous genius relished the chance to beat a challenge that Richards didn't, and agreed to help the X-Men. In the end, it was only when Richards and Doom worked together that they were able to discover the true source of the problem with Shadowcat and helped restore her ability to regain her physical form. This means that Doctor Doom has had plenty of time to analyze Pryde's powers, allowing him to even find workarounds to deal with her and her power set.
While trying to rescue Franklin Richards from Doctor Doom, a specially designed barrier kept the heroes from interfering with his observations on Franklin. Only Shadowcat was able to break through the barrier with her phasing powers -- although it still proved difficult for her. But Doom casually revealed that he had prepared for this by finding a way to reverse Kate's ability to make herself lighter than air by reducing her density, leading Kate to become "heavier than stone" and crash through the ground. Kate was able to restore herself, however, and attack Doom, helping free Franklin from the experiment.
However, this suggests that Doctor Doom has taken some time specifically to try and research the full limits of what Kate Pryde's powers can do, meaning he might actually have some interesting research for the X-Men to potentially gain access to. Kate Pryde has been the only member of the X-Men who hasn't been able to cross the threshold of the Krakoan Gates. There have been suggestions she may not actually even be a mutant at all, and that her failure to pass through the gates might be because of that. If so, then Doom and his analysis of her powers might have given him some very important clues towards finding out what exactly she is.
But considering how poorly things went between the X-Men and Doom, it's unlikely the villain will turn over any useful information to the mutants. However, he could eventually reveal that information down the line, or try to use it to his advantage in the coming battles he predicts will come with the mutants.