I have recently come across a (recent) problem in a different sub-area, by a well-established mathematician, that he deems likely to be intractable, but to which I have found a particularly nice proof (at least I believe so) using the techniques well known in my own sub-area. I think the result itself has a particularly illuminating interpretation so I would like to get it published somehow, but since I am not familiar with the sub-area nor particularly interested in working in it (by which I simply mean I am not suitable for the sub-area), would the best recourse be to contact the author and ask him to write a paper together on the result?
Probably more like: Contact the author, and ask for his reaction. It could be: "Great, let me help you publish it". Or it could be: "You made the following well-nown error."
If you're confident of your result (even if you might not have extensive knowledge of the literature), I would recommend you submit your work to a journal. If your work is so closely based on this well-established mathematician, he or she will likely be contacted by the journal to perform a review. But as @Gerald Edgar points out, be prepared for a review that states: "You make the following well-known error."