Hello, David !
I wrote a message before this one, but as I am not very used on the forums, I dont know if it started.
Thank you for your reply. I doesn't get any proof about the spanish origin of our commun surname. We still thought so because of the letter "J" that does'nt exist in Italian, but exists in Spanish. As you say, Spanish invaders stayed there during many centuries, such as you cand find, still now, in Sicilia, many families with surnames like "Lopez", "Sanchez" and so on. On the other hand, "Bajar" means "go down", and Bajada" meains "down way" in Spanish. So, it seems to me logic to think this way.
Searching on the site of Mormons, I found a Bajata in Mexic. This one is probably one of my relatives, as I know that my grand'father went there and stayed during a few years, at the end of the 19e century, and he probably didn't go there alone…
Anyway, our surname is not so common and spraid, so it is not impossible that we could be relatives. I should be interessed in knowing when your family came to America.