I wondered about that J in Italian records, but there were other names like Mannoja and Ajello (though some of the records read Mannoia and Aiello). The records I'm reading from Sicily have other Spanish surnames like Martinez and Rodriguez.
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There are 8 Bajata's listed in the Sicilian phone book...The earliest Bajata that I've heard of was in Malta in the 1730's. I've never done a search in any Spanish records, but I can give it a try.
I went to a few online translators and found:
baiata means shouting in Italian.
bajada means slope or descent, minimum (taxi fare) and down, in Spanish.
The earliest of my ancestors' from Sicily arrived in the US about 1900.