Fifty years is certainly a long time, but it does not really tell us how intensive and extensive your research has been.
If you have not done so already, you must follow genealogy orthodoxy and search backwards through time for all likely relevant records for both Harry and Aaron. If, as I suspect, Aaron DORF is the one who died in 1935, then his father (as shown on his indexed death certificate on FamilySearch) was Benjamin. Get an actual copy of the death cert. (It may be downloaded from FamilySearch.org at a Family History Center). I suggest you get a photo of Aaron's gravestone to learn what his father's Hebrew/Yiddish name was. Then, confirm it was the same (or similar) for Harry.
If Harry and/or Aaron naturalized, acquire all their paperwork. If Harry filed a declaration of intention, he may have complete the process. His papers may tell you the port, date and ship on which he arrived in the United States. The same for Aaron. Passenger manifests from 1907 on often provide information on last residence and place of birth.
I have one side of my family from Zaleszczyki, Galicia and Bukovina (mostly Radautz, but also Czernowitz). Vital records from Zaleszczyki have not been found. However, vital records from Czernowitz and Radautz have been indexed and may be found at GeneaSearch - https://czernowitz.geneasearch.net/index.php I noted a marriage record for an Aron Dorf and Gitel Zwick in Radautz in 1901. Without doing research for you, I cannot tell if this is your relative. No birth records for Chaim Dorf were noted in that database.
If your people were from Galicia, then searches on JRI-Poland may be useful. I have found that although I cannot find 19th century records from Zaleszczyki, that many relatives moved on from Zaleszczyki in the 1880s and 1890s to other Galician and Bukovinan communities and, sometimes, Vienna.
Best of luck with your research!
Emily Garber, Phoenix, AZ <emilyhgarber@...>