Okay, I think I've found out as much online as I can currently about Julius Pollack. The bad news is that I couldn't find out much about the fashion label. I suspect it was short-lived.
I'm curious about the length of your coat, because during this period you can help date garments by the length. My dress is 46.5 inches long, which suggests a late '40s date. And the only other JP I found was one a vintage dealer listed on Facebook four years ago. Her dress looks late '40s to me, and the length was 46 inches. If your coat is similar, it may be that the business didn't flourish (there was a lot of competition in post-war Melbourne) and he turned to other endeavours.
The only snippet I found about JP relating to fashion was a business listing in 1948 where he registered the name 'Julius Pollack Creations'. But he soon added other names: Fashion Weavers Ltd and Fashion Weavers Pty Ltd and then more that don't appear to be fashion-related, especially Julius Pollack Enterprises, but also Polco Investments and Domain Gate Pty Ltd. The businesses were finally voluntarily liquidated in 2001.
Here's what I can tell you about Julius Pollack about the man:
Born in 1904 in Belarus, he migrated to Australia and married Fay Frieze. They had two sons: Norman and Bruce. Frieze is a famous name in the Melbourne Jewish rag trade, because David Frieze ran 'Frieze' fashions and made trendy menswear, especially tailored suits, from the '60s to the '80s. There may be a family link.
Anyway, Julius was a migrant success story: he became president of the Australian Amateur Fencing Federation and in 1963 was awarded an MBE: Member of the British Empire for his service. In 1966 he was elected to Malvern council and in 1970 (as mentioned above) became mayor, a position he held for two years.
In 1991 he was again awarded, this time with an Order of Australia medal for service to local government and the community. He passed away two years later.
If you wanted to find out more, I think
the Malvern Historical Society would be a good place to start as JP had good roots in the area, and served the community for many decades.