"established fame" is a grammatically correct phrase that can be used in written English.
You can use it to refer to fame that has already been achieved and is now recognized and established. For example, "Throughout his career, the actor had established fame in the entertainment industry."
Despite the established fame of older brother Owen, the industry guy was in no doubt at all: "This kid could be anything".
The lyrics of "Tennis Court" address Lorde's newly established fame and criticise the "high life".
Johnson then introduces Shakespeare: The poet, of whose works I have undertaken the revision, may now begin to assume the dignity of an ancient, and claim the privilege of established fame and prescriptive veneration.
Several of Edmondson's Holby City co-stars have described feeling intimidated by the prospect of working alongside him for the first time, stemming from his already well established fame.
Ms. Yeung has established fame both as a Cantonese pop singer and as an actress.
Yet, as with his contemporary, Christopher Hill, who left the CP at this time, the political trauma of 1956 and the start of a lastingly happy second marriage combined in some way to trigger a sustained and fruitful period of historical writing that was to establish fame and reputation.
There Herrera established his fame and the Herreran style, which was to prevail in Spain for half a century.
In fact, ukiyo-e artists, such as Tōshūsai Sharaku, established their fame by portraying famous Kabuki actors as their subjects.
She established her fame with busts of the philosopher Arthur Schopenhauer, a personal friend, and of King George V of Hanover.
Rubens was commissioned to paint for the Antwerp Town Hall a celebratory Adoration of the Magi (1609), which quickly established his fame at home.
He established his fame as the designer of musha-e ("warrior prints") with his series of prints entitled Tsūzoku Suikoden gōketsu hyakuhachinin ("One Hundred and Eight Popular Warrior Heroes from Shui-hu ch'uan"), published in about 1827.
