What does “big league” in President Trump’s remarks mean? Is it common to use “big league” in English as an adjective or adverb?

Solution 1:

In a comment, Hot Licks wrote:

It is, of course, a Trumpism to a large degree. "Big league", in the US, refers primarily to "major league" baseball teams and games, or at least to adult baseball, as opposed to "Little League" children's baseball. The idiom has been detached from that meaning over the years, though, and now might be used to refer, eg, to a company which has grown to the point that it can compete with others in a national market. Trump abuses it to a degree to simply mean "big", or "bigly" (sic), as you suggest.

Solution 2:

The following is from the Merriam-Webster (https://www.merriam-webster.com/news-trend-watch/did-trump-say-bigly-or-big-league-20160927):

"Did Trump Say 'Bigly' or 'Big League'?

Both are real words, though 'big league' is rarely used as an adverb

Lookups for big and bigly spiked on September 26th and 27th. Many people heard bigly when Trump said big league in the debate:

I'm going to cut taxes big league and you're going to raise taxes big league. End of story.

We've found examples of 'big league' as a figurative adjective from 1916. The adverbial use of big league favored by Trump is not often encountered, however.

Trump often uses the phrase big league in a figurative sense; the manner in which he pronounces these words have caused many people to assume that he was using a single word (bigly), rather than two.

Big league has been in use since 1882 as a term for the major league of professional baseball (particularly in the United States, although it has also been used to refer to the upper echelons of other sports). The term began to see figurative use in the early 20th century.

Our files indicate the big league has been used as a figurative adjective since 1916:

The candidate for governor spoke next, Mr. DesChamps being the first of the “big league” candidates introduced. —The Greenville News (Greenville, SC), 23 Jun., 1916

And the phrase has been in use as a noun in a figurative sense since at least 1930:

“Oh, bottle that in your goodlookin’ mush. You miss those penny-ante punks you thought were big shots in Chi. You’re in the big league now, red head. What they took down is tips to waiters here.” —The Morning News (Wilmington, DE), 14 Nov., 1930

The adverbial use of big league favored by Trump is not often encountered".