This here car doesn't belong to them.
Is the NP This here car a grammatical one? And if so, is "here" an adverb modifying the head of that phrase?
"This here noun " is non-standard, but it is a normal colloquial feature of some dialects. I just now checked the onlne searchable Shakespeare to make sure he didn't use it, and he didn't. I don't know what to call "here" in it.
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"This here noun" is non-standard, but it is a normal colloquial feature of some dialects. I just now checked the onlne searchable Shakespeare to make sure he didn't use it, and he didn't. I don't know what to call "here" in it. It quacks like an adverb, but I can't find its partner.
tkacka15Is the NP This here car a grammatical one?
Not in my variety of English. I have heard people use it, however — usually people in rural areas living near the poverty line. I don't know how that correlation developed historically, but there you have it.
tkacka15is "here" an adverb modifying the head of that phrase?
tkacka15This here car doesn't belong to them.
This use of "This here ..." is a moderately common feature of lower-register colloquial British English or dialect English. I probably say it myself sometimes. It is not considered "proper" English, at least not in modern times. I would imagine that "This here X" is a reordering of "This X here".