Is there a word to define the verbal strategy that requests action of someone without using a direct command—e.g., Could you please open the door?
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Steven Pinker calls it off-record indirect speech acts...
Note that Pinker's definition of indirect speech as quoted there isn't the same as the more well-known reported speech one. But if you Google "indirect speech act" it's obvious in that particular collocation Pinker's meaning is invariably what's meant.
The term cajole is defined as
It is also defined as
Although the AHDEL doesn't license the word request in this usage
re·quest (r-kwst) tr.v. re·quest·ed, re·quest·ing, re·quests
... is as near as it gets ...
request [rɪˈkwɛst] vb (tr) to express a desire for, esp politely; ask for or demand [bolding mine]
Certainly, the expression
Could you please open the door?
contains two hedging devices, and
Could you please just open the door?