What's the difference between these two sentences? How to translate them?
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I believe this is standard -ている (sometimes called progressive tense), i.e.
"Please be relaxed/take it easy" [and stay that way].
ていって has several possible interpretations:
"Please relax/become relaxed"(?)
Although i think -ていく is mostly used in descriptions (e.g.
空が明るくなっていく) and not much in requests so this one is unlikely IMO.
"Please take it easy [before you have to go]"
conjugation of ‐て言う, but it doesn't really work in this example IMO.
a few more options like 入って,云って,要って but these are even less plausible
For more info:
a. You could say this to a person who is trying to help you. ゆっくりしていてください is a polite way to say "Thank you but you don't have to do that. "
b. You say ゆっくりしていってください to a guest of your family etc. Let's say the guest is a colleague of your partner. You intervene their conversation perhaps to say hello or to serve some drinks. ゆっくりしていってください means something like "Have a good time. I won't bother you any more."