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Jul 9 at 10:24 comment added tommy1996q @Hannes unwissen thanks. Especially Ziemer's book goes in great detail, in a very general setting. Actually, I edited the question since I got carried away asking a question way more general than what I really need. Hope that in the simplified case presented there is a more explicit answer. It might be possible to recover it by going through Ziemer's book, but I am not 100% sure
Jul 9 at 10:21 history edited tommy1996q CC BY-SA 4.0
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Jul 8 at 11:42 comment added Hannes It also follows from Theorem 4.2.1 in Ziemer's Weakly differentiable functions which has a whole (and very nice) chapter dedicated to Poincare-type inequalities. (Maybe it is the same theorem as in Lieb-Loss, which I do not have access to right now, if so, then disregard this comment.)
Jul 7 at 21:59 comment added unwissen Yes, this is e.g. included in Theorem 8.11 of Lieb-Loss Analysis.
Jul 7 at 21:17 history asked tommy1996q CC BY-SA 4.0