Dear @Azita, the following link discusses "Analogue of Lebesgue differentiation theorem in Orlicz spaces" and I hope that it leads you to a way of giving the analogues to the Lebesgue Differentiation Theorem for locally compact abelian groups, see the page:

http://math.stackexchange.com/questions/663314/analogue-of-lebesgue-differentiation-theorem-in-orlicz-spaces

The preprint

http://pages.uoregon.edu/mbownik/preprints/mb-kar.pdf

cites a version of the Lebesgue differentiation theorem for locally compact abelian groups. The crucial condition is the existence of a certain decreasing sequence of neighbourhoods of 0.    (I'm no expert in harmonic analysis, it's more of a random hint.)

Yours, S.

Dear Anita,

I'm not an expert on the topic, but I think I know an expert. I recommend to contact Karl Heinrich Hofmann from Darmstadt / Tulane, hofmann [at] mathematik.tu-darmstadt.de. He's already retired, but still very active. He has a perfect overview over the literature.

Dear Abedallah, Ilka, and Stefan, thanks so much for your responses. Ilka: I will contact Karl. Thanks for recommending me him. 

Dear Azita,

I suggest you to have a look at the paper ``The structure of translation invariant spaces on LAC groups" by M. Bownik and K. Ross, Section 7 and the reference therein. I hope it helps you.

Thanks Victoria. 

Hi. Most likely this is a dead thread and you already know the answer. as long as we have a doubling metric measure spaces (balls having non zero finite measure) lebeague diff. Theorem holds. In particular if your left(right) Haar measure is doubling LDT holds. I saw this somewhere that under n-microdoubling condition it also holds(have a look at Tao & Naor paper on Random martingales). Side note: I came across this post as I was trying to study certain operators on some classical LC groups(no abelian in my case). If you are interested, I can share more. Regards