Maximal functions for locally compact groups have been studied. In particular papers of K. Phillips and Phillips-Taibleson (here and here) contain nice results in this direction. It might be too much to ask though and one can consider different types of maximal functions.

Maximal function for locally compact groups in the sense of Phillips. Let $G$ be a locally compact group with left Haar measure $\lambda$. Consider $\{B_n\}_{n\in\mathbb Z}$ a neighborhood base identity $e$ consisting of relatively compact Borel sets satisfying the following

(i)$B_{n+1}\subsetneqq B_n,\,\forall n\in\mathbb Z\text{ and }\displaystyle \lim_{n\to{-\infty}}\lambda(B_n)=\infty;$
(ii)$\lambda(B_nB_n^{-1})<C\lambda(B_n),\,\,C:\text{ constant for all } n\in\mathbb Z$ (this is the so-called Tempelman's condition);

(iii) For each $n\in\mathbb Z$, there exists a $\ell(n)\in\mathbb Z$ such that $B_n^{-1}B_n\subset B_{\ell(n)}$ and $B_n^{-1}B_n\not\subset B_j$ if $j>\ell(n)$. And there exists a constant $\alpha$ such that $\lambda(B_{\ell(n)})<\alpha\lambda(B_n)\,\,\forall n\in\mathbb Z.$ For locally integrable function $f$ define

$$M_nf(x)=\frac{1}{\lambda(B_n)}\int_{x B_n}f\lambda\,\,\,\text{ and } Mf(x)=\sup_{n\in\mathbb Z}M_nf(x).$$ The operator $M$ is what Phillips refers to as maximal function.

Question 1. I was wondering if there are known (sharp) best constants for particular L.C. groups e.g. "ax+b" group, discrete groups,..?

Taking the ambient group to be the additive group of integers, the discrete maximal (centered)function is defined by taken supremum over literally "discrete" averages. More precisely

For $f:\mathbb Z\to \mathbb R$ and $d\lambda$: counting measure on $\mathbb Z$, the discrete maximal is defined by taking the supremum over (literally discrete)averages:
$$Mf(n)=\sup_{r>0}\frac{1}{\lambda\Big(B_r(n)\Big)}\int_{B_r(n)}|f| d\lambda=\sup_{r\in\mathbb N}\frac{1}{2r-1}\sum_{|m|<r}|f(n+m)|,$$ where $B_r(n)$ is the open ball centered at $n$ of radius $r$ and $\lambda\Big(B_r(n)\Big)$ is the number of integers inside this $B_r(n).$ If I remember correctly, the weak $(1-1)$ and strong $(p-p) $p>1$ are due to J. Bourgain (might be Stein, please correct me if I am wrong)

Question 2. Is the best constant known for the weak $(1,1)$ or for strong $(p,p),p>1$ known? J. Bober et al have a nice relevant result but they study variations and not the maximal function.

Remark 1. The best constants depend on the choice of nested neighborhoods (which are denoted in here by $\{B_n\}$). But still it would be really interesting to find the best constant for a sequence of such neighborhoods.

Remark 2. For the case $G=(\mathbb R,+)$ the best constant in known and is a result of Males.link For discrete case(finite graphs) it is also known. I am not aware if there is any generalization of Males' to $\mathbb R ^n$ with $n\geq 2$.

Question 1'. Let's look at a particular example. Consider $X=(0,\infty)$ endowed with the measure $d\mu=\frac{1}{x}dx$ and metric $d(x,y)=|\log x-\log y|$. Its easy to check that $(X,\mu,d)$ is a doubling metric measure space, $B(x,r)=(xe^{-r},xe^r)$ and that $\mu(B(x,r))=2r$ for all $x,r>0$. Now consider the maximal function

$$Mf(x)=\sup_{r>0}\frac{1}{2r}\int_{xe^{-r}}^{xe^r}f(t)\frac{dt}{t},$$ where $f\in L^1_{\text {loc}}(X,\mu)$. I was wondering there are bounds for the best constant for this case? Of course, the general maximal inequality for doubling measure spaces can be used, but it might give way too generous constant.

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    $\begingroup$ Haven't found a definition of maximal function in any of the first two links. The third more recent paper uses "maximal operator", which is possibly more adequate, but writing or at least pointing to a definition would be useful here. $\endgroup$ – YCor Dec 28 '16 at 18:08
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    $\begingroup$ Have you tried to look in Elon Lindenstrauss' thesis? He proves there a general maximal ergodic theorem for amenable group. I think that the same constant should hold in your case also (although you're considering differentiation theorem and not an ergodic theorem). Regarding the ball averages you've mentioned, they are classical (probably Hardy-Littlewood or Vitaly/Besicovitch), the inequalities by Stein and Bourgain deal with spherical maximal functions (and the associated ergodic theorems proven by Jones and Lacey). $\endgroup$ – Asaf Jan 1 '17 at 22:52
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    $\begingroup$ The paper is called "Pointwise Theorems for Amenable Groups, Invent. Math. 146 (2001), no. 2, 259--295", it is linked in his website here - ma.huji.ac.il/~elon/Publications/pointnew.ps $\endgroup$ – Asaf Jan 2 '17 at 17:03
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    $\begingroup$ @Asaf Thanks for the link. I am looking at the paper and seems he actually gives a general maximal inequality with a constant that depends only on the choice of certain neighborhoods. Still finding explicit best constant for particular LCA groups sounds interesting and is not what he was after in his neat paper. $\endgroup$ – BigM Jan 3 '17 at 14:10
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    $\begingroup$ Isn't this an open problem even for the centred maximal function in $\mathbb{R}^n$? I recently heard a talk by Adam Osękowski about optimal constants for the dyadic maximal operator, and I remember him saying that except for dimension one, essentially nothing is known about optimal constants for centred maximal operators on $\mathbb{R}^n$. His paper does not mention the non-dyadic case, however, so you may like to email him. $\endgroup$ – Mateusz Kwaśnicki Aug 9 '18 at 8:32

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