Maximal functions on the infinite rooted k-ary tree

 Abstract: The Hardy-Littlewood maximal function and its variants are classical objects. We start by recalling some classical results regarding the boundedness of the Hardy-Littlewood maximal function on weighted Lebesgue spaces in the Euclidean setting. In the second half,  we introduce the fractional Hardy-Littlewood maximal function on infinite rooted k-ary tree. Since the infinite rooted k-ary tree is very "non-homogeneous" in nature, we illustrate several difficulties in developing the weighted boundedness results on the infinite rooted k-ary tree and finally we finish with some examples of weights for which the fractional the Hardy-Littlewood maximal function satisfies strong type (p, q) estimates.