a modification of hardy-littlewood maximal-function on lie groups

For a real-valued function f on a metric measure space (x, d, µ) the hardy-littlewood centered-ball maximal-function of f is given by the ‘supremum norm’: mf(x) := sup r>0 1/µ(bx,r) ꭍ bx,r |f|dµ. in this note, we replace the supremum-norm on parameters r by lp-norm with weight w on parameters r and define hardy-littlewood integral-function ip,wf. it is shown that ip,wf converges pointwise to mf as p → ∞. boundedness of the sublinear operator ip,w and continuity of the function ip,wf in case that x is a lie group, d is a left-invariant metric, and µ is a left haar-measure (resp. right haar-measure) are studied.