Let X= G/H be a homogeneous space, (X) over tilde = X x [0, infinity), mu a doubling measure on X induced by a Haar measure on the group G, beta a positive measure on (X) over tilde and W a weight on X. Consider the maximal operator given by Mf(x,r) = sup(s >= r)1/mu(B(x,s))integral(B(x,s)) vertical bar f(y)vertical bar d mu(y), (x,r) is an element of (X) over tilde. this paper, we obtain, for each p, q, 1 < p <= q < infinity, a necessary and sufficient condition for the boundedness of the maximal operator M from L-p(X, Wd mu) to L-q((X) over tilde, d beta). As an application, we obtain a necessary and sufficient condition for the boundedness of the Poisson integral of functions defined on the unit sphere S-n of the Euclidian space Rn+1, from L-p(S-n, Wd sigma) to L-q (B, d nu), where sigma is the Lebesgue measure on S-n, W is a weight on S-n and nu is a positive measure on the unit ball B of Rn+1.2716778