Local Lower Norm Estimates for Dyadic Maximal Operators and Related Bellman Functions

We provide lower (L^{q}) and weak (L^{q})-bounds for the localized dyadic maximal operator on (mathbb {R}^{n}), when the local (L^{1}) and the local (L^{p}) norms of the function are given. We actually do that in the more general context of homogeneous trees in probability spaces.