Maximal inequalities and lebesgue's differentiation theorem for best approximant by constant over balls

For f is an element of (P)(R-n), with 1 less than or equal to p < infinity, epsilon > 0 and x is n elment of R-n we denote by T-epsilon(f)(x) the set of every best constant approximant to f in the ball B(x, epsilon). In this paper we extend the operators T-p(epsilon) to the space Lp-1(R-n) + L-infinity(R-n), where L-0 is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators T-P(epsilon) and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem. (C) 2001 Academic Press.