Maximal Inequalities and Lebesgue's Differentiation Theorem for Best Approximant by Constant over Balls

For f∈Lp(Rn), with 1⩽p0 and x∈Rn we denote by Tε(f)(x) the set of every best constant approximant to f in the ball B(x, ε). In this paper we extend the operators Tεp to the space Lp−1(Rn)+L∞(Rn), where L0 is the set of every measurable functions finite almost everywhere. Moreover we consider the maximal operators associated to the operators Tεp and we prove maximal inequalities for them. As a consequence of these inequalities we obtain a generalization of Lebesgue's Differentiation Theorem.