We study the weighted norm inequalities for the minimal operator, a new operator analogous to the Hardy-Littlewood maximal operator which arose in the study of reverse Hölder inequalities. We characterize the classes of weights which govern the strong and weak-type norm inequalities for the minimal operator in the two weight case, and show that these classes are the same. We also show that a generalization of the minimal operator can be used to obtain information about the differentiability of the integral in cases when the associated maximal operator is large, and we give a new condition for this maximal operator to be weak (1, 1).

For 0 < a < oo let Ta f denote one of the operatorsWe characterize the pairs of weights (u, v) [4,16]. As an application we give lower bounds for convolutions d~ * f, where dp is a radially decreasing function. for which Ta is a bounded operator from LP (v) to Lq (u), 0 < p < q < cr This extends to a > 0 the norm inequalities for tr = 0 in

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