Weighted norm inequalities for geometric fractional maximal operators

For 0 less than or equal to alpha < infinity let T-alpha f denote one of th e operators M(alpha,0)f(x) = sup(I is an element of x) \I\(alpha) exp (1/\I\ integral(I ) log\f\), M-alpha,M-o* f(x) = lim(r SE arrow 0) sup(I is an element of x) \I\(alpha) (1/\I\ integral(I) \f\(r))(1/r). We characterize the pairs of weights (u, v) for which T-alpha is a bounded operator from L-p(v) to L-q(u), 0 < p less than or equal to q less than or equal to infinity. This extends to alpha > 0 the norm inequalities for alph a = 0 in [4, 16]. As an application we give lower bounds for convolutions p hi star f, where phi is a radially decreasing function.