Large deviations for martingales and derivatives

Fix a sequence of positive integers ( m n ) and a sequence of positive real numbers ( w n ) . Two closely related sequences of linear operators ( T n ) are considered. One sequence has T n : L 1 ( R ) → L 1 ( R ) given by the Lebesgue derivatives T n f ( x ) = D n f ( x ) = 2 n ∫ 0 1 / 2 n f ( x + t ) d t . The other sequence has T n : L 1 [ 0 , 1 ) → L 1 [ 0 , 1 ) given by the dyadic martingale T n f ( x ) = E ( f | β n ) ( x ) = 2 n ∫ ( l − 1 ) / 2 n l / 2 n f ( t ) d t when ( l − 1 ) / 2 n ⩽ x < l / 2 n for l = 1 , … , 2 n . We prove both positive and negative results concerning the convergence of ∑ n = 1 ∞ m { | T m n f ( x ) | ⩾ w n } .