We establish the following sharp local estimate for the family of Riesz transforms on . For any Borel subset A of and any function ,
, 1 < p < ∞.
Here q = p/(p-1) is the harmonic conjugate to p,
, 1 < p < 2,
and
, 2 ≤ p < ∞.
This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.