Sharp inequalities for Riesz transforms
We establish the following sharp local estimate for the family {Rj}j=1 of Riesz transforms on Rd. For any Borel subset A of Rd and any function f : Rd → R, ∫ A |Rjf(x)|dx ≤ Cp||f ||Lp(Rd)|A| 1/q , 1 < p <∞. Here q = p/(p− 1) is the harmonic conjugate to p, Cp = [ 2q+2Γ(q + 1) πq+1 ∞ ∑ k=0 (−1)k (2k + 1)q+1 ]1/q , 1 < p < 2, and Cp = [ 2q+2Γ(q + 1) πq ∞ ∑ k=0 1 (2k + 1)q ]1/q , 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.