Sharp A₁ bounds for calderón-zygmund operators and the relationship with a problem of muckenhoupt and wheeden

For any Calderón-Zygmund operator T the following sharp estimate is obtained for 1 p <∞ ||T|| Lp(ω)≤CVp||ω|| A₁, where v_p = p²/p-1log(e+1/p-1). In the case where p=2 and T is a classical convolution singular integral, this result is due to R. Fefferman and J. Fipher [7]. Then, we deduce the following weak type (1,1) estimate related to a problem of Muckenhoupt and Wheeden [11]: sup λω{x ∈ⁿ : |Tf(x)>λ} ≤cφ(||ω ||A₁)&Rⁿ|f|ω dx, λ>0 where w ∈ A¹ andψ(t)=t(1+log⁺t(1+log⁺log⁺ t).