BORDERLINE WEIGHTED ESTIMATES FOR COMMUTATORS OF SINGULAR INTEGRALS

In this paper we establish the following estimate: w({x is an element of R-n : vertical bar b,T]f (x) vertical bar > lambda}) <= c(T)/epsilon(2) integral(n)(R) Phi(parallel to b parallel to(BMO) vertical bar f(x)vertical bar/lambda) M-L(log L)(1+epsilon w(x)dx) where w >= 0, 0 < epsilon < 1 and Phi(t) = t(1 + log(+)(t)). This inequality relies upon the following sharp L-p estimate: parallel to b, T]f parallel to(p)(L)(w) <= c(T) (p')(2) p(2) (p - 1/delta)(1/p)' parallel to b parallel to BMO parallel to f parallel to L-p(M-L(log L)(2p - 1 + delta w)) where 1 < p < infinity, w >= 0 and 0 < delta <1. As a consequencewe recover the following estimate essentially contained in [18]: w({x is an element of R-n : vertical bar b,T]f (x) vertical bar > lambda}) <= c(T)[w]A(infinity) (1 + log(+)[w]A(infinity))(2) integral R-n Phi(parallel to b parallel to(BMO) vertical bar f(x)vertical bar/lambda) M w(x)dx. We also obtain the analogue estimates for symbol-multilinear commutators for a wider class of symbols.