Sharp Weighted Estimates for Multilinear Commutators

Multilinear commutators with vector symbol b → =( b 1 ,…, b m ) defined byT b →( f ) ( x ) = ∫R n[∏ j = 1 m( b j ( x ) − b j ( y ) )] K ( x , y ) f ( y ) d yare considered, where K is a Calderón–Zygmund kernel. The following a priori estimates are proved for w ∈ A ∞ . For 0 < p < ∞, there exists a constant C such that‖T ˙b →( f )‖ L p ( w ) ⩽ C ‖b →‖ ‖M L( log L )1 / r( f )‖ L p ( w )andsup t > 01 Φ ( 1 t )w ( { y ∈ R n : | T b →f ( y ) | > t } ) ⩽ C sup t > 01 Φ ( 1 t )w ( { y ∈ R n : M L( log L )1 / r( ‖b →‖ f ) ( y ) > t } ) ,where‖ b → ‖ = ∏ j = 1 m‖ b j ‖o s c exp L r j,Φ ( t ) = t log 1 / r ( e + t ) , 1 r = 1 r 1+ ċ + 1 r m,and M L(log L ) α is an Orlicz type maximal operator. This extends, with a different approach, classical results by Coifman. As a corollary, it is deduced that the operatorsT b →are bounded on L p ( w ) when w ∈ A p , and that they satisfy corresponding weighted L (log L ) 1/r ‐type estimates with w ∈ A 1 .