Commutators of Singular Integral Operators Satisfying a Variant of a Lipschitz Condition

Abstract EN

Let T be a singular integral operator with its kernel satisfying | K ( x - y ) - ∑ k = 1 ℓ B k ( x ) ϕ k ( y ) | ≤ C | y | γ / | x - y | n + γ , | x | > 2 | y | > 0 , where B k and ϕ k ( k = 1 , … , ℓ ) are appropriate functions and γ and C are positive constants.

For b → = ( b 1 , … , b m ) with b j ∈ BMO ( ℝ n ) , the multilinear commutator T b → generated by T and b → is formally defined by T b → f ( x ) = ∫ ℝ n ∏ j = 1 m ( b j ( x ) - b j ( y ) ) K ( x , y ) f ( y ) d y .

In this paper, the weighted L p -boundedness and the weighted weak type L log L estimate for the multilinear commutator T b → are established.