Necessary and Sufficient Conditions for the Bounds of the Commutators with Fractional Differentiations and BMO-Sobolev Spaces on Weighted Lebesgue Space.

Let b ∈ L loc 1 (R n) and α ∈ (0 , 1). In the present paper, we are concerned on commutators with fractional differentiations which are defined by [ b , T α ] f (x) = p. v. ∫ R n Ω (x - y) | x - y | n + α (b (x) - b (y)) f (y) d y. <graphic href="12220_2021_852_Article_Equ63.gif"></graphic> This kind of commutators [ b , T α ] appear in the parabolic boundary value problems. In this paper, we obtain the necessary and sufficient conditions on the function b to guarantee that [ b , T α ] is a bounded operator on L p (w) for 1 < p < ∞ and w ∈ A p. For the sufficiency, we study the quantitative weighted bounds for the commutator with fractional differentiation with rough kernels. Namely, for b ∈ I α (B M O) and 1 ≤ r ′ < p < ∞ , ‖ [ b , T α ] f ‖ L p (w) ≤ C ‖ Ω ‖ L r ‖ D α b ‖ BMO [ w ] A p / r ′ max 1 , 1 p - r ′ [ w ] A p / r ′ 1 r ′ max 1 , r ′ p - r ′ ‖ f ‖ L p (w) , <graphic href="12220_2021_852_Article_Equ64.gif"></graphic> it is the best known quantitative result of this operator. For the necessity, if [ b , T α ] is bounded on L p (w) for some 1 < p < ∞ and w ∈ A p , then b ∈ L i p α. Finally, we apply our general theory to the Hilbert and Riesz transforms, and obtain a characterization of [ b , D α R j ] and [ b , D α H ] on L p (w) for 1 < p < ∞ and w ∈ A p by b ∈ I α (B M O) . [ABSTRACT FROM AUTHOR]