Sharp bounds for the commutators with variable kernels of fractional differentiations and BMO Sobolev spaces. (April 2015) - Chen, Yanping; Ding, Yong
Abstract: For 0 < γ < 1 and b ∈ I γ ( BMO ), we introduce a new class of commutators with fractional differentiations and variable kernels, which is defined by [ b, T γ ] f ( x ) = ∫ R n Ω ( x, x − y ) | x − y | n + γ ( b ( x ) − b ( y ) ) f ( y ) d y . In this paper, we give the sharp L 2 norm inequalities for the rough operators [ b, T γ ] with Ω ( x, z ′ ) ∈ L ∞ ( R n ) × L q ( S n − 1 ) ( q > 2 ( n − 1 ) n ) satisfying the mean zero value condition in its second variable in the sense that the exponent q > 2 ( n − 1 ) / n is optimal. If strengthen the smoothness of Ω ( x, z ′ ) in its second variable, we prove weight norm inequalities for these operators. Our results recover a previous result of Murray and extend a previous result of Calderón.