Commutators of local bilinear maximal operator with Sobolev symbols

Abstract

In the present paper, we introduce the commutators of local bilinear maximal operator, local bilinear maximal commutators and their fractional variants. Under the conditions that the symbol function belongs to the first-order Sobolev spaces, the bounds for the above commutators on the Sobolev spaces are established. These results are based on several new pointwise estimates for the weak gradients of the above commutators.