Gagliardo-Nirenberg Type Inequality And The Boundedness Of The Commutators Of Singular Integral Operators

LetL be the elliptic operator which generates an analytic semigroupe-zL with the kernel of az(x,y) satisfying a Gaussian upper bounds, that is, for v>0,x∈Rn, y∈Rn, and B(x,t1/2) is a ball with radius t1/2centered at x∈Rn. In this paper we prove a version of Gagliardo-Nirenberg type inequality associated to divergence elliptic operators, and the boundedness of the commutators of singular integral operators with non-smooth kernels.There are three chapters in this paper.In the first chapter, we briefly describe some research results of BMO spaces, new BMO spaces associated with differential operators, and singular integral operators, some assumptions used in this paper, and the structure of this paper. We also list our main results of the paper.In the second chapter, we prove a version of Gagliardo-Nirenberg type inequality as-sociated to divergence elliptic operators.In the third chapter, we prove the boundedness of the commutators of singular integral operators with non-smooth kernels.