Estimates for Bilinear Pseudo-Differential Operators and Their Commutators on Product of Homogeneous Spaces

In this paper, the authors prove that bilinear pseudo-differential operatorT sigma is bounded from the product of homogeneous Herz spaces K-q1(alpha 1,p1)(Rn)x K-q2(alpha 2,p2)(R-n) to homogeneous Herz space K-q(alpha,p)(R-n), and bounded from the product of homogenousMorrey-Herz spacesM K-p1,q1(alpha 1,lambda 1)(R-n)xM K-p2,q2(alpha 2,lambda 2)(R-n) to homogeneous Morrey-Herz spaceM K-p,q(alpha,lambda)(R-n). Furthermore, via establishing the sharp maximal estimate for the com-mutator [b(1), b(2), T-sigma] which is generated byT sigma and b(1), b(2)is an element of BMO (R-n), the boundednessof [b(1), b(2), T-sigma] on spaces K-q(alpha,p)(R-n) and on spaces M K-p,q(alpha,lambda)(R-n) is also obtained.