Bilinear Pseudo-Differential Operator and Its Commutator on Generalized Fractional Weighted Morrey Spaces

The aim of this paper is to establish the boundedness of bilinear pseudodifferential operator T-sigma and its commutator [b(1), b(2), T-sigma] generated by T-sigma and b(1), b(2) is an element of BMO(R-n) on generalized fractional weighted Morrey spaces L-p,L-eta,L-phi(omega). Under assumption that a weight satisfies a certain condition, the authors prove that T-sigma is bounded from products of spaces L-1(p),eta(1),phi (omega(1)) x L-2(p),eta(2),phi (omega(2)) into spaces L-p,L-eta,L-phi (omega(1)), where (omega) over right arrow =(omega(1), omega(2)) is an element of A((P) over right arrow), (P) over right arrow=(p(1), p(2)), eta= eta(1) + eta(2) and 1/p = 1/p(1) + 1/p(2) with p(1), p(2) is an element of (1, infinity). Furthermore, the authors show that the [b(1), b(2), T-sigma] is bounded from products of generalized fractional Morrey spaces L-1(p),eta(1), phi(R-n) x L-2(p),eta(2), phi(R-n) into L-p,L-eta,L-phi(R-n). As corollaries, the boundedness of the T-sigma and [b(1), b(2), T-sigma] on generalized weighted Morrey spaces L-p,L-phi(omega) and on generalized Morrey spaces L-p,L-phi(R-n) is also obtained.