Symbolic Calculus and Boundedness of Multi-parameter and Multi-linear Pseudo-differential Operators

Abstract

Abstract Since the work of Hörmander on linear pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, harmonic analysis, theory of several complex variables and other branches of modern analysis (e.g., they are used to construct parametrices and establish the regularity of solutions to PDEs such as the ∂̅ problem, etc.). The work of Coifman and Meyer on multi-linear Fourier multipliers and pseudo-differential operators has stimulated further such applications. In [2], the authors developed a fairly satisfactory theory of symbolic calculus for multi-linear pseudo-differential operators. Motivated by this work [2] and Lp estimates of [34, 35] on multi-parameter and multi-linear Fourier multipliers and of [12] on multi-parameter and multi-linear pseudo-differential operators, we study and carry out the theory of symbolic calculus for multi-parameter and multi-linear pseudo-differential operators. Our results include the symbol estimates of the adjoints, asymptotic behavior, kernel estimates and boundedness properties and extend those in [2] to the multi-parameter and multi-linear setting. The estimates of the distributional kernel of associated multi-parameter and multilinear pseudo-differential operators can be found useful in establishing the boundedness of such multi-parameter and multi-linear pseudo-differential operators.