2023-1 Math Colloquium | Hörmander type theorems for multi-linear and pseudo-differential operators

In this talk we establish a H¨ormander type theorem for multilinear pseudodifferential operators, which is also a generalization of the results to symbols depending on the spatial variable. Most known results for multilinearpseudo-differential operators were obtained by assuming their symbols satisfy Mihlin-type pointwise derivative condition. However, we shall consider multilinear pseudo-differential operators whose symbols have only the firstorder derivative conditions in the spatial variable and lower-order derivative conditions in the frequency variable. We also make use of L2-average condition rather than pointwise derivative conditions for the symbols. Moreover, it can be pointed out that our results are applied to wider classes of symbols which do not belong to the traditional classes of symbols. In part I we present a historical overview of the Mihlin and H¨ormander multiplier theorems for multilinear operators and the classical Mihlin-type condition for multilinear pseudo-differential operators. In part II we prove the H¨ormandertype theorems for multilinear pseudo-differential operators. The second talk is a joint work with Yaryong Heo and Chan Woo Yang.