Yum-Tong Siu, [IBS-KAIST Seminar] Differential Relations for Multiplier Ideal Sheaves in ∂ Estimates

For sums of squares of real vector fields, Hörmander linked subelliptic estimates to the spanning property of iterated Lie brackets of vector fields. Kohn studied the more complicated analogue of subelliptic ​∂​​​​ estimates for weakly pseudoconvex domains, with vector-valued unknowns.

In the weak-solution approach to solving the ∂​​​ equation, multipliers for the test function are introduced so that estimates hold after multiplication by a multiplier. Kohn used the dual formulation of differential forms instead of vector fields so that (i) the Lie brackets of vector fields are replaced by differential relations to generate new multipliers and (ii) the spanning property of iterated Lie brackets is replaced by the constant function 1 being a multiplier.

We will focus on recent problems and results concerning Kohn’s theory of subelliptic estimates in terms of D’Angelo’s condition of finite type. The distant eventual goal is to explore the theory for differential relations to generate multipliers for subelliptic estimates for a general system of equations with compatibility conditions.