Tame Algebra Estimates and Inverses of Product Kernels and Flag Kernels

Abstract

In this thesis, we present two results centered around two algebras of multi-parameter kernels: product kernels and flag kernels under non-commutative group convolution on a direct product of graded Lie groups G₁ × · · · × G_ν. First, we show that product kernels and flag kernels satisfy tame algebra estimates. Second, we obtain an inversion theorem with two distinct proofs. The first proof relies on tools from partial differential equations with the construction of a key a priori smoothing estimate. The second proof relies on tools from the theory of Banach algebras. The key idea here is the application of the tame algebra estimates established earlier in the thesis.