We provide a deep investigation of the notions of - and ${}^{\infty }$-hypoellipticity for partial differential operators with constant Colombeau coefficients. This involves generalized polynomials and fundamental solutions in the dual of a Colombeau algebra. Sufficient conditions and necessary conditions for - and ${}^{\infty }$-hypoellipticity are given

Let $\tilde{f}$, $\tilde{g}$ be ultradistributions in ${𝒵}^{\text{'}}$ and let ${\tilde{f}}_n=\tilde{f}*{\delta }_n$ and ${\tilde{g}}_n=\tilde{g}*{\sigma }_n$ where $\left\{{\delta }_n\right\}$ is a sequence in $𝒵$ which converges to the Dirac-delta function $\delta$. Then the neutrix product $\tilde{f}\diamond \tilde{g}$ is defined on the space of ultradistributions ${𝒵}^{\text{'}}$ as the neutrix limit of the sequence $\left\{\frac{1}{2}({\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n)\right\}$ provided the limit $\tilde{h}$ exist in the sense that $$\underset{n\to \infty }{\mathrm{N}\text{-}lim}\frac{1}{2}\langle {\tilde{f}}_n\tilde{g}+\tilde{f}{\tilde{g}}_n,\psi \rangle =\langle \tilde{h},\psi \rangle$$ for all $\psi$ in $𝒵$. We also prove that the neutrix convolution product $f♦*g$ exist in ${𝒟}^{\text{'}}$, if and only if the neutrix product $\tilde{f}\diamond \tilde{g}$ exist in ${𝒵}^{\text{'}}$ and the exchange formula $$F\left(f♦*g\right)=\tilde{f}\diamond \tilde{g}$$ is then satisfied.

Products $\left[S\right]·\left[T\right]$ and $\left[S\right]·T$, defined by model delta-nets, are equivalent.

We investigate the Hankel transformation and the Hankel convolution on new spaces of generalized functions.

We show that Whitney?s approximation theorem holds in a general setting including spaces of (ultra)differentiable functions and ultradistributions. This is used to obtain real analytic modifications for differentiable functions including optimal estimates. Finally, a surjectivity criterion for continuous linear operators between Fréchet sheaves is deduced, which can be applied to the boundary value problem for holomorphic functions and to convolution operators in spaces of ultradifferentiable functions...