Hilbert transform and singular integrals on the spaces of tempered ultradistributions

The Hilbert transform on the spaces ${\displaystyle S\prime \cdot \left(R^d\right)}$ of tempered ultradistributions is defined, uniquely in the sense of hyperfunctions, as the composition of the classical Hilbert transform with the operators of multiplying and dividing a function by a certain elliptic ultrapolynomial. We show that the Hilbert transform of tempered ultradistributions defined in this way preserves important properties of the classical Hilbert transform. We also give definitions and prove properties of singular integral operators with odd and even kernels on the spaces ${\displaystyle S\prime \cdot \left(R^d\right)}$, whose special cases are the Hilbert transform and Riesz operators.