Hardy, Paley–Wiener and Bernstein spaces in Clifford analysis

Abstract

We describe the relationship between the growth conditions of monogenic extensions of square-integrable functions f in terms of pointwise bounds or bounds on integral averages on the one hand, and the support of the Fourier transform or its annihilation by certain higher-dimensional analogues of the signum function on the other. We review known results involving a function’s monogenic extension and their classical Fourier transform. These results are extended to the Clifford-Fourier transform of Brackx, De Schepper and Sommen. The equivalence of the pointwise bounds and the bounds on the integral averages is observed as a consequence.

Keywords:

• Paley–Wiener theorem
• Hardy spaces
• Clifford algebra
• Clifford Fourier transform

AMS Subject Classifications:

• 42A38
• 30G35
• 15A66

Acknowledgements

The authors are pleased to acknowledge the impact John Ryan has had on their lifes and work. It was while working at the University of Arkansas at the start of the century that Hogan was introduced to Clifford analysis by John as well as to his many deep and original contributions.

Notes

No potential conflict of interest was reported by the authors.

We are delighted to be able to dedicate this paper to him. Happy birthday, John.

Additional information

Funding

Franklin is supported by Australian Postgraduate Award scheme. Hogan is supported by the Australian Research Council [Discovery grant number DP160101537] and through the University of Newcastle’s Centre for Computer Assisted Research in Mathematics and its Applications. This work was completed while Hogan visited the Hausdorff Research Institute for Mathematics in Bonn.