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Bounds on the Phillips Calculus of Abstract First Order Differential Operators

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Abstract

For an operator generating a group on \(L^p\) spaces transference results give bounds on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction of first order differential operators and prove similar spectral multiplier estimates assuming only that the group is bounded on \(L^2\) rather than \(L^p\). We also prove an R-bounded Hörmander calculus result by assuming an abstract Sobolev embedding property and show that the square of a perturbed Hodge–Dirac operator has such calculus.

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Acknowledgements

I am extremely thankful to my supervisor Pierre Portal for his continuous support and encouragement. It was his perseverance and foresight which made me kept working on this problem even after facing so many ups and downs. I am also very grateful to Dorothee Frey for her valuable suggestion to Pierre to look at the paper of Peer Kunstmann. I would like to express my heartfelt thanks to Christoph Kriegler as well for suggesting changes in our Theorem 3.2 and improving the result.

Funding

This research is supported by the Australian Government Research Training Program (AGRTP) scholarship.

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  1. Mathematical Sciences Institute, Australian National University, Canberra, ACT, 2600, Australia

    Himani Sharma

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  1. Himani Sharma
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Correspondence to Himani Sharma.

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Sharma, H. Bounds on the Phillips Calculus of Abstract First Order Differential Operators. Results Math 76, 187 (2021). https://doi.org/10.1007/s00025-021-01496-1

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