### On Herz's extension theorem

Antoine Derighetti

#### Abstract

‎We present a self-contained proof of the following famous extension theorem due to Carl Herz‎. ‎A closed subgroup $H$ of a locally compact group $G$ is a set of $p$-synthesis in $G$ if and only if‎, ‎for every $u\in A_p(H)\cap C_{00}(H)$ and for every $\varepsilon > 0$‎, ‎there is $v\in A_p(G)\cap C_{00}(G)$, an extension of $u$, such that $\|v\|_{A_p(G)} < \|u\|_{A_p(H)}+\varepsilon.$

#### Article information

Source
Adv. Oper. Theory, Volume 4, Number 2 (2019), 529-538.

Dates
Accepted: 16 November 2018
First available in Project Euclid: 1 December 2018

https://projecteuclid.org/euclid.aot/1543633242

Digital Object Identifier
doi:10.15352/aot.1809-1417

Mathematical Reviews number (MathSciNet)
MR3883151

Zentralblatt MATH identifier
1062.43005

#### Citation

Derighetti, Antoine. On Herz's extension theorem. Adv. Oper. Theory 4 (2019), no. 2, 529--538. doi:10.15352/aot.1809-1417. https://projecteuclid.org/euclid.aot/1543633242

#### References

• J. Delaporte and A. Derighetti, On Herz' extension theorem, Boll. Un. Mat. Ital. A (7) 6 (1992), no. 2, 245–247.
• A. Derighetti, Convolution operators on groups, Lecture Notes of the Unione Matematica Italiana, 11. Springer, Heidelberg; UMI, Bologna, 2011.
• C. Herz, Harmonic synthesis for subgroups, Ann. Inst. Fourier (Grenoble) 23 (1973), no. 3, 91–123.
• H. Reiter, Classical harmonic analysis and locally compact groups, Clarendon Press, Oxford, 1968.