##
**Rajchman measures on compact groups.**
*(English)*
Zbl 0673.43005

On a compact group Rajchman measures are Radon measures whose Fourier- Stieltjes transforms have norms \(>\epsilon\) only for finitely many elements of the dual object. The norms on the dual object considered are \(d_{\sigma}^{- 1/p}\| {\hat \mu}\|_{\phi_ p}\), \(1\leq p\leq \infty\). \(\| \cdot \|_{\phi_ p}\) denotes the von Neumann norm. There are essentially only two different classes of Rajchman measures \({\mathcal R}_ p\), one corresponding to the norms for \(p<\infty\) and one to the operator norm \(\| \cdot \|_{\phi_{\infty}}\). It is shown that Rajchman measures are intermediate between continuous and absolutely continuous measures and \(\nu \ll \mu \in {\mathcal R}_ p\Rightarrow \nu \in {\mathcal R}_ p\), which extends theorems of C. F. Dunkl and D. E. Ramirez [Mich. Math. J. 17, 311-319 (1970; Zbl 0188.206), ibid. 19, 65-69 (1972; Zbl 0213.136)], for \({\mathcal R}_{\infty}\). The main theorem characterizes \({\mathcal R}_ p\) by vanishing on a certain class of Borel sets and answers a question of R. Lyons [Ann. Math., II. Ser. 122, 155-170 (1985; Zbl 0583.43006)] how his result on Rajchman measures on l.c. Abelian groups extends to l.c. non Abelian groups for compact groups. A similar characterization holds for Rajchman measures on homogeneous spaces. (Author)

A. Rajchman conjectured (1922) that there exists some class of sets in T (denoted by W) which could be used to characterize those Radon measures \(\mu\) on T of which the Fourier transform \({\hat \mu}\) vanishes at infinity (denote such class by R) in such a way: \(\mu\in R\) iff \(\mu (E)=0\), \(\forall E\in W\). R. Lyons was the first to establish such result for locally compact Abelian groups [Ann. Math., II. Ser. 122, 155- 170 (1985; Zbl 0583.43006)]. This paper establishes such result for compact non-Abelian groups. The author uses \((d_{\sigma}^{-1/p}\| A\|_{\phi_ p)\sigma \in \hat G}\in c_ 0(\hat G)\) to define the class \(R_ p\) of Radon measures \(\mu\) on G, where A is \(\mu\)’s Fourier coefficient operator, and \(\| A\|_{\phi_ p}\) is von Neumann norm of A, \(1\leq p\leq \infty\), and defines the similar classes \(W_ p\) and \(\bar W_ p\) of sets in G, and obtains the corresponding characterization: Theorem 3. \(\mu \in R_ p\) iff \(\mu (E)=0\), \(\forall E\in W_ p\), or equivalently \(\mu (E)=0\), \(\forall E\in \bar W_ p\), \(1\leq p\leq \infty\).

A. Rajchman conjectured (1922) that there exists some class of sets in T (denoted by W) which could be used to characterize those Radon measures \(\mu\) on T of which the Fourier transform \({\hat \mu}\) vanishes at infinity (denote such class by R) in such a way: \(\mu\in R\) iff \(\mu (E)=0\), \(\forall E\in W\). R. Lyons was the first to establish such result for locally compact Abelian groups [Ann. Math., II. Ser. 122, 155- 170 (1985; Zbl 0583.43006)]. This paper establishes such result for compact non-Abelian groups. The author uses \((d_{\sigma}^{-1/p}\| A\|_{\phi_ p)\sigma \in \hat G}\in c_ 0(\hat G)\) to define the class \(R_ p\) of Radon measures \(\mu\) on G, where A is \(\mu\)’s Fourier coefficient operator, and \(\| A\|_{\phi_ p}\) is von Neumann norm of A, \(1\leq p\leq \infty\), and defines the similar classes \(W_ p\) and \(\bar W_ p\) of sets in G, and obtains the corresponding characterization: Theorem 3. \(\mu \in R_ p\) iff \(\mu (E)=0\), \(\forall E\in W_ p\), or equivalently \(\mu (E)=0\), \(\forall E\in \bar W_ p\), \(1\leq p\leq \infty\).

Reviewer: R.Long

### MSC:

43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |

42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |

43A05 | Measures on groups and semigroups, etc. |

### Keywords:

Radon measures; Fourier-Stieltjes transforms; Rajchman measures; continuous measures; Borel sets; compact groups; homogeneous spaces### References:

[1] | Dunkl, C., Ramirez, D.: Translation in measure algebras and the correspondence to Fourier transforms vanishing at infinity. Mich. Math. J.17, 311-319 (1970) · Zbl 0188.20601 · doi:10.1307/mmj/1029000517 |

[2] | Dunkl, C., Ramirez, D.: Helson sets in compact and locally compact groups. Mich. Math. J.19, 65-69 (1971) · Zbl 0213.13602 |

[3] | Hewitt, E., Ross, K.: Abstract harmonic analysis. II. Berlin Heidelberg New York: Springer 1970 · Zbl 0213.40103 |

[4] | Hewitt, E., Stromberg, K.: Real and abstract analysis. Berlin Heidelberg New York: Springer 1965 · Zbl 0137.03202 |

[5] | Lyons, R.: Characterizations of measures whose Fourier-Stieltjes transforms vanish at infinity. Bull. Am. Math. Soc., New Ser.10, 93-96 (1984) · doi:10.1090/S0273-0979-1984-15198-X |

[6] | Lyons, R.: Fourier-Stieltjes coefficients and asymptotic distribution modulo 1. Ann Math.122, 155-170 (1985) · Zbl 0583.43006 · doi:10.2307/1971372 |

[7] | Rajchman, A.: Sur l’unicité du développement trigonométrique. Fundam. Math.3, 287-302 (1922) · JFM 48.0304.03 |

[8] | Révész, P.: On a problem of Stainhaus. Acta Math. Acad. Sci. Hung.16, 310-318 (1965) |

[9] | ?reîder, J.: On the ring of functions with bounded variation (Russian). Usp. Mat. Nauk.24, 224-225 (1948) |

[10] | Vilenkin, N.: Special functions and the theory of group representations. Vol. 22. Providence: Am. Math. Soc. 1968 · Zbl 0172.18404 |

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