The spectral measure and Hilbert transform of a measure-preserving transformation

Authors:
James Campbell and Karl Petersen

Journal:
Trans. Amer. Math. Soc. **313** (1989), 121-129

MSC:
Primary 28D05; Secondary 47A35, 47A60

MathSciNet review:
958884

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: V. F. Gaposhkin gave a condition on the spectral measure of a normal contraction on sufficient to imply that the operator satisfies the pointwise ergodic theorem. We prove that unitary operators which come from measure-preserving transformations satisfy a stronger version of this condition. This follows from the fact that the rotated ergodic Hubert transform is a continuous function of its parameter. The maximal inequality on which the proof depends follows from an analytic inequality related to the Carleson-Hunt Theorem on the a.e. convergence of Fourier series.

**[J]**James T. Campbell,*Spectral analysis of the ergodic Hilbert transform*, Indiana Univ. Math. J.**35**(1986), no. 2, 379–390. MR**833401**, 10.1512/iumj.1986.35.35023**[M]**Mischa Cotlar,*A unified theory of Hilbert transforms and ergodic theorems*, Rev. Mat. Cuyana**1**(1955), 105–167 (1956) (English, with Spanish summary). MR**0084632****[R]**Richard Duncan,*Some pointwise convergence results in 𝐿^{𝑝}(𝜇), 1<𝑝<∞*, Canad. Math. Bull.**20**(1977), no. 3, 277–284. MR**0499074****[V]**V. F. Gaposhkin,*An individual ergodic theorem for normal operators in 𝐿₂*, Funktsional. Anal. i Prilozhen.**15**(1981), no. 1, 18–22, 96 (Russian). MR**609791****[L]**Hörmander, [1973]*An introduction to complex analysis in several variables*, North-Holland, Amsterdam,**[C]**Carlos E. Kenig and Peter A. Tomas,*Maximal operators defined by Fourier multipliers*, Studia Math.**68**(1980), no. 1, 79–83. MR**583403****[U]**Ulrich Krengel,*Ergodic theorems*, de Gruyter Studies in Mathematics, vol. 6, Walter de Gruyter & Co., Berlin, 1985. With a supplement by Antoine Brunel. MR**797411****[K]**Karl Petersen,*Another proof of the existence of the ergodic Hilbert transform*, Proc. Amer. Math. Soc.**88**(1983), no. 1, 39–43. MR**691275**, 10.1090/S0002-9939-1983-0691275-2**[N]**Norbert Wiener and Aurel Wintner,*Harmonic analysis and ergodic theory*, Amer. J. Math.**63**(1941), 415–426. MR**0004098**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
28D05,
47A35,
47A60

Retrieve articles in all journals with MSC: 28D05, 47A35, 47A60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0958884-4

Keywords:
Spectral measure,
ergodic Hubert transform,
maximal inequality

Article copyright:
© Copyright 1989
American Mathematical Society