On the almost everywhere existence of the ergodic Hilbert transform
HTML articles powered by AMS MathViewer
- by Diego Gallardo and F. J. Martín-Reyes PDF
- Proc. Amer. Math. Soc. 105 (1989), 636-643 Request permission
Abstract:
Let $(X,\mathfrak {M},\mu )$ be a finite measure space, $T$ an invertible measure-preserving transformation and $\upsilon$ a positive measurable function. For $p = 1$, we prove that the ergodic Hubert transform $Hf(X) = {\text {li}}{{\text {m}}_{n \to \infty }}\sum \nolimits _{i = - n}^n {’f({T^i}x)/i}$ exists a.e. for every $f$ in ${L^1}(\upsilon d\mu )$ if and only if ${\text {in}}{{\text {f}}_{i \geq 0}}\upsilon ({T^i}x) > 0$ a.e. We also solve the problem for $1 < p \leq 2$. In this case the condition is ${\text {su}}{{\text {p}}_{k \geq 1}}{k^{ - 1}}\sum \nolimits _{i - 0}^{k - 1} {{\upsilon ^{ - 1/(p - 1)}}} ({T^i}x) < \infty$ a.e. If the transformation $T$ is ergodic, the characterizing conditions become that $1/\upsilon \in {L^\infty }$ and ${\upsilon ^{ - 1/(p - 1)}} \in {L^1}(\mu )$, respectively. These characterizations, together with some recent results, give, for $1 \leq p \leq 2$, that $Hf(x)$ exists a.e. for every $f$ in ${L^p}(\upsilon d\mu )$ if and only if the sequence of the Césàro-averages ${k^{ - 1}}(f(x) + f(Tx) + \ldots f({T^{k - 1}}x))$ converge a.e. for every $f$ in ${L^p}(\upsilon d\mu )$. This equivalence has recently been obtained by Jajte for a unitary operator, not necessarily positive, acting on ${L^2}$.References
- Mischa Cotlar, A unified theory of Hilbert transforms and ergodic theorems, Rev. Mat. Cuyana 1 (1955), 105–167 (1956) (English, with Spanish summary). MR 84632
- John E. Gilbert, Nikišin-Stein theory and factorization with applications, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 233–267. MR 545313
- R. Jajte, On the existence of the ergodic Hilbert transform, Ann. Probab. 15 (1987), no. 2, 831–835. MR 885148
- F. J. Martín-Reyes, Inequalities for the ergodic maximal function and convergence of the averages in weighted $L^p$-spaces, Trans. Amer. Math. Soc. 296 (1986), no. 1, 61–82. MR 837798, DOI 10.1090/S0002-9947-1986-0837798-1
- F. J. Martín-Reyes and A. de la Torre, Weighted weak type inequalities for the ergodic maximal function and the pointwise ergodic theorem, Studia Math. 87 (1987), no. 1, 33–46. MR 924759, DOI 10.4064/sm-87-1-33-46 —, On the almost everywhere convergence of the ergodic averages, to appear in Ergodic Theorey and Dynamical Systems.
- Julian Musielak, Orlicz spaces and modular spaces, Lecture Notes in Mathematics, vol. 1034, Springer-Verlag, Berlin, 1983. MR 724434, DOI 10.1007/BFb0072210
- Karl Petersen, Another proof of the existence of the ergodic Hilbert transform, Proc. Amer. Math. Soc. 88 (1983), no. 1, 39–43. MR 691275, DOI 10.1090/S0002-9939-1983-0691275-2
- J. Woś, A remark on the existence of the ergodic Hilbert transform, Colloq. Math. 53 (1987), no. 1, 97–101. MR 890844, DOI 10.4064/cm-53-1-97-101
Additional Information
- © Copyright 1989 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 105 (1989), 636-643
- MSC: Primary 28D05; Secondary 47A35
- DOI: https://doi.org/10.1090/S0002-9939-1989-0939964-1
- MathSciNet review: 939964