Weighted norm inequalities for Hilbert transforms and conjugate functions of even and odd functions
HTML articles powered by AMS MathViewer
- by Kenneth F. Andersen PDF
- Proc. Amer. Math. Soc. 56 (1976), 99-107 Request permission
Abstract:
It is well known that the Hilbert tranformation and the conjugate function operator restricted to even (odd) functions define bounded linear operators on weighted ${L^p}$ spaces under more general conditions than is the case for the unrestricted operators. In analogy with recent results of Hunt, Muckenhoupt and Wheeden for the Hilbert transform and the conjugate function operator, we obtain necessary and sufficient conditions in order that these restricted operators should satisfy weighted weak-type inequalities and hence also necessary and sufficient conditions in order that these operators should be bounded on weighted ${L^p}$ spaces for $1 < p < \infty$.References
- K. I. Babenko, On conjugate functions, Doklady Akad. Nauk SSSR (N. S.) 62 (1948), 157–160 (Russian). MR 0027093
- Kien-Kwong Chen, On the absolute Cesàro summability of negative order for a Fourier series at a given point, Amer. J. Math. 66 (1944), 299–312. MR 10207, DOI 10.2307/2371989
- Yung-Ming Chen, On conjugate functions, Canadian J. Math. 15 (1963), 486–494. MR 160066, DOI 10.4153/CJM-1963-053-5
- T. M. Flett, Some theorems on odd and even functions, Proc. London Math. Soc. (3) 8 (1958), 135–148. MR 113092, DOI 10.1112/plms/s3-8.1.135
- G. H. Hardy and J. E. Littlewood, Some more theorems concerning Fourier series and Fourier power series, Duke Math. J. 2 (1936), no. 2, 354–382. MR 1545928, DOI 10.1215/S0012-7094-36-00228-4
- Richard Hunt, Benjamin Muckenhoupt, and Richard Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227–251. MR 312139, DOI 10.1090/S0002-9947-1973-0312139-8
- Benjamin Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207–226. MR 293384, DOI 10.1090/S0002-9947-1972-0293384-6
- P. G. Rooney, On the ranges of certain fractional integrals, Canadian J. Math. 24 (1972), 1198–1216. MR 310147, DOI 10.4153/CJM-1972-130-9
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 56 (1976), 99-107
- DOI: https://doi.org/10.1090/S0002-9939-1976-0399739-2
- MathSciNet review: 0399739