Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A commutator theorem and weighted BMO


Author: Steven Bloom
Journal: Trans. Amer. Math. Soc. 292 (1985), 103-122
MSC: Primary 42A50; Secondary 42B20, 46E40, 47B47
DOI: https://doi.org/10.1090/S0002-9947-1985-0805955-5
MathSciNet review: 805955
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The main result of this paper is a commutator theorem: If $ \mu $ and $ \lambda $ are $ {A_p}$ weights, then the commutator $ H$, $ {M_b}$ is a bounded operator from $ {L^p}(\mu )$ into $ {L^p}(\lambda )$ if and only if $ b \in {\operatorname{BMO} _{{{(\mu {\lambda ^{ - 1}})}^{1/p}}}}$. The proof relies heavily on a weighted sharp function theorem. Along the way, several other applications of this theorem are derived, including a doubly-weighted $ {L^p}$ estimate for BMO. Finally, the commutator theorem is used to obtain vector-valued weighted norm inequalities for the Hilbert transform.


References [Enhancements On Off] (What's this?)

  • [1] S. Bloom, Weighted norm inequalities for vector-valued functions, Ph. D. dissertation, Washington University, St. Louis, Missouri, 1981.
  • [2] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
  • [3] R. R. Coifman, R. Rochberg and G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. 103 (1976), 611-635. MR 0412721 (54:843)
  • [4] H. Helson and G. Szegö, A problem in prediction theory, Ann. Math. Pura. Appl. 51 (1960), 107-138. MR 0121608 (22:12343)
  • [5] R. A. Hunt, B. Muckenhoupt and R. L. Wheeden, Weighted norm inqualities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 0312139 (47:701)
  • [6] E. Lotkowski and R. L. Wheeden, The equivalence of various Lipschitz conditions on the weighted oscillation of a function, Proc. Amer. Math. Soc. 61 (1976), 323-328. MR 0427552 (55:583)
  • [7] B. Muckenhoupt, The equivalence of two conditions for weights, Studia Math. 49 (1974), 101-106. MR 0350297 (50:2790)
  • [8] -, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
  • [9] H. Pousson, Systems of Toeplitz operators on $ {H^2}$. I, Proc. Amer. Math. Soc. 19 (1968), 603-608; II, Trans. Amer. Math. Soc. 133 (1968), 527-536.
  • [10] R. Rabindranathan, On the inversion of Töeplitz operators, J. Math. Mech. 19 (1969), 195-206. MR 0251558 (40:4785)
  • [11] A. Cordoba and C. Fefferman, A weighted norm inequality for singular integrals, Studia Math. 57 (1976), 97-101. MR 0420115 (54:8132)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 42A50, 42B20, 46E40, 47B47

Retrieve articles in all journals with MSC: 42A50, 42B20, 46E40, 47B47


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0805955-5
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society