Optimal factorization of Muckenhoupt weights
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- by Michael Brian Korey
- Trans. Amer. Math. Soc. 352 (2000), 5251-5262
- DOI: https://doi.org/10.1090/S0002-9947-00-02547-2
- Published electronically: July 18, 2000
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Correction: Trans. Amer. Math. Soc. 353 (2001), 839-851.
Abstract:
Peter Jones’ theorem on the factorization of $A_p$ weights is sharpened for weights with bounds near $1$, allowing the factorization to be performed continuously near the limiting, unweighted case. When $1<p<\infty$ and $w$ is an $A_p$ weight with bound $A_p(w)=1+\varepsilon$, it is shown that there exist $A_1$ weights $u,v$ such that both the formula $w=uv^{1-p}$ and the estimates $A_1(u), A_1(v)=1+\mathcal O(\sqrt \varepsilon )$ hold. The square root in these estimates is also proven to be the correct asymptotic power as $\varepsilon \to 0$.References
- R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
- R. Coifman, Peter W. Jones, and José L. Rubio de Francia, Constructive decomposition of BMO functions and factorization of $A_{p}$ weights, Proc. Amer. Math. Soc. 87 (1983), no. 4, 675–676. MR 687639, DOI 10.1090/S0002-9939-1983-0687639-3
- Radu Bǎdescu, On a problem of Goursat, Gaz. Mat. 44 (1939), 571–577. MR 0000087
- John B. Garnett and Peter W. Jones, BMO from dyadic BMO, Pacific J. Math. 99 (1982), no. 2, 351–371. MR 658065, DOI 10.2140/pjm.1982.99.351
- Peter J. Holden, Extension theorems for functions of vanishing mean oscillation, Pacific J. Math. 142 (1990), no. 2, 277–295. MR 1042047, DOI 10.2140/pjm.1990.142.277
- Peter W. Jones, Factorization of $A_{p}$ weights, Ann. of Math. (2) 111 (1980), no. 3, 511–530. MR 577135, DOI 10.2307/1971107
- F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415–426. MR 131498, DOI 10.1002/cpa.3160140317
- M. B. Korey, Ideal weights: doubling and absolute continuity with asymptotically optimal bounds, Ph.D. Thesis, University of Chicago, 1995.
- M. B. Korey, Ideal weights: asymptotically optimal versions of doubling, absolute continuity, and mean oscillation, J. Fourier Anal. Appl. 4 (1998), 491–519.
- 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
- A. Politis, Sharp results on the relation between weight spaces and BMO, Ph.D. Thesis, University of Chicago, 1995.
- José Luis Rubio de Francia, Factorization and extrapolation of weights, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 2, 393–395. MR 663793, DOI 10.1090/S0273-0979-1982-15047-9
- José L. Rubio de Francia, Factorization theory and $A_{p}$ weights, Amer. J. Math. 106 (1984), no. 3, 533–547. MR 745140, DOI 10.2307/2374284
- Walter Rudin, Functional analysis, McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. MR 0365062
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
- Kôsaku Yosida, Functional analysis, Die Grundlehren der mathematischen Wissenschaften, Band 123, Academic Press, Inc., New York; Springer-Verlag, Berlin, 1965. MR 0180824
Bibliographic Information
- Michael Brian Korey
- Affiliation: Institut für Mathematik, Universität Potsdam, 14415 Potsdam, Germany
- Email: mike@math.uni-potsdam.de
- Received by editor(s): February 3, 1999
- Published electronically: July 18, 2000
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 352 (2000), 5251-5262
- MSC (1991): Primary 42B25; Secondary 26D15, 46E30
- DOI: https://doi.org/10.1090/S0002-9947-00-02547-2
- MathSciNet review: 1694375