Transactions of the American Mathematical Society

Author(s): Michael Christ; Michael Goldberg
Journal: Trans. Amer. Math. Soc. 353 (2001), 1995-2002.
MSC (2000): Primary 42B25
Posted: January 5, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

An analogue of the Hardy-Littlewood maximal function is introduced, for functions taking values in finite-dimensional Hilbert spaces. It is shown to be

bounded with respect to weights in the class

of Treil, thereby extending a theorem of Muckenhoupt from the scalar to the vector case.

1.: M. Christ, Weighted norm inequalities and Schur's lemma, Studia Math. 78 (1984), 309-319. MR 86h:42032
2.: M. Christ and R. Fefferman, A note on weighted norm inequalities for the Hardy-Littlewood maximal operator, Proc. Amer. Math. Soc. 87 (1983), 447-448. MR 84g:42017
3.: Coifman, R. R., Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 50:10670
4.: R. Fefferman, C. E. Kenig, and J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. Math. 134 (1991), 65-124. MR 93h:31010
5.: M. Goldberg, manuscript in preparation.
6.: R. Hunt, B. Muckenhoupt, and R. Wheeden, Weighted norm inequalities for conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 47:701
7.: B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 45:2461
8.: F. L. Nazarov and S. R Tre ${\u{\i}}\kern.15em$ l, The hunt for a Bellman function: applications to estimates for singular integral operators and to other classical problems of harmonic analysis, (Russian) Algebra i Analiz 8 (1996), no. 5, 32-162; translation in St. Petersburg Math. J. 8 (1997), no. 5, 721-824. MR 99d:42026
9.: -, The weighted norm inequalities for Hilbert transform are now trivial, C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), no. 7, 717-722. MR 99j:42010
10.: F. Nazarov, S. R Treil, and A. Volberg, Counterexample to the infinite-dimensional Carleson embedding theorem, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 4, 383-388. MR 98d:46039
11.: F. Nazarov, S. R. Treil, and A. Volberg, The Bellman functions and two-weight inequalities for Haar multipliers, J. Amer. Math. Soc. 12 (1999), no. 4, 909-928. MR 2000k:42009
12.: E. M. Stein, Harmonic Analysis, Princeton University Press, Princeton, NJ, 1993. MR 95c:42002
13.: S. R. Treil, The geometric approach to weight estimates of the Hilbert transform, (Russian) Funktsional. Anal. i Prilozhen. 17 (1983), no. 4, 90-91. MR 85e:42012
14.: S. R. Treil, An operator approach to weighted estimates of singular integrals, (Russian) Investigations on linear operators and the theory of functions, XIII. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 135 (1984), 150-174. MR 86g:42032
15.: S. R. Treil, Geometric methods in spectral theory of vector valued functions: Some recent results, Operator Theory: Adv. Appl. 42 (1989), 209-280. MR 91j:47036
16.: S. R. Treil and A. L. Volberg, Weighted embeddings and weighted norm inequalities for the Hilbert transform and the maximal operator, Algebra i Analiz 7 (1995), no. 6, 205-226; translation in St. Petersburg Math. J. 7 (1996), no. 6, 1017-1032. MR 97c:42017
17.: S. Treil and A. Volberg, Wavelets and the angle between past and future, J. Funct. Anal. 143 (1997), no. 2, 269-308. MR 99k:42073
18.: S. Treil and A. Volberg, Continuous frame decomposition and a vector Hunt-Muckenhoupt-Wheeden theorem, Ark. Mat. 35 (1997), no. 2, 363-386. MR 98m:42026
19.: S. Treil, A. Volberg, and D. Zheng, Hilbert transform, Toeplitz operators and Hankel operators, and invariant $A\sb \infty$ weights, Rev. Mat. Iberoamericana 13 (1997), no. 2, 319-360. MR 2000e:47061
20.: A. Volberg, Matrix $A\sb p$ weights via -functions, J. Amer. Math. Soc. 10 (1997), no. 2, 445-466. MR 98a:42013

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B25

Michael Christ
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: mchrist@math.berkeley.edu

Michael Goldberg
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720-3840
Email: mikeg@math.berkeley.edu

DOI: 10.1090/S0002-9947-01-02759-3
PII: S 0002-9947(01)02759-3
Received by editor(s): June 22, 2000
Posted: January 5, 2001
Additional Notes: The first author was supported in part by NSF grant DMS-9970660. He thanks the staff of the Bamboo Garden hotel in Shenzhen, PRC, for the hospitable atmosphere in which a portion of this work was done
The second author was supported by an NSF graduate fellowship
Copyright of article: Copyright 2001, American Mathematical Society

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2008, American Mathematical Society Privacy Statement		Search the AMS