Solving weighted norm inequalities using the Rubio de Francia algorithm

Bloom, Steve

doi:10.1090/S0002-9939-1987-0902547-1

Solving weighted norm inequalities using the Rubio de Francia algorithm
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Proc. Amer. Math. Soc. 101 (1987), 306-312 Request permission

Abstract:

Given an operator $T$ bounded on a weighted ${L^p}$ space, the factorization technique of Rubio de Francia forces strong conditions on the weight. This algorithm is extended to two weight problems, and is shown to yield not just necessary but sufficient conditions in a wide range of settings.

References

Steven Bloom, A commutator theorem and weighted BMO, Trans. Amer. Math. Soc. 292 (1985), no. 1, 103–122. MR 805955, DOI 10.1090/S0002-9947-1985-0805955-5

Applications of commutator theory to weighted BMO and matrix analogs of

C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
Peter W. Jones, Factorization of $A_{p}$ weights, Ann. of Math. (2) 111 (1980), no. 3, 511–530. MR 577135, DOI 10.2307/1971107
Benjamin Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. MR 311856, DOI 10.4064/sm-44-1-31-38
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
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
Eric T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1–11. MR 676801, DOI 10.4064/sm-75-1-1-11
Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972

Various remarks concerning Rubio de Francia’s proof of Peter Jones’ factorization theorem and some applications of the ideas in this proof