Boundedness of maximal functions and singular integrals in weighted 𝐿^{𝑝} spaces

Rubio de Francia, José L.

doi:10.1090/S0002-9939-1981-0630035-3

Boundedness of maximal functions and singular integrals in weighted $L^{p}$ spaces
HTML articles powered by AMS MathViewer

by José L. Rubio de Francia

Proc. Amer. Math. Soc. 83 (1981), 673-679

DOI: https://doi.org/10.1090/S0002-9939-1981-0630035-3

PDF | Request permission

Abstract:

Given a weight $w(x) > 0$ in ${{\mathbf {R}}^n}$, necessary and sufficient conditions are found for the boundedness of the Hardy-Littlewood maximal function and singular integral operators from ${L^p}(w)$ to some other weighted ${L^p}$ space. The dual question is also considered and partially answered.

References

A. Benedek, A.-P. Calderón, and R. Panzone, Convolution operators on Banach space valued functions, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 356–365. MR 133653, DOI 10.1073/pnas.48.3.356
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
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
Adriano M. Garsia, Topics in almost everywhere convergence, Lectures in Advanced Mathematics, No. 4, Markham Publishing Co., Chicago, Ill., 1970. MR 0261253
John E. Gilbert, Nikišin-Stein theory and factorization with applications, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 233–267. MR 545313
Paul Koosis, Moyennes quadratiques pondérées de fonctions périodiques et de leurs conjuguées harmoniques, C. R. Acad. Sci. Paris Sér. A-B 291 (1980), no. 4, A255–A257 (French, with English summary). MR 591744
Bernard Maurey, Théorèmes de factorisation pour les opérateurs linéaires à valeurs dans les espaces $L^{p}$, Astérisque, No. 11, Société Mathématique de France, Paris, 1974 (French). With an English summary. MR 0344931
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
Benjamin Muckenhoupt, Weighted norm inequalities for classical operators, Harmonic analysis in Euclidean spaces (Proc. Sympos. Pure Math., Williams Coll., Williamstown, Mass., 1978) Proc. Sympos. Pure Math., XXXV, Part, Amer. Math. Soc., Providence, R.I., 1979, pp. 69–83. MR 545240
José L. Rubio de Francia, Vector-valued inequalities for operators in $L^{p}$-spaces, Bull. London Math. Soc. 12 (1980), no. 3, 211–215. MR 572104, DOI 10.1112/blms/12.3.211
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095