A boundedness criterion for general maximal operators. (English) Zbl 1183.42024

Summary: We consider maximal operators \(M_{\mathcal B}\) with respect to a basis \({\mathcal B}\). In the case when \(M_{\mathcal B}\) satisfies a reversed weak type inequality, we obtain a boundedness criterion for \(M_{\mathcal B}\) on an arbitrary quasi-Banach function space \(X\). Being applied to specific \({\mathcal B}\) and \(X\) this criterion yields new and short proofs of a number of well-known results. Our principal application is related to an open problem on the boundedness of the two-dimensional one-sided maximal function \(M^{+}\) on \(L^p_w\).


42B25 Maximal functions, Littlewood-Paley theory
Full Text: DOI Euclid


[1] M. A. Ariño and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320(2) (1990), 727\Ndash735. · Zbl 0716.42016
[2] C. Bennett and R. Sharpley, “Interpolation of operators” , Pure and Applied Mathematics 129 , Academic Press, Inc., Boston, MA, 1988. · Zbl 0647.46057
[3] M. J. Carro, J. A. Raposo, and J. Soria, Recent developments in the theory of Lorentz spaces and weighted inequalities, Mem. Amer. Math. Soc. 187(877) (2007), 128 pp. · Zbl 1126.42005
[4] R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241\Ndash250. · Zbl 0291.44007
[5] J. Duoandikoetxea, “Fourier analysis” , Translated and revised from the 1995 Spanish original by David Cruz-Uribe, Graduate Studies in Mathematics 29 , American Mathematical Society, Providence, RI, 2001. · Zbl 0969.42001
[6] R. Fefferman, Some weighted norm inequalities for Córdoba’s maximal function, Amer. J. Math. 106(5) (1984), 1261\Ndash1264. · Zbl 0575.42022
[7] L. Forzani, F. J. Martín-Reyes, and S. Ombrosi, Weighted inequalities for the two-dimensional one-sided Hardy-Littlewood maximal function, Preprint, available at http://webpersonal.uma.es/\(\sim\)MARTIN_REYES/preprints.htm. · Zbl 1218.42008
[8] M. de Guzmán, “Differentiation of integrals in \(R^{n}\)” , with appendices by Antonio Córdoba, and Robert Fefferman, and two by Roberto Moriyón, Lecture Notes in Mathematics 481 , Springer-Verlag, Berlin-New York, 1975.
[9] B. Jawerth, Weighted inequalities for maximal operators: linearization, localization and factorization, Amer. J. Math. 108(2) (1986), 361\Ndash414. · Zbl 0608.42012
[10] B. Jawerth and A. Torchinsky, The strong maximal function with respect to measures, Studia Math. 80(3) (1984), 261\Ndash285. · Zbl 0565.42008
[11] N. J. Kalton, N. T. Peck, and J. W. Roberts, “An \(F\)-space sampler” , London Mathematical Society Lecture Note Series 89 , Cambridge University Press, Cambridge, 1984. · Zbl 0556.46002
[12] A. K. Lerner, S. Ombrosi, and C. Pérez, Sharp \(A_ 1\) bounds for Calderón-Zygmund operators and the relationship with a problem of Muckenhoupt and Wheeden, Int. Math. Res. Not. IMRN 6 (2008), Art. ID rnm161, 11 pp. · Zbl 1237.42012
[13] A. K. Lerner and C. Pérez, A new characterization of the Muckenhoupt \(A_ p\) weights through an extension of the Lorentz-Shimogaki theorem, Indiana Univ. Math. J. 56(6) (2007), 2697\Ndash2722. · Zbl 1214.42021
[14] F. J. Martín-Reyes, New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Proc. Amer. Math. Soc. 117(3) (1993), 691\Ndash698. · Zbl 0771.42011
[15] F. J. Martín-Reyes, On the one-sided Hardy-Littlewood maximal function in the real line and in dimensions greater than one, in: “Fourier analysis and partial differential equations” (Miraflores de la Sierra, 1992), Stud. Adv. Math., CRC, Boca Raton, FL, 1995, pp. 237\Ndash250. · Zbl 0895.42007
[16] F. J. Martín-Reyes and A. de la Torre, Two weight norm inequalities for fractional one-sided maximal operators, Proc. Amer. Math. Soc. 117(2) (1993), 483\Ndash489. · Zbl 0769.42010
[17] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207\Ndash226. · Zbl 0236.26016
[18] C. J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35(2) (1991), 429\Ndash447. · Zbl 0746.42014
[19] S. Ombrosi, Weak weighted inequalities for a dyadic one-sided maximal function in \(\mathbb{R}^ n\), Proc. Amer. Math. Soc. 133(6) (2005), 1769\Ndash1775 (electronic). · Zbl 1063.42011
[20] J. L. Rubio de Francia, Factorization theory and \(A_{p}\) weights, Amer. J. Math. 106(3) (1984), 533\Ndash547. · Zbl 0558.42012
[21] E. Sawyer, Weighted inequalities for the one-sided Hardy\guioLittlewood maximal functions, Trans. Amer. Math. Soc. 297(1) (1986), 53\Ndash61. · Zbl 0627.42009
[22] P. Sjögren, A remark on the maximal function for measures in \(\mathbb{R}^{n}\), Amer. J. Math. 105(5) (1983), 1231\Ndash1233. · Zbl 0528.42007
[23] E. M. Stein, Note on the class \(L\log L\), Studia Math. 32 (1969), 305\Ndash310. · Zbl 0182.47803
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.