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## Studia Mathematica

1995 | 115 | 1 | 1-22
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### Two-weight mixed ф-inequalities for the one-sided maximal function

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Suppose u, v, w, and t are weight functions on an appropriate measure space (X,μ), and $Φ_1$, $Φ_2$ are Young functions satisfying a certain relationship. Let T denote an operator to be specified below. The main purpose of this paper is to characterize (i) the strong type mixed Φ-inequality $Φ^{-1}_{2}(ʃ_{X} Φ_{2}(T(fv))wdμ) ≤ Φ^{-1}_{1} (ʃ_X Φ_{1}(Cf)vdμ)$, (ii) the weak type mixed Φ-inequality $Φ^{-1}_2 (ʃ_{|Tf|>λ}$ Φ_{2}(λw)tdμ) ≤ Φ^{-1}_{1} (ʃ_{X} Φ_{1}(Cfu)vdμ)$and (iii) the extra-weak type mixed Φ-inequality$|{x ∈ X : |Tf(x)| > λ}|_{wdμ} ≤ Φ_{2}Φ^{-1}_{1} (ʃ_{X} Φ_{1}(Cfu/λ)vdμ)$, when T is the one-sided maximal function$M^{+}_{g}$; as well to characterize (iii) for the Fefferman-Stein type fractional maximal operator and the Hardy-type operator. Słowa kluczowe EN Kategorie tematyczne Czasopismo Rocznik Tom Numer Strony 1-22 Opis fizyczny Daty wydano 1995 otrzymano 1992-04-07 poprawiono 1995-01-30 Twórcy autor • Department of Pure Mathematics, The University of Leeds, Leeds LS2 9JT, U.K. Bibliografia • [1] S. Bloom and R. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991), 135-141. • [2] S. Bloom and R. Kerman, Weighted$L_Φ$integral inequalities for operators of Hardy type, Studia Math. 110 (1994), 35-52. • [3] S. Bloom and R. Kerman, Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator, ibid., 149-167. • [4] R. Coifman et G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971. • [5] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. • [6] M. A. Krasnosel'skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961. • [7] Q. Lai, Weighted weak type inequalities for fractional maximal operator on spaces of homogeneous type, Acta Math. Sinica 32 (1989), 448-456. • [8] Q. Lai, Two weight mixed Φ-inequalities for the Hardy operator and the Hardy-Littlewood maximal operator, J. London Math. Soc. (2) 48 (1992), 301-318. • [9] Q. Lai, Two weight Φ-inequalities for the Hardy operator, Hardy-Littlewood maximal operator and fractional integrals, Proc. Amer. Math. Soc. 118 (1993), 129-142. • [10] Q. Lai, Weighted integral inequalities for the Hardy type operator and the fractional maximal operator, J. London Math. Soc. (2) 49 (1994), 244-266. • [11] Q. Lai, A note on the weighted norm inequality for the one-sided maximal operator, Proc. Amer. Math. Soc., to appear. • [12] Q. Lai, Weighted$L^p\$-estimates for the one-sided maximal operator, preprint.
• [13] Q. Lai, A note on maximal inequalities, preprint.
• [14] F. J. Martín-Reyes, New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, preprint.
• [15] F. J. Martín-Reyes, P. Ortega Salvador and A. de la Torre, Weighted inequalities for one-sided maximal functions, Trans. Amer. Math. Soc. 319 (1990), 517-534.
• [16] P. Ortega Salvador, Weighted inequalities for one-sided maximal function in Orlicz classes, preprint.
• [17] P. Ortega Salvador and L. Pick, Two weight weak and extra-weak type inequalities for the one-sided maximal operator, preprint.
• [18] L. Pick, Two-weight weak type maximal inequalities in Orlicz classes, Studia Math. 100 (1991), 207-218.
• [19] E. Sawyer, A characterization of a two-weight norm inequality for maximal operator, ibid. 75 (1982), 1-11.
• [20] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53-61.
• [21] T. Shimogaki, Hardy-Littlewood majorants in function spaces, J. Math. Soc. Japan 17 (1965), 365-373.
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