Two-weight mixed ф-inequalities for the one-sided maximal function

Qinsheng Lai

ArticleOriginal scientific text

Title

Two-weight mixed ф-inequalities for the one-sided maximal function

Authors ¹

Affiliations

Department of Pure Mathematics, The University of Leeds, Leeds LS2 9JT, U.K.

Abstract

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 (f v)) w d μ) \leq Φ^{- 1}_{1} (ʃ_{X} Φ_{1} (C f) v d μ)

, (ii) the weak type mixed Φ-inequality

Φ^{- 1}_2 (ʃ_{| T f | > λ}

Φ_{2}(λw)tdμ) ≤ Φ^{-1}_{1} (ʃ_{X} Φ_{1}(Cfu)vdμ)

and (i i i) t h e e x t r a - w e a k t y p e m i x e d Φ - \in e q u a l i t y

|{x ∈ X : |Tf(x)| > λ}|_{wdμ} ≤ Φ_{2}Φ^{-1}_{1} (ʃ_{X} Φ_{1}(Cfu/λ)vdμ)

, w h e n T i s t h e o \neq - s d max i m a l f u n c t i o n

M^{+}_{g}!$!; as well to characterize (iii) for the Fefferman-Stein type fractional maximal operator and the Hardy-type operator.

Keywords

ENG

Young function, one-sided maximal function, Fefferman-Stein type fractional operator, Hardy-type operator

S. Bloom and R. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113 (1991), 135-141.
S. Bloom and R. Kerman, Weighted $L_{Φ}$ integral inequalities for operators of Hardy type, Studia Math. 110 (1994), 35-52.
S. Bloom and R. Kerman, Weighted Orlicz space integral inequalities for the Hardy-Littlewood maximal operator, ibid., 149-167.
R. Coifman et G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.
M. A. Krasnosel'skiĭ and Ya. B. Rutickiĭ, Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961.
Q. Lai, Weighted weak type inequalities for fractional maximal operator on spaces of homogeneous type, Acta Math. Sinica 32 (1989), 448-456.
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.
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.
Q. Lai, Weighted integral inequalities for the Hardy type operator and the fractional maximal operator, J. London Math. Soc. (2) 49 (1994), 244-266.
Q. Lai, A note on the weighted norm inequality for the one-sided maximal operator, Proc. Amer. Math. Soc., to appear.
Q. Lai, Weighted $L^{p}$ -estimates for the one-sided maximal operator, preprint.
Q. Lai, A note on maximal inequalities, preprint.
F. J. Martín-Reyes, New proofs of weighted inequalities for the one-sided Hardy-Littlewood maximal functions, preprint.
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.
P. Ortega Salvador, Weighted inequalities for one-sided maximal function in Orlicz classes, preprint.
P. Ortega Salvador and L. Pick, Two weight weak and extra-weak type inequalities for the one-sided maximal operator, preprint.
L. Pick, Two-weight weak type maximal inequalities in Orlicz classes, Studia Math. 100 (1991), 207-218.
E. Sawyer, A characterization of a two-weight norm inequality for maximal operator, ibid. 75 (1982), 1-11.
E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53-61.
T. Shimogaki, Hardy-Littlewood majorants in function spaces, J. Math. Soc. Japan 17 (1965), 365-373.

Title

Two-weight mixed ф-inequalities for the one-sided maximal function

Affiliations

Abstract

Keywords

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