Some weighted inequalities for general one-sided maximal operators

Martín-Reyes, F.; de la Torre, A.

Studia Mathematica, Tome 122 (1997), p. 1-14 / Harvested from The Polish Digital Mathematics Library

Résumé

We characterize the pairs of weights on ℝ for which the operators $M_{h, k}^{+} f (x) = s u p_{c > x} h (x, c) ʃ_{x}^{c} f (s) k (x, s, c) d s$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on $(x, c) : x < c$ , while k is defined on $(x, s, c) : x < s < c$ . If $h (x, c) = {(c - x)}^{- β}$ , $k (x, s, c) = {(c - s)}^{α - 1}$ , 0 ≤ β ≤ α ≤ 1, we obtain the operator $M_{α, β}^{+} f = s u p_{c > x} 1 / {(c - x)}^{β} ʃ_{x}^{c} f (s) / {(c - s)}^{1 - α} d s$ . For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal weights and 1 < p < ∞, weak and strong type are equivalent. If we take α = β we characterize the strong type weights for the operator $M_{α, α}^{+}$ introduced by W. Jurkat and J. Troutman in the study of $C_{α}$ differentiation of the integral.

Publié le : 1997-01-01

Zbl 0866.42012

EUDML-ID : urn:eudml:doc:216358

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     author = {F. Mart\'\i n-Reyes and A. de la Torre},
     title = {Some weighted inequalities for general one-sided maximal operators},
     journal = {Studia Mathematica},
     volume = {122},
     year = {1997},
     pages = {1-14},
     zbl = {0866.42012},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv122i1p1bwm}
}

Martín-Reyes, F.; de la Torre, A. Some weighted inequalities for general one-sided maximal operators. Studia Mathematica, Tome 122 (1997) pp. 1-14. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv122i1p1bwm/

Bibliographie

[00000] [A] K. F. Andersen, Weighted inequalities for maximal functions associated with general measures, Trans. Amer. Math. Soc. 326 (1991), 907-920. | Zbl 0736.42013

[00001] [AM] K. F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9-26. | Zbl 0501.47011

[00002] [AS] K. F. Andersen and E. T. Sawyer, Weighted norm inequalities for the Riemann-Liouville and Weyl fractional integral operators, Trans. Amer. Math. Soc. 308 (1988), 547-557. | Zbl 0664.26002

[00003] [CHS] A. Carbery, E. Hernandez and F. Soria, Estimates for the Kakeya maximal operator and radial functions in $ℝ^{n}$ , in: Harmonic Analysis (Sendai, 1990), ICM-90 Satell. Conf. Proc., Springer, Tokyo, 1991, 41-50.

[00004] [JT] W. Jurkat and J. Troutman, Maximal inequalities related to generalized a.e. continuity, Trans. Amer. Math. Soc. 252 (1979), 49-64. | Zbl 0441.42023

[00005] [KG] V. Kokilashvili and M. Gabidzashvili, Two weight weak type inequalities for fractional type integrals, Math. Inst. Czech. Acad. Sci. Prague 45 (1989).

[00006] [LT] M. Lorente and A. de la Torre, Weighted inequalities for some one-sided operators, Proc. Amer. Math. Soc. 124 (1996), 839-848. | Zbl 0895.26002

[00007] [MOT] 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. | Zbl 0696.42013

[00008] [MPT] F. J. Martín-Reyes, L. Pick and A. de la Torre, $A_{\infty}^{+}$ condition, Canad. J. Math. 45 (1993), 1231-1244.

[00009] [MT] F. J. Martín-Reyes and A. de la Torre, Two weight norm inequalities for fractional one-sided maximal operators, Proc. Amer. Math. Soc. 117 (1993), 483-489. | Zbl 0769.42010

[00010] [S] E. T. Sawyer, Weighted inequalities for the one sided Hardy-Littlewood maximal functions, Trans. Amer. Math. Soc. 297 (1986), 53-61. | Zbl 0627.42009

[00011] [SW] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Pres 1971.