On a converse inequality for maximal functions in Orlicz spaces

Kita, H.

Kita, H.

Studia Mathematica, Tome 119 (1996), p. 1-10 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $Φ (t) = ʃ_{0}^{t} a (s) d s$ and $Ψ (t) = ʃ_{0}^{t} b (s) d s$ , where a(s) is a positive continuous function such that $ʃ_{1}^{\infty} a (s) / s d s = \infty$ and b(s) is quasi-increasing and $l i m_{s \to \infty} b (s) = \infty$ . Then the following statements for the Hardy-Littlewood maximal function Mf(x) are equivalent: (j) there exist positive constants $c_{1}$ and $s_{0}$ such that $ʃ_{1}^{s} a (t) / t d t \geq c_{1} b (c_{1} s)$ for all $s \geq s_{0}$ ; (jj) there exist positive constants $c_{2}$ and $c_{3}$ such that $ʃ_{0}^{2 π} Ψ ((c_{2}) / {(| ⨍ |}_{}) | ⨍ (x) |) d x \leq c_{3} + c_{3} ʃ_{0}^{2 π} {Φ (1 / (| ⨍ |}_{})) M f (x) d x$ for all $⨍ \in L^{1} ()$ .

Publié le : 1996-01-01

Zbl 0845.42008

EUDML-ID : urn:eudml:doc:216260

@article{bwmeta1.element.bwnjournal-article-smv118i1p1bwm,
     author = {H. Kita},
     title = {On a converse inequality for maximal functions in Orlicz spaces},
     journal = {Studia Mathematica},
     volume = {119},
     year = {1996},
     pages = {1-10},
     zbl = {0845.42008},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv118i1p1bwm}
}

Kita, H. On a converse inequality for maximal functions in Orlicz spaces. Studia Mathematica, Tome 119 (1996) pp. 1-10. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv118i1p1bwm/

Bibliographie

[00000] [1] M. de Guzmán, Differentiation of Integrals in $ℝ^{n}$ , Lecture Notes in Math. 481, Springer, Berlin, 1975.

[00001] [2] H. Kita, On maximal functions in Orlicz spaces, submitted. | Zbl 0864.42007

[00002] [3] H. Kita and K. Yoneda, A treatment of Orlicz spaces as a ranked space, Math. Japon. 37 (1992), 775-802. | Zbl 0766.46008

[00003] [4] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Sci., 1991. | Zbl 0751.46021

[00004] [5] G. Moscariello, On the integrability of the Jacobian in Orlicz spaces, Math. Japon. 40 (1994), 323-329. | Zbl 0805.46026

[00005] [6] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.

[00006] [7] E. M. Stein, Note on the class L log L, Studia Math. 31 (1969), 305-310. | Zbl 0182.47803

[00007] [8] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1985.

[00008] [9] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1959. | Zbl 0085.05601