On a converse inequality for maximal functions in Orlicz spaces

• Autorzy: H. Kita
• Języki publikacji: EN

Abstrakty

EN

Let $Φ(t) = ʃ_{0}^{t} a(s)ds$ and $Ψ(t) = ʃ_{0}^{t} b(s)ds$, where a(s) is a positive continuous function such that $ʃ_{1}^{∞} a(s)/s ds = ∞$ and b(s) is quasi-increasing and $lim_{s→∞}b(s) = ∞$. 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 $ʃ_{1}^{s} a(t)/t dt ≥ c_{1}b(c_{1}s)$ for all $s ≥ s_{0}$; (jj) there exist positive constants $c_2$ and $c_3$ such that $ʃ_{0}^{2π} Ψ((c_2)/(|⨍|_{𝕋}) |⨍(x)|) dx ≤ c_3 + c_{3} ʃ_{0}^{2π} Φ(1/(|⨍|_{𝕋})) Mf(x) dx$ for all $⨍ ∈ L^{1}(𝕋)$.

Słowa kluczowe

EN

Hardy-Littlewood maximal function; Orlicz space

Kategorie tematyczne

• 46E30: Spaces of measurable functions ( L p -spaces, Orlicz spaces, K\"othe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
• 42B25: Maximal functions, Littlewood-Paley theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 118
• Numer: 1
• Strony: 1-10

Daty

wydano

1996

otrzymano

1993-12-28

poprawiono

1994-07-15

Twórcy

autor

H. Kita

• Department of Mathematics, Faculty of Education, Oita University, 700 Dannoharu, Oita 870-11, Japan

Bibliografia

• [1] M. de Guzmán, Differentiation of Integrals in $ℝ^n$, Lecture Notes in Math. 481, Springer, Berlin, 1975.
• [2] H. Kita, On maximal functions in Orlicz spaces, submitted.
• [3] H. Kita and K. Yoneda, A treatment of Orlicz spaces as a ranked space, Math. Japon. 37 (1992), 775-802.
• [4] V. Kokilashvili and M. Krbec, Weighted Inequalities in Lorentz and Orlicz Spaces, World Sci., 1991.
• [5] G. Moscariello, On the integrability of the Jacobian in Orlicz spaces, Math. Japon. 40 (1994), 323-329.
• [6] M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker, 1991.
• [7] E. M. Stein, Note on the class L log L, Studia Math. 31 (1969), 305-310.
• [8] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, Academic Press, New York, 1985.
• [9] A. Zygmund, Trigonometric Series, Cambridge Univ. Press, Cambridge, 1959.