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On a converse inequality for maximal functions in Orlicz spaces

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Kita H.

Studia Mathematica

1996

tom 118

nr 1

1-10

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 $⨍ ∈ L^{1}(𝕋)$.

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1 Studia Mathematica

1 Kita H.

1 1996

Wyniki wyszukiwania

On a converse inequality for maximal functions in Orlicz spaces