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On a converse inequality for maximal functions in Orlicz spaces

H. Kita (1996)

Studia Mathematica

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 a ( s ) / s d s = and b(s) is quasi-increasing and l i m 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 d t 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 2 π Ψ ( ( c 2 ) / ( | | ) | ( x ) | ) d x c 3 + c 3 ʃ 0 2 π Φ ( 1 / ( | | ) ) M f ( x ) d x for all L 1 ( ) .

On a decomposition of non-negative Radon measures

Bérenger Akon Kpata (2019)

Archivum Mathematicum

We establish a decomposition of non-negative Radon measures on d which extends that obtained by Strichartz [6] in the setting of α -dimensional measures. As consequences, we deduce some well-known properties concerning the density of non-negative Radon measures. Furthermore, some properties of non-negative Radon measures having their Riesz potential in a Lebesgue space are obtained.

On a gap series of Mark Kac

Katusi Fukuyama (1999)

Colloquium Mathematicae

Mark Kac gave an example of a function f on the unit interval such that f cannot be written as f(t)=g(2t)-g(t) with an integrable function g, but the limiting variance of n - 1 / 2 k = 0 n - 1 f ( 2 k t ) vanishes. It is proved that there is no measurable g such that f(t)=g(2t)-g(t). It is also proved that there is a non-measurable g which satisfies this equality.

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