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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 Fourier asymptotics of a generalized Cantor measure

Bérenger Akon KpataIbrahim FofanaKonin Koua — 2010

Colloquium Mathematicae

Let d be a positive integer and μ a generalized Cantor measure satisfying μ = j = 1 m a j μ S j - 1 , where 0 < a j < 1 , j = 1 m a j = 1 , S j = ρ R + b j with 0 < ρ < 1 and R an orthogonal transformation of d . Then ⎧1 < p ≤ 2 ⇒ ⎨ s u p r > 0 r d ( 1 / α ' - 1 / p ' ) ( J x r | μ ̂ ( y ) | p ' d y ) 1 / p ' D ρ - d / α ' , x d , ⎩ p = 2 ⇒ infr≥1 rd(1/α’-1/2) (∫J₀r|μ̂(y)|² dy)1/2 ≥ D₂ρd/α’ , where J x r = i = 1 d ( x i - r / 2 , x i + r / 2 ) , α’ is defined by ρ d / α ' = ( j = 1 m a j p ) 1 / p and the constants D₁ and D₂ depend only on d and p.

Necessary condition for measures which are ( L q , L p ) multipliers

Bérenger Akon KpataIbrahim FofanaKonin Koua — 2009

Annales mathématiques Blaise Pascal

Let G be a locally compact group and ρ the left Haar measure on G . Given a non-negative Radon measure μ , we establish a necessary condition on the pairs q , p for which μ is a multiplier from L q G , ρ to L p G , ρ . Applied to n , our result is stronger than the necessary condition established by Oberlin in [14] and is closely related to a class of measures defined by Fofana in [7]. When G is the circle...

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