Abstract
In this paper, we are interested in the Laguerre hypergroup \(\mathbb{K} = [0,\infty ) \times \mathbb{R}\) which is the fundamental manifold of the radial function space for the Heisenberg group. So, we consider the generalized shift operator generated by the dual of the Laguerre hypergroup which can be topologically identified with the so-called Heisenberg fan, the subset of ℝ2:
by means of which the maximal function is investigated. For 1 < p ≤ ∞, the L p ()-boundedness and weak L 1()-boundedness result for the maximal function is obtained.
Реэюме
В данной работе мы рассматриваем гипергруппу Лагерра \(\mathbb{K} = [0,\infty ) \times \mathbb{R}\), являюшуюся фундаментальным многообраэием пространства радиальных функций для группы Гейэенберга. Таким обраэом, мы рассматриваем оператор обобшенного сдвига, порождённый дуальной гипергруппой Лагерра , которая топологически может быть отождествлена с так наэываемым «вентилятором» Гейэенберга, то есть следуюшим подмножеством на ℝ2:
в терминах которого иэучается максимальная функция. Для 1 < p < оо, получен реэультат L p ()-ограниченности и слабой L 1()-ограниченности для максимальной функции.
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Miloud, A., Atef, R. & Ahmed, T. Maximal function on the dual of Laguerre hypergroup. Anal Math 38, 161–171 (2012). https://doi.org/10.1007/s10476-012-0301-6
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DOI: https://doi.org/10.1007/s10476-012-0301-6