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A WEIGHTED MAXIMAL WEAK‐TYPE INEQUALITY
Author(s) -
Osȩkowski Adam,
Rapicki Mateusz
Publication year - 2021
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/mtk.12065
Subject(s) - mathematics , inequality , type (biology) , combinatorics , pure mathematics , discrete mathematics , mathematical analysis , biology , ecology
Let w be a dyadic A p weight ( 1 ⩽ p < ∞ ), and let M D be the dyadic Hardy–Littlewood maximal function on R d . The paper contains the proof of the estimate w { x ∈ R d : M D f ( x ) > w ( x ) } ⩽ C p[ w ] A p∫ R d| f | d x , where the constant C p does not depend on the dimension d . Furthermore, the linear dependence on[ w ] A pis optimal, which is a novel result for 1 < p < ∞ . The estimate is shown to hold in a wider context of probability spaces equipped with an arbitrary tree‐like structure. The proof rests on the Bellman function method: we construct an abstract special function satisfying certain size and concavity requirements.