Geometric Maximal Operators and $$\mathrm {{BMO}}{}{}{}$$ BMO on Product Bases
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Geometric Maximal Operators and BMO on Product Bases Galia Dafni1 · Ryan Gibara2 · Hong Yue3 Received: 17 July 2020 © Mathematica Josephina, Inc. 2020
Abstract We consider the problem of the boundedness of maximal operators on BMO on shapes in Rn . We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO. Keywords Geometric maximal operator · Bounded mean oscillation Mathematics Subject Classification 42B25 · 42B35
1 Introduction The uncentred Hardy–Littlewood maximal function, M f , of a function f ∈ L 1loc (Rn ) is defined as G.D. was partially supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, the Centre de recherches mathématiques (CRM), and the Fonds de recherche du Québec – Nature et technologies (FRQNT). R.G. was partially supported by the Centre de recherches mathématiques (CRM), the Institut des sciences mathématiques (ISM), and the Fonds de recherche du Québec – Nature et technologies (FRQNT). H.Y. was partially supported by GCSU Faculty Development Funds.
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Ryan Gibara [email protected] Galia Dafni [email protected] Hong Yue [email protected]
1
Department of Mathematics and Statistics, Concordia University, Montréal, QC H3G 1M8, Canada
2
Département de mathématiques et de statistique, Université Laval, Québec, QC G1V 0A6, Canada
3
Department of Mathematics, Georgia College and State University, Milledgeville, GA 31061, USA
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1 M f (x) = sup − | f | = sup | f |, Qx Q Qx |Q| Q
(1.1)
where the supremum is taken over all cubes Q containing the point x and |Q| is the measure of the cube. Note that, unless otherwise stated, cubes in this paper will mean cubes with sides parallel to the axes. The well-known Hardy–Littlewood–Wiener theorem states that the operator M is bounded from L p (Rn ) to L p (Rn ) for 1 < p ≤ ∞ and from L 1 (Rn ) to L 1,∞ (Rn ) (see Stein [29]). This maximal function is a classical object of study in real analysis due to its connection with differentiation of the integral. When the cubes in (1.1) are replaced by rectangles (the Cartesian product of intervals), we have the strong maximal function, Ms , which is also bounded from L p (Rn ) to L p (Rn ) for 1 < p ≤ ∞ but is not bounded from L 1 (Rn ) to L 1,∞ (Rn ). Its connection to what is known as strong differentiation of the integral is also quite classical (see Jessen–Marcinkiewicz–Zygmund [20]). When the cubes in (1.1) are replaced by more general sets taken from a basis S , we obtain a geometric maximal operator, MS (we follow the nomenclature of [18]). Here, the subscript S emphasizes that the behaviour of this opera
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