Geometric maximal operators and BMO on product bases

Abstract

We consider the problem of the boundedness of maximal operators on BMO on shapes in R n . 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. 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.