Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 215-226

THE MAXIMAL BEURLING TRANSFORM ASSOCIATED WITH SQUARES

Anna Bosch-Camós, Joan Mateu and Joan Orobitg

Universitat Autònoma de Barcelona, Departament de Matemàtiques
08193, Bellaterra, Barcelona, Catalonia; annaboschcamos 'at' gmail.com

Universitat Autònoma de Barcelona, Departament de Matemàtiques
08193, Bellaterra, Barcelona, Catalonia; mateu 'at' mat.uab.cat

Universitat Autònoma de Barcelona, Departament de Matemàtiques
08193, Bellaterra, Barcelona, Catalonia; orobitg 'at' mat.uab.cat

Abstract. It is known that the improved Cotlar's inequality B^∗f(z) ≤ CM(Bf)(z), z ∈ C, holds for the Beurling transform B, the maximal Beurling transform B^∗f(z) = sup_ε>0 |∫_|w|>εf(z - w) 1/w² dw|, z ∈ C, and the Hardy–Littlewood maximal operator M. In this note we consider the maximal Beurling transform associated with squares, namely, B^∗_Sf(z) = sup_ε>0 |∫_w∉Q(0,ε)f(z - w) 1/w² dw|, z ∈ C, Q(0,ε) being the square with sides parallel to the coordinate axis of side length ε. We prove that B^∗_Sf(z) ≤ CM²(Bf)(z), z ∈ C, where M² = M o M is the iteration of the Hardy–Littlewood maximal operator, and that M² cannot be replaced by M.

2010 Mathematics Subject Classification: Primary 42B20, 42B25.

Key words: Cotlar's inequality, maximal Beurling transform, Calderón–Zygmund operators.

Reference to this article: A. Bosch-Camós, J. Mateu and J. Orobitg: The maximal Beurling transform associated with squares. Ann. Acad. Sci. Fenn. Math. 40 (2015), 215-226.

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doi:10.5186/aasfm.2015.4016