On directional maximal operators in higher dimensions

2012/05/16 by Javier Parcet, Keith M. Rogers, Parcet, Javier +1
Mathematics · #26B05 #42B25 #Advanced Banach Space Theory #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.1205.3606

openalex publication_date 2012/05/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce a notion of (finite order) lacunarity in higher dimensions for which we can bound the associated directional maximal operators in L^p(ℝⁿ), with p>1. In particular, we are able to treat the classes previously considered by Nagel--Stein--Wainger, Sjögren--Sjölin and Carbery. Closely related to this, we find a characterisation of the sets of directions which give rise to bounded maximal operators. The bounds enable Lebesgue type differentiation of integrals in L_loc^p(ℝⁿ), replacing balls by tubes which point in these directions.

Citations

• The Multiplier Problem for the Ball

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