Triangular maximal operators on locally finite trees

2023/12/11 by Stefano Meda, Meda, Stefano, Federico Santagati +1 · 1 citation
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2312.06595

openalex publication_date 2023/12/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We introduce the centred and the uncentred triangular maximal operators \mathcal T and \mathcal U, respectively, on any locally finite tree in which each vertex has at least three neighbours. We prove that both \mathcal T and \mathcal U are bounded on L^p for every p in (1,∞], that \mathcal T is also bounded on L¹(\mathfrak T), and that \mathcal U is not of weak type (1,1) on homogeneous trees. Our proof of the L^p boundedness of \mathcal U hinges on the geometric approach of A. Córdoba and R. Fefferman. We also establish L^p bounds for some related maximal operators. Our results are in sharp contrast with the fact that the centred and the uncentred Hardy--Littlewood maximal operators (on balls) may be unbounded on L^p for every p<∞ even on some trees where the number of neighbours is uniformly bounded.

Cited by

• Harmonic Bergman spaces on locally finite trees

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