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Directional square functions

Natalia Accomazzo, Francesco Di Plinio, Paul Hagelstein, Ioannis Parissis and Luz Roncal

Vol. 16 (2023), No. 7, 1651–1699

Quantitative formulations of Fefferman’s counterexample for the ball multiplier are naturally linked to square function estimates for conical and directional multipliers. We develop a novel framework for these square function estimates, based on a directional embedding theorem for Carleson sequences and multiparameter time-frequency analysis techniques. As applications we prove sharp or quantified bounds for Rubio-de Francia-type square functions of conical multipliers and of multipliers adapted to rectangles pointing along N directions. A suitable combination of these estimates yields a new and currently best-known logarithmic bound for the Fourier restriction to an N-gon, improving on previous results of A. Córdoba. Our directional Carleson embedding extends to the weighted setting, yielding previously unknown weighted estimates for directional maximal functions and singular integrals.

directional operators, directional square functions, Rubio de Francia inequalities, directional Carleson embedding theorems, polygon multiplier
Mathematical Subject Classification
Primary: 42B20
Secondary: 42B25
Received: 28 June 2021
Revised: 9 December 2021
Accepted: 24 January 2022
Published: 21 September 2023
Natalia Accomazzo
Department of Mathematics
University of British Columbia
Vancouver, BC
Francesco Di Plinio
Dipartimento di Matematica e Applicazioni
Università di Napoli
Paul Hagelstein
Department of Mathematics
Baylor University
Waco, TX
United States
Ioannis Parissis
Departamento de Matemáticas
Universidad del País Vasco
Basque Foundation for Science
Luz Roncal
BCAM - Basque Center for Applied Mathematics
Basque Foundation for Science

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