Duality in Harmonic Analysis and beyond
Abstract
In this short note, we develop a duality-based, purely operator-theoretic method for proving several classical maximal inequalities on Lp spaces for p∈(1,∞). The method applies to maximal ergodic operators, Hardy--Littlewood maximal functions, and maximal Radon averaging operators, and derives their boundedness directly from structural properties of the underlying operators. Further, we show that the same duality principle extends beyond maximal functions. We develop a theory of r-variational and jump inequalities for sequences of bounded positive monotone operators on Hilbert spaces, using soft arguments that apply to various classical operators in analysis. In particular, answering a question that had been open for some time, we give direct proofs of r-variational and jump inequalities for convolution operators associated with lacunary dilations of smooth bumps on ℝd, completely avoiding any appeal to martingale inequalities.
Related
BibTeX
@misc{langowski2026duality,
title = {{Duality in Harmonic Analysis and beyond}},
author = {Bartosz Langowski and Mariusz Mirek and James Wright and Błażej Wróbel},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29803},
eprint = {2610.29803},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29803},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Duality in Harmonic Analysis and beyond
AU - Bartosz Langowski
AU - Mariusz Mirek
AU - James Wright
AU - Błażej Wróbel
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29803
UR - https://arxiv.org/abs/2610.29803
AB - In this short note, we develop a duality-based, purely operator-theoretic method for proving several classical maximal inequalities on L^p spaces for p∈(1,∞). The method applies to maximal ergodic operators, Hardy--Littlewood maximal functions, and maximal Radon averaging operators, and derives their boundedness directly from structural properties of the underlying operators. Further, we show that the same duality principle extends beyond maximal functions. We develop a theory of r-variational and jump inequalities for sequences of bounded positive monotone operators on Hilbert spaces, using soft arguments that apply to various classical operators in analysis. In particular, answering a question that had been open for some time, we give direct proofs of r-variational and jump inequalities for convolution operators associated with lacunary dilations of smooth bumps on ℝ^d, completely avoiding any appeal to martingale inequalities.
ER -