An upper bound for the Menchov-Rademacher operator for right triangles

Vagharshakyan, Armen

doi:10.1090/proc/15950

An upper bound for the Menchov-Rademacher operator for right triangles
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by Armen Vagharshakyan

Proc. Amer. Math. Soc. 150 (2022), 3959-3971

DOI: https://doi.org/10.1090/proc/15950

Published electronically: April 7, 2022

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Abstract:

The Menchov-Rademacher inequality is an inequality in harmonic analysis that bounds the $L_2$ norm of a certain maximal operator. It was first established in order to prove almost everywhere convergence of a one-parameter series of orthogonal functions. When two-parameter series of orthogonal functions is considered, the exact way the series is grouped becomes essential. We will consider grouping of a two-parameter series, generated by a sequence of right triangles with a vertex at the origin, who might be non-equilateral, and prove almost everywhere convergence when the eccentricity of those triangles is bounded. In order to carry out the proof, we will derive an analogue of the Menchov-Rademacher inequality for right triangles.

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