Sparse domination via the helicoidal method

TL;DR

This paper demonstrates how the helicoidal method can be used to derive sparse estimates for complex operators, including vector-valued extensions and Fourier multipliers with singular symbols, advancing harmonic analysis techniques.

Contribution

It introduces a novel approach to obtain sparse domination using localized estimates within the helicoidal method framework, applicable to multiple operators.

Findings

Sparse domination achieved for vector-valued operator extensions

Applied to an n-linear Fourier multiplier with singular symbol

Extended to the variational Carleson operator

Abstract

Using exclusively the localized estimates upon which the helicoidal method was built, we show how sparse estimates can also be obtained. This approach yields a sparse domination for multiple vector-valued extensions of operators as well. We illustrate these ideas for an $n$-linear Fourier multiplier whose symbol is singular along a $k$-dimensional subspace of $\Gamma=\lbrace \xi_1+\ldots+\xi_{n+1}=0 \rbrace$, where $k<\dfrac{n+1}{2}$, and for the variational Carleson operator.