Sparse domination via the helicoidal method

Abstract

Using exclusively the localized estimates upon which the helicoidal method was built by the authors, we show how sparse estimates can also be obtained. This approach yields a sparse domination for scalar and multiple vector-valued extensions of operators alike. We illustrate these ideas for an $n$-linear Fourier multiplier whose symbol is singular along a $k$-dimensional subspace of $\Gamma =\{{\xi }_1+\cdots +{\xi }_{n+1}=0\}$, where $k<(n+1)/2$, and for the variational Carleson operator.