Mixed-norm estimates via the helicoidal method

TL;DR

This paper develops new mixed-norm estimates for multilinear operators with singular symbols in higher dimensions, expanding the understanding of their boundedness properties especially when involving $L^ olinebreak ext{infinity}$ spaces and non-isotropic operators.

Contribution

It introduces novel mixed-norm estimates for multilinear singular integral operators and variants of the Hardy-Littlewood maximal function, extending previous methods to non-isotropic and purely mixed-norm contexts.

Findings

Established mixed-norm bounds for multilinear operators with singular symbols.

Derived Loomis-Whitney-type inequalities for singular integrals in mixed-norm spaces.

Provided examples of operators with purely mixed-norm estimates, not reducible to classical $L^p$ bounds.

Abstract

We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $\rr R^d$, more precisely for multilinear operators $T_k$ associated to a symbol singular along a $k$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $d \geq 2$, the input functions are not necessarily in $L^p(\rr R^d)$ and can instead be elements of mixed-norm spaces $L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$. Such a result has interesting consequences especially when $L^\infty$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type inequalities for singular integrals, as well as the boundedness of multilinear operators associated to certain rational symbols. We also present examples of operators that are not susceptible to isotropic rescaling, which only satisfy ``purely mixed-norm estimates" and no classical $L^p$ estimates. Relying on previous estimates implied by the helicoidal method, we also prove (non-mixed-norm) estimates for generic singular Brascamp-Lieb-type inequalities.