Multiple vector-valued, mixed norm estimates for Littlewood-Paley square functions

TL;DR

This paper establishes new multiple vector-valued mixed norm estimates for tensor products of Littlewood-Paley square functions on Euclidean spaces, extending classical Littlewood-Paley theory using the helicoidal method.

Contribution

It proves the boundedness of tensor product Littlewood-Paley square functions in mixed norm spaces for all componentwise positive exponents, answering an open question and extending classical results.

Findings

Validates mixed norm estimates for tensor product square functions.

Completes the classical Littlewood-Paley theory in a new vector-valued setting.

Introduces the helicoidal method for these estimates.

Abstract

We prove that for any $L^Q$-valued Schwartz function $f$ defined on $\mathbb{R}^d$, one has the multiple vector-valued, mixed norm estimate $$ \| f \|_{L^P(L^Q)} \lesssim \| S f \|_{L^P(L^Q)} $$ valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying $0 < P, Q < \infty$ componentwise. Here $S:= S_{d_1}\otimes ... \otimes S_{d_N}$ is a tensor product of several Littlewood-Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces $\mathbb{R}^{d_j}$ for $1\leq j\leq N$, with the property that $d_1 + ... + d_N = d$. This answers a question that came up implicitly in our recent works and completes in a natural way classical results of the Littlewood-Paley theory. The proof is based on the \emph{helicoidal method} introduced by the authors.