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

Abstract

We prove that for any $L^Q$-valued Schwartz function $f$ defined on ${ℝ}^d$, one has the multiple vector-valued, mixed-norm estimate

$$\parallel f{\parallel }_{L^P(L^Q)}\lesssim \parallel Sf{\parallel }_{L^P(L^Q)}$$

valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying componentwise. Here $S:=S_{d_1}\otimes \cdots \otimes S_{d_N}$ is a tensor product of several Littlewood–Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces ${ℝ}^{d_j}$ for $1\le j\le N$, with the property that $d_1+\cdots +d_N=d$. This answers a question that came up implicitly in our recent works [2], [3], [5] and completes in a natural way classical results of Littlewood–Paley theory. The proof is based on the helicoidal method introduced by the authors in the aforementioned papers.