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 ${\mathbb{R}}^d$, one has the multiple vector-valued, mixed norm estimate $$\Vert f{\Vert }_{L^P(L^Q)}\lesssim \Vert Sf{\Vert }_{L^P(L^Q)}$$ 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 $1\le j\le N$, 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.

38 pages, technical revision of Proposition4.1