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

2018/08/09 by Cristina Benea, Benea, Cristina, Camil Muscalu +1
Mathematics · #Advanced Harmonic Analysis Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical Approximation and Integration #math.CA

paper · pdf · doi:10.48550/arxiv.1808.03248

38 pages, technical revision of Proposition4.1

openalex publication_date 2018/08/09 · arxiv created 2019/08/06 · arxiv updated 2019/08/08 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28

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 ‖ 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 < ∞ componentwise. Here S:= S_d1⊗ ... ⊗ S_dN is a tensor product of several Littlewood-Paley square functions S_dj defined on arbitrary Euclidean spaces ℝ^d_j for 1≤ j≤ N, with the property that d₁ + ... + 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 helicoidal method introduced by the authors.

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