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

Cristina Benea, Camil Muscalu

• We prove that for any LQ-valued Schwartz function f defined on Rd, one has the multiple vector-valued, mixed-norm estimate kfkLP (LQ) . kSfkLP (LQ) valid for every d-tuple P and every n-tuple Q satisfying 0 < P, Q < ∞ componentwise. Here S := Sd1 ⊗ · · · ⊗ SdN is a tensor product of several Littlewood–Paley square functions Sdj defined on arbitrary Euclidean spaces R dj for 1 ≤ j ≤ N, with the property that d1 + · · · + dN = 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.