The n-linear embedding theorem for dyadic rectangles

2017/10/23 by Hitoshi Tanaka, Tanaka, Hitoshi, Kôzô Yabuta +1 · 1 citation
Mathematics · #42B25 #42B35 #Advanced Harmonic Analysis Research #Differential Equations and Boundary Problems #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1710.08059

openalex publication_date 2017/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let \sg_i, i=1,…,n, denote reverse doubling weights on \R^d, let \cdr(\R^d) denote the set of all dyadic rectangles on \R^d (Cartesian products of usual dyadic intervals) and let K: \cdr(\R^d)→[0,\8) be a~map. In this paper we give the n-linear embedding theorem for dyadic rectangles. That is, we prove the n-linear embedding inequality for dyadic rectangles ∑_{R∈\cdr(\R^d)} K(R)∏_i=1ⁿ\lt|∫_Rf_i \rm d\sg_i\rt| ≤ C ∏_i=1ⁿ ‖f_i‖_L^pi(\sgi) can be characterized by simple testing condition K(R)∏_i=1ⁿ\sg_i(R) ≤ C ∏_i=1ⁿ\sg_i(R)^(1)/(p_i) R∈\cdr(\R^d), in the range 11. As a~corollary to this theorem, for reverse doubling weights, we verify a~necessary and sufficient condition for which the weighted norm inequality for the multilinear strong positive dyadic operator and for strong fractional integral operator to hold.

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