Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 29-39

THE n-LINEAR EMBEDDING THEOREM FOR DYADIC RECTANGLES

Hitoshi Tanaka and Kôzô Yabuta

National University Corporation Tsukuba University of Technology
Research and Support Center on Higher Education for the Hearing and Visually Impaired
Kasuga 4-12-7, Tsukuba 305-8521, Japan; htanaka 'at' k.tsukuba-tech.ac.jp

Kwansei Gakuin University, Research center for Mathematical Sciences
Gakuen 2-1, Sanda 669-1337, Japan; kyabuta3 'at' kwansei.ac.jp

Abstract. Let σ_i, i = 1,...,n, be reverse doubling weights on R^d, DR(R^d) be the set of all dyadic rectangles on R^d (Cartesian products of usual dyadic intervals) and K : DR(R^d) → [0,∞) be a map. In this paper we give the n-linear embedding theorem for dyadic rectangles. That is, we prove that the n-linear embedding inequality for dyadic rectangles

∑_{R ∈ DR(R^d)} K(R )∏ _{i = 1}ⁿ|∫_Rf_i dσ_i| ≤ C∏_i=1ⁿ ||f_i||_{L^pi(σ_i)}

can be characterized by simple testing condition

K(R)∏_i=1ⁿσ_i(R) ≤ C∏_i=1ⁿσ_i(R)^1/pi R ∈ DR(R^d),

in the range 1 < p_i < ∞ with ∑_i=1ⁿ1/p_i > 1. As a corollary to this theorem, for reverse doubling weights, we verify a necessary and sufficient condition for which weighted norm inequality for multilinear strong positive dyadic operator and for multilinear strong fractional integral operator to hold.

2010 Mathematics Subject Classification: Primary 42B25, 42B35.

Key words: Multilinear strong fractional integral operator, multilinear strong positive dyadic operator, n-linear embedding theorem, reverse doubling weight.

Reference to this article: H. Tanaka and K. Yabuta: The n-linear embedding theorem for dyadic rectangles. Ann. Acad. Sci. Fenn. Math. 44 (2019), 29-39.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4404