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The trilinear embedding theorem

Hitoshi Tanaka — 2015

Studia Mathematica

Let σ i , i = 1,2,3, denote positive Borel measures on ℝⁿ, let denote the usual collection of dyadic cubes in ℝⁿ and let K: → [0,∞) be a map. We give a characterization of a trilinear embedding theorem, that is, of the inequality Q K ( Q ) i = 1 3 | Q f i d σ i | C i = 1 3 | | f i | | L p i ( d σ i ) in terms of a discrete Wolff potential and Sawyer’s checking condition, when 1 < p₁,p₂,p₃ < ∞ and 1/p₁ + 1/p₂ + 1/p₃ ≥ 1.

The John-Nirenberg type inequality for non-doubling measures

Yoshihiro SawanoHitoshi Tanaka — 2007

Studia Mathematica

X. Tolsa defined a space of BMO type for positive Radon measures satisfying some growth condition on d . This new BMO space is very suitable for the Calderón-Zygmund theory with non-doubling measures. Especially, the John-Nirenberg type inequality can be recovered. In the present paper we introduce a localized and weighted version of this inequality and, as applications, we obtain some vector-valued inequalities and weighted inequalities for Morrey spaces.

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