1. Sharp bilinear decomposition for products of both anisotropic Hardy spaces and their dual spaces with its applications to endpoint boundedness of commutators.
- Author
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Liu, Jun, Yang, Dachun, and Zhang, Mingdong
- Abstract
Let a → : = (a 1 , ... , a n) ∈ [ 1 , ∞) n , p ∈ (0 , 1) , and α:= 1/p − 1. For any x ∈ ℝ
n and t ∈ [0, ∞), let Φ p (x , t) : = { t 1 + (t [ x ] a → ν) 1 − p if ν α ∉ ℕ , t 1 + (t [ x ] a → ν) 1 − p [ log (e + ∣ x ∣ a →) ] p if ν α ∈ ℕ , where [ ⋅ ] a → := 1 + ∣ ⋅ ∣ a → , ∣ ⋅ ∣ a → denotes the anisotropic quasi-homogeneous norm with respect to a → , and ν:= a1 +...+ an . Let H a → p (ℝ n) , ℒ α a → (ℝ n) , and H a → Φ p (ℝ n) be, respectively, the anisotropic Hardy space, the anisotropic Campanato space, and the anisotropic Musielak-Orlicz Hardy space associated with Φp on ℝn . In this article, via first establishing the wavelet characterization of anisotropic Campanato spaces, we prove that for any f ∈ H a → p (ℝ n) and g ∈ ℒ α a → (ℝ n) , the product of f and g can be decomposed into S(f, g) + T(f, g) in the sense of tempered distributions, where S is a bilinear operator bounded from H a → p (ℝ n) × ℒ α a → (ℝ n) to L1 (ℝn ) and T is a bilinear operator bounded from H a → p (ℝ n) × ℒ α a → (ℝ n) to H a → Φ p (ℝ n) . Moreover, this bilinear decomposition is sharp in the dual sense that any Y ⊂ H a → Φ p (ℝ n) that fits into the above bilinear decomposition should satisfy (L 1 ( R n) + Y) ∗ = (L 1 ( R n) + H a → Φ p ( R n)) ∗ . As applications, for any non-constant b ∈ ℒ α a → (ℝ n) and any sublinear operator T satisfying some mild bounded assumptions, we find the largest subspace of H a → p (ℝ n) , denoted by H a → , b p (ℝ n) , such that the commutator [b, T] is bounded from H a → , b p (ℝ n) to L1 (ℝn ). In addition, when T is an anisotropic Calderón-Zygmund operator, the boundedness of [b, T] from H a → , b p (ℝ n) to L1 (ℝn )(or to H a → 1 (ℝ n) ) is also presented. The key of their proofs is the wavelet characterization of function spaces under consideration. [ABSTRACT FROM AUTHOR]- Published
- 2024
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