Product of functions in BMO and H 1 in non-homogeneous spaces

Abstract

Abstract. Under the assumption that the underlying measure is a nonnegative Radon measure which only satisfies some growth condition and may not be doubling, we define as in [BIJZ] the product of functions in the regular BM O and the atomic block H 1 in the sense of distribution, and show that this product may be split into two parts, one in L 1 and the other in some Hardy-Orlicz space.

Products of Functions in $$\mathrm {BMO}({\mathcal X})$$ BMO ( X ) and $$H^1_\mathrm{at}({\mathcal X})$$ H at 1 ( X ) via Wavelets Over Spaces of Homogeneous Type

“…Let (X , d, µ) be an RD-space. The problem about the product of f ∈ H 1 at (X , d, µ) and g ∈ BMO(X , d, µ) was first studied by Feuto [9]. In [9], Feuto showed that the product of f ∈ H 1 at (X , d, µ) and g ∈ BMO(X , d, µ), viewed as a distribution, can be written as a sum of an integrable function and a distribution in some adapted Hardy-Orlicz space.…”

“…The problem about the product of f ∈ H 1 at (X , d, µ) and g ∈ BMO(X , d, µ) was first studied by Feuto [9]. In [9], Feuto showed that the product of f ∈ H 1 at (X , d, µ) and g ∈ BMO(X , d, µ), viewed as a distribution, can be written as a sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. Recently, Ky [27] improved the above result via showing that the product g × f can be written into a sum of two linear operators and via replacing the Hardy-Orlicz space by H log (X , d, µ) which is a smaller space than the aforementioned Hardy-Orlicz space and is known to be optimal even when X = R D endowed with the D-dimensional Lebesgue measure.…”

Weighted norm inequalities for multilinear strongly singular Calderón–Zygmund operators on RD‐spaces

“…Definition 2.3. The triple (, 𝑑, 𝜇) is called an RD-space, if it is a space of homogeneous type and 𝜇 satisfies the reverse doubling condition: There exist constants 𝜅 ∈ (0, ∞) and 𝐶 2 ∈ (0, 1] such that, for all 𝑥 ∈ , 0 < 𝑟 < 2 diam() and 1 ≤ 𝜆 < 2 diam()∕𝑟, Similar to [14], we will also assume that there exists a positive nondecreasing function 𝜑 defined on [0, ∞) such that for all 𝑥 ∈  and 𝑟 > 0, 𝜇(𝐵(𝑥, 𝑟)) ≈ 𝜑(𝑟).…”