[Paper Review] Operator-valued local Hardy spaces

Runlian Xia, Xiao Xiong|arXiv (Cornell University)|Mar 27, 2018

Advanced Harmonic Analysis Research37 references3 citations

TL;DR

This paper introduces and systematically studies operator-valued local Hardy spaces on $\mathbb{R}^d$, establishing their duality with operator-valued BMO spaces, interpolation properties, and a full atomic decomposition. It extends classical local Hardy space theory to the noncommutative setting using truncated Littlewood-Paley and Lusin area functions, proving that the Poisson kernel can be replaced by any reasonable test function and that the atomic and functional norms are equivalent.

ABSTRACT

This paper gives a systematic study of operator-valued local Hardy spaces. These spaces are localizations of the Hardy spaces defined by Tao Mei, and share many properties with Mei's Hardy spaces. We prove the ${ m h}_1$-$ m bmo$ duality, as well as the ${ m h}_p$-${ m h}_q$ duality for any conjugate pair $(p,q)$ when $1

Motivation & Objective

• To develop a theory of operator-valued local Hardy spaces as a noncommutative, inhomogeneous analogue of classical local Hardy spaces.
• To establish the duality between ${\rm h}_p^c(\mathbb{R}^d,\mathcal{M})$ and ${\rm bmo}_q^c(\mathbb{R}^d,\mathcal{M})$ for $1 \leq p < 2$ and conjugate $q$.
• To show that ${\rm h}_1^c(\mathbb{R}^d,\mathcal{M})$ and ${\rm bmo}^c(\mathbb{R}^d,\mathcal{M})$ are interpolation endpoints for $L_p(L_\infty(\mathbb{R}^d)\overline{\otimes}\mathcal{M})$.
• To develop a Calderón-Zygmund theory adapted to the local setting and prove that the Poisson kernel in the norm definition can be replaced by any reasonable test function.
• To establish the atomic decomposition of ${\rm h}_1^c(\mathbb{R}^d,\mathcal{M})$ and prove equivalence of atomic and functional norms.

Proposed method

• The paper defines operator-valued local Hardy spaces ${\rm h}_p^c(\mathbb{R}^d,\mathcal{M})$ via truncated versions of the Littlewood-Paley $g$-function and Lusin area integral using the Poisson kernel.
• It proves duality by constructing a bounded map from ${\rm bmo}_q^c(\mathbb{R}^d,\mathcal{M})$ to the dual of ${\rm h}_p^c(\mathbb{R}^d,\mathcal{M})$ and showing equivalence for $p=1$ and $2<q<\infty$.
• Interpolation results are derived by reducing the local case to Mei’s non-local interpolation theory via complex and real interpolation methods.
• A local Calderón-Zygmund theory is developed, showing that operators satisfying the Hörmander condition are bounded on ${\rm h}_p^c(\mathbb{R}^d,\mathcal{M})$ under modified conditions.
• The paper uses discrete characterizations involving convolutions with smooth, compactly supported functions $\Phi$ and $\phi$ to control norms in the atomic decomposition.
• The atomic decomposition is proven by showing that ${\rm h}_{1,{\rm at}}^c(\mathbb{R}^d,\mathcal{M}) \subset {\rm h}_1^c(\mathbb{R}^d,\mathcal{M})$ and the reverse inclusion via duality and norm equivalence.

Experimental results

Research questions

• RQ1 Can the classical local Hardy space duality and atomic decomposition be extended to the noncommutative, operator-valued setting?
• RQ2 How do operator-valued local Hardy spaces behave under complex and real interpolation?
• RQ3 Can the Poisson kernel in the definition of the local Hardy norm be replaced by any reasonable test function?
• RQ4 What is the relationship between the atomic and functional characterizations of ${\rm h}_1^c(\mathbb{R}^d,\mathcal{M})$?
• RQ5 How does the Calderón-Zygmund theory adapt to the local, truncated setting in the noncommutative framework?

Key findings

• The paper establishes ${\rm h}_1^c(\mathbb{R}^d,\mathcal{M})$-${\rm bmo}^c(\mathbb{R}^d,\mathcal{M})$ duality, generalizing the classical Fefferman-Stein theorem to the operator-valued local setting.
• For $1 < p < \infty$, the dual of ${\rm h}_p^c(\mathbb{R}^d,\mathcal{M})$ is ${\rm h}_q^c(\mathbb{R}^d,\mathcal{M})$ with $q$ the conjugate index, and ${\rm h}_q^c(\mathbb{R}^d,\mathcal{M}) = {\rm bmo}_q^c(\mathbb{R}^d,\mathcal{M})$ for $2 < q < \infty$.
• The local Hardy spaces ${\rm h}_1^c(\mathbb{R}^d,\mathcal{M})$ and ${\rm bmo}^c(\mathbb{R}^d,\mathcal{M})$ are interpolation endpoints: $({\rm bmo}^c, {\rm h}_1^c)_{1/p} = {\rm h}_p^c$ for $1 < p < \infty$.
• The Poisson kernel in the norm definition of ${\rm h}_p^c(\mathbb{R}^d,\mathcal{M})$ can be replaced by any reasonable test function, as shown via the local Calderón-Zygmund theory.
• The atomic Hardy space ${\rm h}_{1,{\rm at}}^c(\mathbb{R}^d,\mathcal{M})$ coincides with ${\rm h}_1^c(\mathbb{R}^d,\mathcal{M})$ with equivalent norms, establishing a full atomic decomposition.
• The proof of atomic equivalence relies on discrete characterizations and norm estimates using Cauchy-Schwarz and $L_2$-based $\mathcal{M}$-valued norms.