[Paper Review] Weak Hardy Spaces $WH_L^p({\mathbb R}^n)$ Associated to Operators Satisfying $k$-Davies-Gaffney Estimates

Jun Cao, Der‐Chen Chang+2 more|arXiv (Cornell University)|Aug 25, 2013

Advanced Harmonic Analysis Research31 references7 citations

TL;DR

This paper introduces weak Hardy spaces $WH_L^p(\mathbb{R}^n)$ associated with operators $L$ satisfying $k$-Davies-Gaffney estimates, establishing a weak molecular characterization and proving boundedness of Riesz transforms and fractional powers between these spaces. The key contribution is identifying the dual space of $WH_L^p(\mathbb{R}^n)$ via mean oscillations, a result previously unknown even for the Laplacian.

ABSTRACT

Let $L$ be a one-to-one operator of type $ω$ having a bounded $H_\infty$ functional calculus and satisfying the $k$-Davies-Gaffney estimates with $k\in{\mathbb N}$. In this paper, the authors introduce the weak Hardy space $WH_L^p(\mathbb{R}^n)$ associated to $L$ for $p\in (0,\,1]$ via the non-tangential square function $S_L$ and establish a weak molecular characterization of $WH_L^p(\mathbb{R}^n)$. Typical examples of such operators include the $2k$-order divergence form homogeneous elliptic operator $L_1:=(-1)^k\sum_{|α|=k=|β|}\partial^β(a_{α,β}\partial^α)$, where $\{a_{α,β}\}_{|α|=k=|β|}$ are complex bounded measurable functions, and the $2k$-order Schrödinger type operator $L_2:= (-Δ)^k+V^k$, where $Δ$ is the Laplacian operator and $0\le V\in L^k_{\mathop\mathrm{loc}}(\mathbb{R}^n)$. As applications, for $i\in\{1,\,2\}$ and $p\in(\frac{n}{n+k},\,1]$, the authors prove that the associated Riesz transform $ abla^k (L_i^{-1/2})$ is bounded from $WH^p_{L_i}(\mathbb{R}^n)$ to the classical weak Hardy space $WH^p(\mathbb{R}^n)$ and, for all $0

Motivation & Objective

• To define and characterize weak Hardy spaces $WH_L^p(\mathbb{R}^n)$ associated with operators $L$ satisfying $k$-Davies-Gaffney estimates for $p \in (0,1]$.
• To establish a weak molecular characterization of $WH_L^p(\mathbb{R}^n)$ using the non-tangential square function $S_L$.
• To prove boundedness of the Riesz transform $\nabla^k L^{-1/2}$ from $WH_L^p(\mathbb{R}^n)$ to the classical weak Hardy space $WH^p(\mathbb{R}^n)$ for $p \in (\frac{n}{n+k}, 1]$.
• To establish an interpolation theorem showing $L^2(\mathbb{R}^n) \cap WH_L^p(\mathbb{R}^n)$ are intermediate spaces in real interpolation between $L^2(\mathbb{R}^n) \cap H_L^p(\mathbb{R}^n)$ for different $p \in (0,1]$.
• To identify the dual space of $WH_L^p(\mathbb{R}^n)$ for $p \in (0,1]$, defined via mean oscillations based on subtle coverings, extending to cases where $L$ is a nonnegative self-adjoint operator satisfying Davies-Gaffney estimates.

Proposed method

• Define $WH_L^p(\mathbb{R}^n)$ via the non-tangential square function $S_L(f)(x) = \left(\int_{\Gamma(x)} |t^k \nabla^k e^{-t^{2k}L}f(y)|^2 \frac{dy\,dt}{t^{n+1}}\right)^{1/2}$.
• Establish a weak molecular characterization by constructing molecules adapted to the operator $L$, satisfying size, cancellation, and square function decay conditions.
• Use real interpolation theory to show that $L^2(\mathbb{R}^n) \cap WH_L^p(\mathbb{R}^n)$ are intermediate spaces between $L^2(\mathbb{R}^n) \cap H_L^p(\mathbb{R}^n)$ for varying $p \in (0,1]$.
• Employ subtle covering arguments based on dyadic cubes and annuli to define mean oscillation spaces $W\Lambda_L^{\alpha}(\mathbb{R}^n)$ and relate them to the dual of $WH_L^p(\mathbb{R}^n)$.
• Prove boundedness of $\nabla^k L^{-1/2}$ from $WH_L^p(\mathbb{R}^n)$ to $WH^p(\mathbb{R}^n)$ using weak-type estimates and square function domination.
• Establish boundedness of the fractional power $L^{-\alpha/(2k)}$ from $WH_L^p(\mathbb{R}^n)$ to $WH_L^r(\mathbb{R}^n)$ for $0 < p < r \leq 1$ and $\alpha = n(1/p - 1/r)$ via interpolation and functional calculus.

Experimental results

Research questions

• RQ1 How can weak Hardy spaces $WH_L^p(\mathbb{R}^n)$ be defined and characterized for operators $L$ satisfying $k$-Davies-Gaffney estimates?
• RQ2 What is the weak molecular characterization of $WH_L^p(\mathbb{R}^n)$, and how does it relate to the square function $S_L$?
• RQ3 Is the Riesz transform $\nabla^k L^{-1/2}$ bounded from $WH_L^p(\mathbb{R}^n)$ to the classical weak Hardy space $WH^p(\mathbb{R}^n)$, and for which $p$?
• RQ4 Can the dual space of $WH_L^p(\mathbb{R}^n)$ be identified, and what is its structure in terms of mean oscillations?
• RQ5 How do the spaces $WH_L^p(\mathbb{R}^n)$ behave under real interpolation, and what intermediate spaces do they generate?

Key findings

• The Riesz transform $\nabla^k L^{-1/2}$ is bounded from $WH_L^p(\mathbb{R}^n)$ to $WH^p(\mathbb{R}^n)$ for all $p \in (\frac{n}{n+k}, 1]$.
• The fractional power $L^{-\alpha/(2k)}$ is bounded from $WH_L^p(\mathbb{R}^n)$ to $WH_L^r(\mathbb{R}^n)$ for $0 < p < r \leq 1$ and $\alpha = n(1/p - 1/r)$.
• The space $L^2(\mathbb{R}^n) \cap WH_L^p(\mathbb{R}^n)$ is the intermediate space in the real interpolation method between $L^2(\mathbb{R}^n) \cap H_L^p(\mathbb{R}^n)$ for different $p \in (0,1]$.
• The dual space of $WH_L^p(\mathbb{R}^n)$ for $p \in (0,1]$ is identified as $W\Lambda_L^{n(1/p - 1)}(\mathbb{R}^n)$, defined via mean oscillations over coverings of bounded open sets.
• Even for the Laplacian ($L = -\Delta$), the dual space of $WH_L^p(\mathbb{R}^n)$ is newly identified as $W\Lambda_L^{n(1/p - 1)}(\mathbb{R}^n)$, resolving a previously unknown case.
• The dual space characterization is independent of the parameter $\mathcal{N}$ as long as $\mathcal{N} > n(1/p - 1/2)$, allowing the notation $W\Lambda_L^{\alpha}(\mathbb{R}^n)$ without dependence on $\mathcal{N}$.