[Paper Review] Maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type

Anh Tuan Bui, Xuan Thinh Duong+1 more|arXiv (Cornell University)|Sep 1, 2016

Advanced Harmonic Analysis Research28 references3 citations

TL;DR

This paper establishes maximal function characterizations for new local Hardy type spaces on spaces of homogeneous type associated with a nonnegative self-adjoint operator 𝔏 satisfying Gaussian estimates. It introduces a critical function ρ and proves nontangential and radial maximal function characterizations for local Hardy spaces linked to 𝔏 and ρ, extending Coifman-Weiss theory and enabling applications to Schrödinger operators on manifolds and nilpotent Lie groups.

ABSTRACT

Let $X$ be a space of homogeneous type and let $\mathfrak{L}$ be a nonnegative self-adjoint operator on $L^2(X)$ enjoying Gaussian estimates. The main aim of this paper is twofold. Firstly, we prove the (local) nontangential and radial maximal function charaterization for the local Hardy spaces associated to $\mathfrak{L}$. This deduces the maximal function charaterization for local Hardy spaces in the sense of Coifman and Weiss provided that $\mathfrak{L}$ satisfies certain extra conditions. Secondly, we introduce the local Hardy space associated to the critical function $ ho$ which is motivated by the theory of Hardy spaces related to Schrodinger operators and includes the local Hardy spaces of Coifman and Weiss as a special case. Then we prove that these local Hardy spaces can be characterized by the the local nontangential and radial maximal function charaterization related to $\mathfrak{L}$ and $ ho$ and the global maximal function charaterizations associated to `perturbations' of $\mathfrak{L}$. As applications, we apply our theory to obtain a number of new results on maximal characterizations for the local Hardy type spaces in various settings ranging from Shrodinger operators on manifolds to Shrodinger operators on connected and simply connected nilpotent Lie groups.

Motivation & Objective

• To establish nontangential and radial maximal function characterizations for local Hardy spaces associated with a nonnegative self-adjoint operator 𝔏 on a space of homogeneous type.
• To extend the Coifman-Weiss local Hardy space theory by proving maximal function characterizations under additional conditions on 𝔏.
• To introduce a new local Hardy space based on a critical function ρ, generalizing the Coifman-Weiss framework.
• To prove global and local maximal function characterizations for these new Hardy spaces using perturbations of 𝔏.
• To apply the theory to Schrödinger operators on manifolds and connected, simply connected nilpotent Lie groups.

Proposed method

• Utilize Gaussian upper bounds for the heat kernel associated with 𝔏 to control the growth of functions in the Hardy space setting.
• Define the local Hardy space via the critical function ρ derived from the operator 𝔏, generalizing the classical size condition in Coifman-Weiss theory.
• Establish nontangential and radial maximal function characterizations using maximal operators adapted to the geometry of the space and the operator 𝔏.
• Introduce perturbations of 𝔏 to derive global maximal function characterizations, linking local and global behavior.
• Apply the theory to specific geometric settings, including Riemannian manifolds and nilpotent Lie groups, via the structure of 𝔏 and the associated heat semigroup.
• Use the theory of singular integrals and maximal functions in the context of spaces of homogeneous type to prove boundedness and equivalence of norms.

Experimental results

Research questions

• RQ1 Can nontangential and radial maximal functions characterize local Hardy spaces associated with a nonnegative self-adjoint operator 𝔏 on a space of homogeneous type?
• RQ2 Under what conditions does the maximal function characterization of Coifman and Weiss extend to operators 𝔏 with Gaussian estimates?
• RQ3 How can a critical function ρ be used to define a generalized local Hardy space that includes the Coifman-Weiss space as a special case?
• RQ4 What is the relationship between local and global maximal function characterizations when perturbations of 𝔏 are introduced?
• RQ5 To what extent can this theory be applied to Schrödinger operators on manifolds and nilpotent Lie groups?

Key findings

• The local Hardy space associated with the critical function ρ is characterized by both nontangential and radial maximal functions related to the operator 𝔏.
• The maximal function characterization of Coifman and Weiss is recovered when additional conditions on 𝔏 are satisfied, extending their theory to a broader class of operators.
• Global maximal function characterizations are established for perturbations of 𝔏, linking local and global behavior in the Hardy space framework.
• The theory applies to Schrödinger operators on Riemannian manifolds, yielding new maximal function characterizations in that setting.
• The framework extends to connected, simply connected nilpotent Lie groups, providing maximal function characterizations for Hardy spaces in these non-compact, homogeneous settings.
• The critical function ρ plays a central role in defining the size and decay properties of functions in the new Hardy space, generalizing the classical Lebesgue space-based size conditions.