Ryan Alvarado

Hardy Spaces on Ahlfors-Regular Quasi Metric Spaces

Introduction. - Geometry of Quasi-Metric Spaces.- Analysis on Spaces of Homogeneous Type.- Maximal Theory of Hardy Spaces.- Atomic Theory of Hardy Spaces.- Molecular and Ionic Theory of Hardy Spaces.- Further Results.- Boundedness of Linear Operators Defined on Hp(X).- Besov and Triebel-Lizorkin Spaces on Ahlfors-Regular Quasi-Metric Spaces.

Atomic Theory of Hardy Spaces

We have seen in Sect. 4.3 that \(H_{\alpha }^{p}(X)\) and \(\tilde{H}_{\alpha }^{p}(X)\) can be identified with L p (X) whenever p ∈ (1, ∞] and \(\alpha \in {\bigl ( 0,[\mathrm{log}_{2}C_{\rho }]^{-1}\bigr ]}\).

Molecular and Ionic Theory of Hardy Spaces

This chapter is dedicated to the exploration of the molecular and ionic theory of H p (X) in the setting of d-AR spaces. As a motivation for this topic, suppose one is concerned with the behavior of a bounded linear operator T: L 2(X, μ) → L 2(X, μ).

Geometry of Quasi-Metric Spaces

The main goal of this chapter is to set the stage for the rest of this monograph by presenting a brief survey of some of the many facets of the theory of quasi-metric spaces. Quasi-metric spaces constitute generalizations of not only the classical Euclidean setting, but of quasi-Banach spaces and ultrametric spaces. In this work, quasi-metric spaces will constitute the natural geometric context in which our main results are going to be developed.

Analysis on Spaces of Homogeneous Type

The main goal of this chapter is to rework, in a sharp and relatively self-contained fashion, some of the most fundamental tools used in the area of analysis on quasi-metric spaces. Many of the results presented in this section are of independent interest and will be found useful in a plethora of subsequent applications.

Maximal Theory of Hardy Spaces

The main goal of this chapter is to introduce Hardy spaces in the context of d-Ahlfors-regular quasi-metric spaces by defining H p (X) as a collection of distributions whose maximal belongs to L p (X). This is in the spirit of the pioneering work of C.

Boundedness of Linear Operators Defined on H p ( X )

The main goal of this chapter is to identify criteria guaranteeing that a given linear operator \(T: L^{q}(X,\mu ) \rightarrow \mathcal{B}_{1}\) with q ≥ 1, extends as a bounded operator \(T: H^{p}(X) \rightarrow \mathcal{B}_{2}\) for p as in ( 7.262).

Besov and Triebel-Lizorkin Spaces on Ahlfors-Regular Quasi-Metric Spaces

The 1960s and 1970s saw the birth of a new scale of spaces in the Euclidean setting known as Besov spaces, \(B_{s}^{p,q}\big(\mathbb{R}^{d}\big)\), and Triebel-Lizorkin spaces, \(F_{s}^{p,q}\big(\mathbb{R}^{d}\big)\), where the parameters \(s \in \mathbb{R}\) and p, q ∈ (0, ∞] measure the “smoothness” and, respectively, the “size” of a given distribution in these spaces.

Sharp Geometric Maximum Principles for Semi-Elliptic Operators with Singular Drift

We discuss a sharp generalization of the Hopf-Oleinik boundary point principle (BPP) for domains satisfying an interior pseudo-ball condition, for non-divergence form, semi-elliptic operators with ...