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Characterizations of Hardy spaces on tube domains over polyhedral cones

Zunwei Fu, Loukas Grafakos, Wei Wang, Qingyan Wu

Abstract

This paper is devoted to the equivalence of various characterizations of holomorphic $H^1$ Hardy spaces on tube domains over polyhedral cones. We establish a new iterated Poisson integral formula which reproduces holomorphic functions on such domains. However, this formula shows that holomorphic $H^1$ functions have boundary values in a new type of Hardy space of real variables on their Shilov boundaries $\mathbb{R}^n$, which cannot be treated by standard classical multi-parameter harmonic analysis. We overcome this difficulty by developing techniques suitably adapted in this setting. Using the iterated Poisson integral as our approximation to the identity, and employing a lifting technique, we introduce various notions of multi-parameter analysis adapted to tube domains, such as twisted rectangles, new non-tangential approach regions, non-tangential maximal functions and Littlewood-Paley type functions. All these notions exhibit new geometric features associated with polyhedral cones and involve hidden parameters, as in the flag setting. We develop the necessary multi-parameter tools to investigate these new Hardy spaces. In particular, we apply these tools to obtain equivalent characterizations of the holomorphic $H^1$ Hardy spaces on tube domains in terms of non-tangential maximal, Lusin-Littlewood-Paley area and Littlewood-Paley $g$-functions.

Characterizations of Hardy spaces on tube domains over polyhedral cones

Abstract

This paper is devoted to the equivalence of various characterizations of holomorphic Hardy spaces on tube domains over polyhedral cones. We establish a new iterated Poisson integral formula which reproduces holomorphic functions on such domains. However, this formula shows that holomorphic functions have boundary values in a new type of Hardy space of real variables on their Shilov boundaries , which cannot be treated by standard classical multi-parameter harmonic analysis. We overcome this difficulty by developing techniques suitably adapted in this setting. Using the iterated Poisson integral as our approximation to the identity, and employing a lifting technique, we introduce various notions of multi-parameter analysis adapted to tube domains, such as twisted rectangles, new non-tangential approach regions, non-tangential maximal functions and Littlewood-Paley type functions. All these notions exhibit new geometric features associated with polyhedral cones and involve hidden parameters, as in the flag setting. We develop the necessary multi-parameter tools to investigate these new Hardy spaces. In particular, we apply these tools to obtain equivalent characterizations of the holomorphic Hardy spaces on tube domains in terms of non-tangential maximal, Lusin-Littlewood-Paley area and Littlewood-Paley -functions.
Paper Structure (18 sections, 30 theorems, 225 equations, 2 figures)

This paper contains 18 sections, 30 theorems, 225 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Partial convolutions along lines and partial Laplace equations
  4. A formula of integration by parts
  5. New maximal functions associated to twisted rectangles and iterated Poisson integral formula
  6. Twisted rectangles, iterated and twisted maximal functions
  7. Boundary growth estimates for iterated Poisson integrals and holomorphic $H^p$-functions
  8. Proof of iterated Poisson integral formula and harmonic control
  9. The maximal function characterization
  10. Reduction of the Fefferman-Stein type good-$\lambda$ inequality to an integral estimate
  11. Estimates of integrals by differential identities
  12. Estimates in the general case
  13. Proof of the Fefferman-Stein type good-$\lambda$ inequality
  14. Littlewood-Paley $g$ function and $S$ functions
  15. The Plancherel-Pólya type inequality for $\mathfrak{H}$-valued functions
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose that $T_\Omega$ is a tube domain over a polyhedral cone as above. For $F \in H^p(T_\Omega)$ with $p>1$, there exists $F^b\in L^p(\mathbb{R}^n)$ such that for all $( x,\mathbf{t})\in\mathbb{R}^n\times(\mathbb{R}_+)^m$. Moreover, if $F$ is continuous on $\overline{T_\Omega}$, then $F^b=\lim_{\mathbf{t}\rightarrow 0} F ( x+\mathbf{i}\pi(\mathbf{t}))$.

Figures (2)

  • Figure 1: Polyhedral cone
  • Figure 2: Domains $W_\beta$ and $\widetilde{ W}_\beta$

Theorems & Definitions (59)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 3.1
  • ...and 49 more