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An $H^1$ multiplier theorem on anisotropic spaces

Yiyu Tang

TL;DR

The paper establishes an anisotropic version of the Sledd-Stegenga H^1 to L^p multiplier theorem for Hardy spaces $H^1_A$ associated with a dilation matrix $A$. It proves a sharp, two-regime criterion: for $1\le p<2$, a discrete $\ell^{2/(2-p)}$-type bound on $\mu$ over translated anisotropic rectangles, and for $2\le p<\infty$, a finite measure condition on anisotropic spheres $\{x: \rho_*(x)=b^k\}$. The sufficiency parts combine atomic decomposition, Fourier decay of atoms, and anisotropic covering arguments, while the necessity parts rely on Bochner–Riesz-type constructions and square-function techniques to extract necessary bounds. Together, these results extend the classical multiplier theory to anisotropic Hardy spaces, providing explicit, practically usable criteria for $H^1_A\to L^p$ boundedness in terms of the measure $\mu$ and the dilation geometry.

Abstract

A parallel result of (the classical) Sledd--Stegenga's $H^1\rightarrow L^1$ multiplier theorem was obtained on the $H^1$ space under the anisotropic settings. Based on the same technique, an $H^1\rightarrow L^p$ multiplier theorem is also proved for $1\leq p<\infty$.

An $H^1$ multiplier theorem on anisotropic spaces

TL;DR

The paper establishes an anisotropic version of the Sledd-Stegenga H^1 to L^p multiplier theorem for Hardy spaces associated with a dilation matrix . It proves a sharp, two-regime criterion: for , a discrete -type bound on over translated anisotropic rectangles, and for , a finite measure condition on anisotropic spheres . The sufficiency parts combine atomic decomposition, Fourier decay of atoms, and anisotropic covering arguments, while the necessity parts rely on Bochner–Riesz-type constructions and square-function techniques to extract necessary bounds. Together, these results extend the classical multiplier theory to anisotropic Hardy spaces, providing explicit, practically usable criteria for boundedness in terms of the measure and the dilation geometry.

Abstract

A parallel result of (the classical) Sledd--Stegenga's multiplier theorem was obtained on the space under the anisotropic settings. Based on the same technique, an multiplier theorem is also proved for .

Paper Structure

This paper contains 17 sections, 11 theorems, 124 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgement
  3. Preliminaries
  4. Dilation matrix and its homogeneous norm
  5. Anisotropic $H^1$, and the atomic decomposition
  6. Duality, and anisotropic $\mathrm{BMO}$
  7. Use of notations and abbreviations
  8. Main theorem and its proof
  9. Proof of the sufficient part
  10. Proof of the necessary part
  11. Further results
  12. Proof of Theorem \ref{['The H^1 to L^p boundedness criterion']}, when $1\leq p<2$
  13. The sufficient part
  14. The necessary part
  15. Proof of Theorem \ref{['The H^1 to L^p boundedness criterion']}, when $2\leq p<\infty$
  16. ...and 2 more sections

Key Result

Proposition 3.1

(see Lemarie-Rieusset, Appendix B) Suppose that $\rho_A$ is a homogenous quasi-norm associated with the matrix $A$, then there is a constant $c_A>0$, so that Here $\zeta_{\pm}\coloneqq \ln\lambda_{\pm}/\ln b$, and $c_A$ only depends on $\zeta_{\pm}$.

Figures (3)

  • Figure 1:
  • Figure 2: The rectangle $y+\frac{1}{2}R^\ast$ contains $r_1$, whenever $y\in r_1^\prime$.
  • Figure :

Theorems & Definitions (17)

  • Proposition 3.1
  • Proposition 3.2
  • Theorem 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Remark 1
  • Proposition 4.4
  • Remark 2
  • proof : proof of Proposition \ref{['Prop: control of the ell^2 norm over the translation of cubes']}
  • Proposition 4.5
  • ...and 7 more