### Abstract

Let L be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that L has a bounded holomorphic functional calculus on L^{2} (R^{n} ). In this paper, we construct a frame decomposition for the functions belonging to the Hardy space H_{L}^{1} (R^{n} ) associated to L, and for functions in the Lebesgue spaces L^{p} , 1<p<∞. We then show that the corresponding H_{L}^{1} (R ^{n} )-norm (resp. L^{p} (R^{n} )-norm) of a function f in terms of the frame coefficients is equivalent to the H_{L}^{1} (R^{n} )-norm (resp. L^{p} (R^{n} )-norm) of f. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space H_{L}^{1} (R^{n} ) under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of L.

Language | English |
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Pages | 45-85 |

Number of pages | 41 |

Journal | Journal of Approximation Theory |

Volume | 243 |

DOIs | |

Publication status | Published - 1 Jul 2019 |

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### Keywords

- Frame decomposition
- Functional calculus
- Gaussian estimate
- Hardy space
- Heat semigroup
- Radial maximal function

### Cite this

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*Journal of Approximation Theory*, vol. 243, pp. 45-85. https://doi.org/10.1016/j.jat.2019.03.006

**Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators.** / Duong, Xuan Thinh; Li, Ji; Song, Liang; Yan, Lixin.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - Frame decomposition and radial maximal semigroup characterization of Hardy spaces associated to operators

AU - Duong, Xuan Thinh

AU - Li, Ji

AU - Song, Liang

AU - Yan, Lixin

PY - 2019/7/1

Y1 - 2019/7/1

N2 - Let L be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that L has a bounded holomorphic functional calculus on L2 (Rn ). In this paper, we construct a frame decomposition for the functions belonging to the Hardy space HL1 (Rn ) associated to L, and for functions in the Lebesgue spaces Lp , 1L1 (R n )-norm (resp. Lp (Rn )-norm) of a function f in terms of the frame coefficients is equivalent to the HL1 (Rn )-norm (resp. Lp (Rn )-norm) of f. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space HL1 (Rn ) under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of L.

AB - Let L be the generator of an analytic semigroup whose kernels satisfy Gaussian upper bounds and Hölder's continuity. Also assume that L has a bounded holomorphic functional calculus on L2 (Rn ). In this paper, we construct a frame decomposition for the functions belonging to the Hardy space HL1 (Rn ) associated to L, and for functions in the Lebesgue spaces Lp , 1L1 (R n )-norm (resp. Lp (Rn )-norm) of a function f in terms of the frame coefficients is equivalent to the HL1 (Rn )-norm (resp. Lp (Rn )-norm) of f. As an application of the frame decomposition, we establish the radial maximal semigroup characterization of the Hardy space HL1 (Rn ) under the extra condition of Gaussian upper bounds on the gradient of the heat kernels of L.

KW - Frame decomposition

KW - Functional calculus

KW - Gaussian estimate

KW - Hardy space

KW - Heat semigroup

KW - Radial maximal function

UR - http://www.scopus.com/inward/record.url?scp=85063409449&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2019.03.006

DO - 10.1016/j.jat.2019.03.006

M3 - Article

VL - 243

SP - 45

EP - 85

JO - Journal of Approximation Theory

T2 - Journal of Approximation Theory

JF - Journal of Approximation Theory

SN - 0021-9045

ER -