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Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group

Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the... AbstractThis paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group Hn{{\mathbb{H}}}^{n}. A sharp strong estimate for TΦm{T}_{\Phi }^{m}is obtained. As an application, we derive the sharp constant for the product Hardy operator on Hn{{\mathbb{H}}}^{n}. Some weak-type (p,q)\left(p,q)(1≤p≤∞)\left(1\le p\le \infty )estimates for TΦ,β{T}_{\Phi ,\beta }are also obtained. As applications, we calculate some sharp weak constants for the fractional Hausdorff operator on the Heisenberg group. Besides, we give an explicit weak estimate for TΦ,β→m{T}_{\Phi ,\overrightarrow{\beta }}^{m}under some mild assumptions on Φ\Phi . We extend the results of Guo et al. [Hausdorff operators on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714] to the fractional setting. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

Weak and strong estimates for linear and multilinear fractional Hausdorff operators on the Heisenberg group

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References (21)

Publisher
de Gruyter
Copyright
© 2021 Yangkendi Deng et al., published by De Gruyter
ISSN
2391-5455
eISSN
2391-5455
DOI
10.1515/math-2021-0016
Publisher site
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Abstract

AbstractThis paper is devoted to the weak and strong estimates for the linear and multilinear fractional Hausdorff operators on the Heisenberg group Hn{{\mathbb{H}}}^{n}. A sharp strong estimate for TΦm{T}_{\Phi }^{m}is obtained. As an application, we derive the sharp constant for the product Hardy operator on Hn{{\mathbb{H}}}^{n}. Some weak-type (p,q)\left(p,q)(1≤p≤∞)\left(1\le p\le \infty )estimates for TΦ,β{T}_{\Phi ,\beta }are also obtained. As applications, we calculate some sharp weak constants for the fractional Hausdorff operator on the Heisenberg group. Besides, we give an explicit weak estimate for TΦ,β→m{T}_{\Phi ,\overrightarrow{\beta }}^{m}under some mild assumptions on Φ\Phi . We extend the results of Guo et al. [Hausdorff operators on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 11, 1703–1714] to the fractional setting.

Journal

Open Mathematics – de Gruyter

Published: May 19, 2021

Keywords: Hausdorff operator; Heisenberg group; multilinear; sharp bound; 42B20; 42B35

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