Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators

Nguyen Ngoc, Trong; Le Xuan, Truong; Tan Duc, Do

doi:10.5802/crmath.213

Analyse fonctionnelle, Analyse harmonique

Boundedness of second-order Riesz transforms on weighted Hardy and

B M O

spaces associated with Schrödinger operators

Nguyen Ngoc, Trong ¹ ; Le Xuan, Truong ¹ ; Tan Duc, Do ¹

¹ University of Economics Ho Chi Minh City, Vietnam

Comptes Rendus. Mathématique, Tome 359 (2021) no. 6, pp. 687-717.

Résumé

Let $d \in {3, 4, 5, ...}$ and a weight $w \in A_{\infty}^{ρ}$ . We consider the second-order Riesz transform $T = \nabla^{2} L^{- 1}$ associated with the Schrödinger operator $L = - Δ + V$ , where $V \in R H_{σ}$ with $σ > \frac{d}{2}$ . We present three main results. First $T$ is bounded on the weighted Hardy space $H_{w, L}^{1} (ℝ^{d})$ associated with $L$ if $w$ enjoys a certain stable property. Secondly $T$ is bounded on the weighted $B M O$ space $B M O_{w, ρ} (ℝ^{d})$ associated with $L$ if $w$ also belongs to an appropriate doubling class. Thirdly $B M O_{w, ρ} (ℝ^{d})$ is the dual of $H_{w, L}^{1} (ℝ^{d})$ when $w \in A_{1}^{ρ}$ .

Reçu le : 2020-11-11
Révisé le : 2021-03-08
Accepté le : 2021-04-12
Publié le : 2021-08-25

DOI : 10.5802/crmath.213

Classification : 30H35, 42B20, 42B25, 42B30, 42B37

Affiliations des auteurs :

Nguyen Ngoc, Trong ¹ ; Le Xuan, Truong ¹ ; Tan Duc, Do ¹

¹ University of Economics Ho Chi Minh City, Vietnam

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     author = {Nguyen Ngoc, Trong and Le Xuan, Truong and Tan Duc, Do},
     title = {Boundedness of second-order {Riesz} transforms on weighted {Hardy} and $BMO$ spaces associated with {Schr\"odinger} operators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {687--717},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {359},
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     year = {2021},
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     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/crmath.213/}
}

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Nguyen Ngoc, Trong; Le Xuan, Truong; Tan Duc, Do. Boundedness of second-order Riesz transforms on weighted Hardy and $BMO$ spaces associated with Schrödinger operators. Comptes Rendus. Mathématique, Tome 359 (2021) no. 6, pp. 687-717. doi : 10.5802/crmath.213. http://archive.numdam.org/articles/10.5802/crmath.213/

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