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Carleson measure characterizations of the Campanato type space associated with Schrödinger operators on stratified Lie groups
Forum Mathematicum ( IF 0.8 ) Pub Date : 2020-09-01 , DOI: 10.1515/forum-2019-0224 Yixin Wang 1 , Yu Liu 2 , Chuanhong Sun 1 , Pengtao Li 1
Forum Mathematicum ( IF 0.8 ) Pub Date : 2020-09-01 , DOI: 10.1515/forum-2019-0224 Yixin Wang 1 , Yu Liu 2 , Chuanhong Sun 1 , Pengtao Li 1
Affiliation
Abstract Let ℒ = - Δ 𝔾 + V {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} be a Schrödinger operator on the stratified Lie group 𝔾 {\mathbb{G}} , where Δ 𝔾 {{\Delta}_{\mathbb{G}}} is the sub-Laplacian and the nonnegative potential V belongs to the reverse Hölder class B q 0 {B_{q_{0}}} with q 0 > 𝒬 / 2 {q_{0}>\mathcal{Q}/2} and 𝒬 {\mathcal{Q}} is the homogeneous dimension of 𝔾 {\mathbb{G}} . In this article, by Campanato type spaces Λ ℒ α ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} , we introduce Hardy type spaces associated with ℒ {\mathcal{L}} denoted by H ℒ p ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} and prove the atomic characterization of H ℒ p ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} . Further, we obtain the following duality relation: Λ ℒ 𝒬 ( 1 / p - 1 ) ( 𝔾 ) = ( H ℒ p ( 𝔾 ) ) ∗ , 𝒬 / ( 𝒬 + δ ) < p < 1 for δ = min { 1 , 2 - 𝒬 / q 0 } . \Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{\mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)
中文翻译：
与 Schrödinger 算子在分层李群上相关的 Campanato 型空间的 Carleson 测度特征
摘要 令 ℒ = - Δ 𝔾 + V {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} 是分层李群 𝔾 {\mathbb{G}} 上的薛定谔算子，其中 Δ 𝔾 {{\Delta}_{\mathbb{G}}} 是亚拉普拉斯算子，非负势 V 属于反向 Hölder 类 B q 0 {B_{q_{0}}} 且 q 0 > 𝒬 / 2 {q_{0}>\mathcal{Q}/2} 和 𝒬 {\mathcal{Q}} 是 𝔾 {\mathbb{G}} 的齐次维度。在本文中，通过 Campanato 类型空间 Λ ℒ α ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} ，我们介绍了与 ℒ {\ mathcal{L}} 表示为 H ℒ p ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} 并证明原子H ℒ p ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} 的表征。进一步，我们得到以下对偶关系：Λ ℒ 𝒬 ( 1 / p - 1 ) ( 𝔾 ) = ( H ℒ p ( 𝔾 ) ) ∗ , 𝒬 / ( 𝒬 + δ ) < p < 1 for ⁒ δ { - 𝔾 = min , ❔ / q 0 } 。\Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{ \mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)
更新日期：2020-09-01
中文翻译：
与 Schrödinger 算子在分层李群上相关的 Campanato 型空间的 Carleson 测度特征
摘要 令 ℒ = - Δ 𝔾 + V {\mathcal{L}=-{\Delta}_{\mathbb{G}}+V} 是分层李群 𝔾 {\mathbb{G}} 上的薛定谔算子，其中 Δ 𝔾 {{\Delta}_{\mathbb{G}}} 是亚拉普拉斯算子，非负势 V 属于反向 Hölder 类 B q 0 {B_{q_{0}}} 且 q 0 > 𝒬 / 2 {q_{0}>\mathcal{Q}/2} 和 𝒬 {\mathcal{Q}} 是 𝔾 {\mathbb{G}} 的齐次维度。在本文中，通过 Campanato 类型空间 Λ ℒ α ( 𝔾 ) {\Lambda^{\alpha}_{\mathcal{L}}(\mathbb{G})} ，我们介绍了与 ℒ {\ mathcal{L}} 表示为 H ℒ p ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} 并证明原子H ℒ p ( 𝔾 ) {H^{{p}}_{\vphantom{\varepsilon}{\mathcal{L}}}(\mathbb{G})} 的表征。进一步，我们得到以下对偶关系：Λ ℒ 𝒬 ( 1 / p - 1 ) ( 𝔾 ) = ( H ℒ p ( 𝔾 ) ) ∗ , 𝒬 / ( 𝒬 + δ ) < p < 1 for ⁒ δ { - 𝔾 = min , ❔ / q 0 } 。\Lambda_{\mathcal{L}}^{\mathcal{Q}(1/p-1)}(\mathbb{G})=(H^{{p}}_{\vphantom{% \varepsilon}{ \mathcal{L}}}(\mathbb{G}))^{\ast},\quad\mathcal{Q}/(\mathcal{Q}+% \delta)
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