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Semigroup and Riesz transform for the Dunkl–Schrödinger operators
Semigroup Forum ( IF 0.7 ) Pub Date : 2020-05-28 , DOI: 10.1007/s00233-020-10106-5
Béchir Amri , Amel Hammi

Let $$L_k=-\Delta _k+V$$ L k = - Δ k + V be the Dunkl–Schrödinger operators, where $$\Delta _k=\sum _{j=1}^dT_j^2$$ Δ k = ∑ j = 1 d T j 2 is the Dunkl Laplace operator associated to the Dunkl operators $$T_j$$ T j on $$\mathbb {R}^d$$ R d and V is a nonnegative potential function. In the first part of this paper we introduce the Riesz transform $$R_j= T_j L_k^{-1/2}$$ R j = T j L k - 1 / 2 as a $$L^2$$ L 2 -bounded operator and we first prove that it is of weak type (1, 1) and then that it is bounded on $$L^p(\mathbb {R}^d,d\mu _k(x))$$ L p ( R d , d μ k ( x ) ) for $$1

中文翻译:

Dunkl–Schrödinger 算子的半群和 Riesz 变换

令 $$L_k=-\Delta _k+V$$ L k = - Δ k + V 为 Dunkl–Schrödinger 算子,其中 $$\Delta _k=\sum _{j=1}^dT_j^2$$ Δ k = ∑ j = 1 d T j 2 是与在 $$\mathbb {R}^d$$ R d 上的 Dunkl 算子 $$T_j$$ T j 相关联的 Dunkl Laplace 算子,V 是非负势函数。在本文的第一部分中,我们介绍了 Riesz 变换 $$R_j= T_j L_k^{-1/2}$$ R j = T j L k - 1 / 2 作为 $$L^2$$ L 2 -有界算子,我们首先证明它是弱类型 (1, 1) 然后证明它有界于 $$L^p(\mathbb {R}^d,d\mu _k(x))$$ L p ( R d , d μ k ( x ) ) 1 美元
更新日期:2020-05-28
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