论文标题
Riesz转换和与部分谐波振荡器相关的Sobolev空间
Riesz transforms and Sobolev spaces associated to the partial harmonic oscillator
论文作者
论文摘要
在本文中，我们的目标是建立与部分谐波振荡器相关的Sobolev空间。根据其热核的估计，我们首先给出了部分谐波振荡器$$ \ ah \ ah = - \partial_ρ^2-Δ_x+| x |^2的定义，并证明其负功率在$ l^p（\ nathb r r^{d+1} $ p for $ p for $ p（$ p）上。然后，我们定义关联的Riesz转换，并表明它们通过符号的演算在经典的Sobolev空间上界定。 其次，通过对操作员$ \ ah $的分解，我们定义了两个具有正整数索引的Sobolev空间家族，并通过Riesz Transforms的界限显示了它们之间的等价性。此外，改编后的符号演算还意味着Riesz类型在与部分谐波振荡器$ \ ah $相关的Sobolev空间上转换的界限。 最后，作为我们结果的应用，我们获得了修订后的Hardy-Littlewood-Sobolev不平等，Gagliardo-Nirenberg-Sobolev不平等现象以及Hardy在潜在空间中的不平等$ l _ {\ ah}^{α{α，P} $。
In this paper, our goal is to establish the Sobolev space associated to the partial harmonic oscillator. Based on its heat kernel estimate, we firstly give the definition of the fractional powers of the partial harmonic oscillator $$\AH=-\partial_ρ^2-Δ_x+|x|^2,$$ and show that its negative powers are well defined on $L^p(\mathbb R^{d+1})$ for $p\in [1,\infty]$. We then define associated Riesz transforms and show that they are bounded on classical Sobolev spaces by the calculus of symbols. Secondly, by a factorization of the operator $\AH$, we define two families of Sobolev spaces with positive integer indices, and show the equivalence between them by the boundedness of Riesz transforms. Moreover, the adapted symbolic calculus also implies the boundedness of Riesz type transforms on the Sobolev spaces associated to the partial harmonic oscillator $\AH$. Lastly, as applications of our results, we obtain the revised Hardy-Littlewood-Sobolev inequality, the Gagliardo-Nirenberg-Sobolev inequality, and Hardy's inequality in the potential space $L_{\AH}^{α, p}$.
