论文标题
光谱乘数定理用于UMD晶格上的抽象谐波振荡器
Spectral multiplier theorems for abstract harmonic oscillators on UMD lattices
论文作者
论文摘要
我们考虑在UMD Banach Lattice $ X $上作用的操作员，其代数结构与与谐波振荡器$ - \frac12δ+ \ frac12 | x |^{2} $相同的位置和动量运算符相同。更准确地说，我们考虑表单$ \ frac12 \ sum _ {j = 1} ^{d}（a_ {j} ^{2}+b_ {j} ^{2} ^{2}）$的抽象谐波振荡器（a_ {j} ^{2}+b_ {2} ^{2}） $ b =（b_ {k}）_ {k = 1} ^{d} $，其中$ ia_j $和$ ib_k $被假定生成$ c_ {0} $组并满足规范的换向器关系。我们证明了这些抽象的谐波振荡器的功能性计算结果，这些谐波振荡器与谐波振荡器$ - \frac12Δ+ \ frac12 | x | x |^{2} $ on $ l^{p}（p}（\ sathbb {rathbb {r}^d}^d}）$匹配经典的Hörmander光谱乘数估计值。这涵盖了基础度量空间并没有加倍的情况，并且使用功能空间并不特别适合于外推参数。例如，作为应用程序，我们在混合规范的Bargmann-Fock空间上处理谐波振荡器。我们的方法基于海森堡集团的Schrödinger代表的转移原理，该原理使我们能够将问题减少到Bochner空间上的扭曲的Laplacian的研究$ l^{2}（\ Mathbb {r}^r}^{2d} {2d}; x）$。这可以看作是对不是希尔伯特空间的umd lattices $ x $的石冯·诺伊曼定理的概括。
We consider operators acting on a UMD Banach lattice $X$ that have the same algebraic structure as the position and momentum operators associated with the harmonic oscillator $-\frac12Δ+ \frac12|x|^{2} $ acting on $L^{2}(\mathbb{R}^{d})$. More precisely, we consider abstract harmonic oscillators of the form $\frac12 \sum _{j=1} ^{d}(A_{j}^{2}+B_{j}^{2})$ for tuples of operators $A=(A_{j})_{j=1} ^{d}$ and $B=(B_{k})_{k=1} ^{d}$, where $iA_j$ and $iB_k$ are assumed to generate $C_{0}$ groups and to satisfy the canonical commutator relations. We prove functional calculus results for these abstract harmonic oscillators that match classical Hörmander spectral multiplier estimates for the harmonic oscillator $-\frac12Δ+ \frac12|x|^{2}$ on $L^{p}(\mathbb{R}^{d})$. This covers situations where the underlying metric measure space is not doubling and the use of function spaces that are not particularly well suited to extrapolation arguments. For instance, as an application we treat the harmonic oscillator on mixed norm Bargmann-Fock spaces. Our approach is based on a transference principle for the Schrödinger representation of the Heisenberg group that allows us to reduce the problem to the study of the twisted Laplacian on the Bochner spaces $L^{2}(\mathbb{R}^{2d};X)$. This can be seen as a generalisation of the Stone-von Neumann theorem to UMD lattices $X$ that are not Hilbert spaces.
