Sobolev空间上偏代数的广群刻画
Groupoid Characterization of Partial Algebras on Sobolev Spaces
AI总结：
本研究利用李广群框架刻画了在光滑代数𝒦(Ω)偏作用下不变的Sobolev空间W^{k,p}(Ω)，分析了其上偏代数与偏动力系统的关联，并通过酉表示验证了该刻画的简化作用。
摘要：
当p≠∞时，L^p空间构成带有函数逐点乘法的偏代数(L^p(Ω), Γ, ·)。作为L^p空间中由弱导数定义的子空间，Sobolev空间W^{k,p}(Ω)被证明包含由光滑代数𝒦(Ω)通过卷积在Banach空间L^p(Ω)上的偏作用推广而来的偏代数(L^p(Ω), Γ, ·)。我们利用李广群框架刻画了在𝒦(Ω)偏作用下不变的Sobolev空间W^{k,p}(Ω)，并将该偏代数作为定义与弱微分算子相关的L^p空间上偏动力系统的对象进行研究。局部凸偏*-代数(L^p(Ω), Γ, ·, *)定义了与李广群的局部二分截面对应的稳定局部流。所得李广群𝒲⇉W^{k,p}(Ω)在相关希尔伯特丛上的酉表示证明了该刻画带来的简化效果。
英文摘要：
The $L^p$-spaces, with $p \not = \infty$, form a partial algebra $(L^p(Ω), Γ, \cdot)$ with pointwise multiplication of functions. The Sobolev spaces $W^{k,p}(Ω)$, delineated by weak derivatives as subspaces of $L^p$-spaces is shown to contain the partial algebra $(L^p(Ω), Γ, \cdot)$ generalized by the partial action of the smooth algebra $\mathscr{K}(Ω)$ by convolution on the Banach spaces $L^p(Ω)$. We characterised the Sobolev space $W^{k,p}(Ω)$, invariant under $\mathscr{K}(Ω)$ partial action, using Lie groupoid framework, and study the partial algebra as defining the partial dynamical systems on the $L^p$-space associated with the weak differential operators. The locally convex partial $^*$-algebra $(L^p(Ω), Γ, \cdot,^*)$ defines the stable local flows coinciding with local bisections of the Lie groupoid. The unitary representation of resulting Lie groupoid $\mathscr{W} \rightrightarrows W^{k,p}(Ω)$ on the associated Hilbert bundle demonstrates the simplification achieved by the characterisation.
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