Our paper begins with a revision of the spectral theory for commutative Banach algebras, which enables us to prove the \(L^p_{\omega }\)-conjecture for locally compact abelian groups. We follow an alternative approach to the one known in the literature. In particular, we do not resort to any structural theorems for locally compact groups at this stage. Subsequently, we discuss nilpotent, locally compact groups. The climax of the paper is the proof of the \(L^p_{\omega }\)-conjecture for these groups.

中文翻译：

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局部紧幂零组上的加权$$ L ^ p $$ Lp-空间

我们的文章从对交换Banach代数的谱理论的修正开始，这使我们能够证明局部紧的阿贝尔群的\（L ^ p _ {\ omega} \）猜想。对于文献中已知的方法，我们采用了另一种方法。特别是，在此阶段，我们不对局部紧凑型群体求助于任何结构定理。随后，我们讨论了无能的局部紧凑型群。本文的高潮是这些群体\（L ^ p _ {\ omega} \）猜想的证明。