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An algebra associated with the generalized sublaplacian. (English) Zbl 0672.46025

Summary: We construct and investigate a commutative Banach algebra associated with the differential operator \[ L=-(\frac{\partial^ 2}{\partial x^ 2}+\frac{2\alpha -1}{x}\frac{\partial}{\partial x}+x^ 2\frac{\partial ^ 2}{\partial t^ 2}),\quad \alpha \geq 1, \] acting on \({\mathbb{R}}_+\times {\mathbb{R}}\). The construction has been inspired by the existence of the well-known algebras of integrable radial functions on the Heisenberg groups. In consequence we also describe an example of Urbanik’s generalized convolution which lives on the semigroup \({\mathbb{R}}_+\times {\mathbb{R}}\cup \{(0,0)\}\).

MSC:

46J10 Banach algebras of continuous functions, function algebras
60B99 Probability theory on algebraic and topological structures
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