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arXiv · 1901.11021

Sturm-Liouville hypergroups without the compactness axiom

Rúben Sousa·Manuel Guerra·Semyon Yakubovich
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Abstract

We establish a positive product formula for the solutions of the Sturm-Liouville equation $\ell(u) = λu$, where $\ell$ belongs to a general class which includes singular and degenerate Sturm-Liouville operators. Our technique relies on a positivity theorem for possibly degenerate hyperbolic Cauchy problems and on a regularization method which makes use of the properties of the diffusion semigroup generated by the Sturm-Liouville operator. We show that the product formula gives rise to a convolution algebra structure on the space of finite measures, and we discuss whether this structure satisfies the basic axioms of the theory of hypergroups. We introduce the notion of a degenerate hypergroup of full support and improve the known existence theorems for Sturm-Liouville hypergroups. Convolution-type integral equations on weighted Lebesgue spaces are also introduced, and a solvability condition is established.

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Rúben Sousa, Manuel Guerra, Semyon Yakubovich. 2019-03-04. Sturm-Liouville hypergroups without the compactness axiom. https://arxiv.org/abs/1901.11021

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