We establish a positive product formula for the solutions of the Sturm–Liouville equation \(\ell (u) = \lambda u\), 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 probability preserving convolution algebra structure on the space of finite measures which satisfies the basic axioms for developing harmonic analysis on the convolution algebra. Unlike previous works, our framework includes a subfamily of Sturm–Liouville operators for which the support of the convolution of Dirac measures is noncompact. The connection with hypergroup theory is discussed. Convolution-type integral equations on weighted Lebesgue spaces are also discussed, and a solvability condition is established.

我们为Sturm–Liouville方程\（\ ell（u）= \ lambda u \）的解建立一个正积公式，其中\（\ ell \）属于一般类，其中包括奇异和简并的Sturm–Liouville算子。我们的技术依靠正定理来解决可能的简并双曲柯西问题，并依靠利用Sturm–Liouville算子生成的扩散半群性质的正则化方法。我们证明了乘积公式在有限度量空间上产生了一个保留概率的卷积代数结构，它满足了在卷积代数上进行调和分析的基本公理。与以前的工作不同，我们的框架包括Sturm–Liouville算子的一个子族，对此Dirac测度的卷积的支持并不紧凑。讨论了与超群理论的联系。还讨论了加权Lebesgue空间上的卷积型积分方程，