A unified construction of product formulas and convolutions for Sturm–Liouville operators

Abstract

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.