  We introduce the notion of a family of convolution operators associated with
a given elliptic partial differential operator. Such a convolution structure is
shown to exist for a general class of Laplace-Beltrami operators on
two-dimensional manifolds endowed with cone-like metrics. This structure gives
rise to a convolution semigroup representation for the Markovian semigroup
generated by the Laplace-Beltrami operator.
  In the particular case of the operator $\mathcal{L} = \partial_x^2 + {1 \over
2x} \partial_x + {1 \over x} \partial_\theta^2$ on $\mathbb{R}^+ \times
\mathbb{T}$, we deduce the existence of a convolution structure for a
two-dimensional integral transform whose kernel and inversion formula can be
written in closed form in terms of confluent hypergeometric functions. The
results of this paper can be interpreted as a natural extension of the theory
of one-dimensional generalized convolutions to the framework of multiparameter
eigenvalue problems.
