Locally Vladimirov operators on the $p$-adic points of a Mumford curve

After an introduction to Mumford curves as a kind of p-adic analogue of Riemann surfaces, new psedodifferential operators generalising the Vladimirov operator are introduced. They induce a wavelet decomposition of the Hilbert space of functions on the $p$-adic rational points on a Mumford curve. The Cauchy problem for the heat equation on those points of Mumford curves, in a special case is also addressed.