A Pseudo-Differential Operator of the $p$-Adic Multiplicative Calculus

We introduce and study a pseudo-differential operator $W^{\alpha }$, defined via the $p$-adic Mellin transformation, acting on real- or complex-valued functions on the multiplicative group of the field of $p$-adic numbers. The operator $W^{\alpha }$ is maximally dissipative. An estimate of its heat kernel is given.