Fundamental solutions of pseudo-differential operators over \(p\)-adic fields (Q2568766)

Let \(K\) be a \(p\)-adic field. The author proves the following theorem: Every \(p\)-adic pseudodifferential equation \(f(\partial,\beta)u=g\), with \(f(x)\in K[x_1,\ldots,x_n]\setminus K\), \(g\in{\mathcal S}(K^n)\), and \(\beta\in C\), \(\Re\, \beta >0\), has a fundamenntal solution \(E\in{\mathcal S}^{\prime}(K^n)\). Such \(p\)-adic pseudodifferential operators occur naturally in \(p\)-adic quantum field theory. The connection between work by Igusa on the local zeta function \(G(x)\) and \(p\)-adic pseudodifferential equations is used in the proof of the theorem.