We introduce three representation formulas for the fractional $p$-Laplace operator in whole range of parameters $0<s<1$ and $1<p<\infty$. Note that $p\ne 2$ this a nonlinear operator. The first is based on splitting procedure combines renormalized nonlinearity with linear heat semigroup. second adapts to Caffarelli-Silvestre extension technique. third one corresponding version Balakrishnan form...

Let k be a finite extension of Qp and Ek be the set of the extensions of degree p over k whose normal closure is a p-extension. For a fixed discriminant, we show how many extensions there are in EQp with such discriminant, and we give the discriminant and the Galois group (together with its filtration of the ramification groups) of their normal closure. We show how this method can be generalize...

Abstract We calculate the extension groups between simple modules of pro- p -Iwahori Hecke algebras.