By Ortega R., Tarallo M.

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**Additional info for Almost periodic equations and conditions of Ambrosetti-Prodi type**

**Sample text**

1) K Suppose that the functional equation (1) holds and let PROOF. u,ve L1(K\G/K). Then (u)4(v)= fGJfGu(g)v(g)0(g)0(h)dhdg J = JIGJGu(g)v(h)O(gkh)dhdgdk= fGJfGu(g)v(h)o(gh)dhdg J 1)dh= Vv*u). I J (We have used the fact that a K-bi-invariant function has the = I fG u(g)v(g lh)¢(h)dgdh= J same value at g and g-1. ) Conversely, if 7(v*u)=4(uWv), then The Gelfand pair (G,K) I 49 fG v(h)u(gh 1)O(g)dgdh=JGJGv(h)u(g)O(h)¢(g)dgdh, J GJ which implies, for every kEK, fGJ fGv(h)u(gh'k 1)0(g)dgdh= fGJ Gv(h)u(g)o(gkh)dhdg J J = fGJfGv(h)u(g)0(h)O(g)dgdh.

GnK that x acts transitively on Q. Therefore GnK acts transitively on the sets x Six={y: d(x,y)=n}, for every n. n Let yEG(x), and let n=d(x,y). Then SixcG(x). Therefore G(x) is the union G(x)= U Sixn n , where E nEE is a set of nonnegative integers. Let k be the smallest integer other than zero occurring in E, Furthermore K and let Then yEG(x). is a conjugate subgroup of K nG. Therefore K nG x y acts transitively on 0 and on the sets Siy={z: d(z,y)=n}. This n means that every element having distance k from y belongs to 28 G(x).

There exists hEG, In the first case, if gEr, such that g(x)=h(x), which implies g-1 hEK x and gEG. Thus G contains every faithful transitive subgroup and it must be all of Aut (If) . In the second case G contains r+ and therefore G=Aut(3f)+ because Aut(X)+ has index 2 in Aut(30 (in fact Aut(30+={gEAut(X): d(x,g(x)) is even}). ' We remark that the proposition above implies that Aut(3f)+ is generated by Kx and a nontrivial rotation not belonging to K . In particular, if {a,b} is an edge, Aut(X)+ is generated by x the subgroups Ka and Kb.

### Almost periodic equations and conditions of Ambrosetti-Prodi type by Ortega R., Tarallo M.

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