Mean value theorems on symmetric spaces

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Mean value theorems on symmetric spaces François Rouvière to Sigur?ur Helgason on his 85th birthday. Abstract. Revisiting some mean value theorems by F. John, respectively S. Helgason, we study their extension to general Riemannian symmetric spaces, resp. their restatement in a more detailed form, with emphasis on their relation to the in?nitesimal structure of the symmetric space. 1. An old formula by Fritz John In his inspiring 1955 book Plane waves and spherical means [7], Fritz John considers the mean value operator on spheres in the Euclidean space Rn: (1.1) MXu(p) = Z K u(p + k X)dk = Mxu(p) with x = kXk where X; p 2 Rn, u is a continuous function on Rn, dk is the normalized Haar measure on the orthogonal group K = SO(n) and dot denotes the natural action of this group on Rn. This average of u over the sphere with center p and radius x = kXk (the Euclidean norm of Rn) only depends on p and this radius; it may be written Mxu(p) as well. For X;Y; p 2 Rn the iterated spherical mean is (1.2) MXMY u(p) = Z K MX+kY u(p)dk, as easily checked.

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Lie bracket Continuous function

Mean value theorems on symmetric spaces

François Rouvière

to Sigurur Helgason on his 85th birthday.

Abstract.Revisiting some mean value theorems by F. John, respectively S. Helgason, we study their extension to general Riemannian symmetric spaces, resp. their restatement in a more detailed form, with emphasis on their relation to the innitesimal structure of the symmetric space.

An old formula by Fritz John

In his inspiring 1955 bookPlane waves and spherical means[7], Fritz John n considers the mean value operator on spheres in the Euclidean spaceR: Z X x (1.1)M u(p) =u(p+kX)dk=M u(p)withx=kXk K n n whereX; p2R,uis a continuous function onR,dkis the normalized Haar measure on the orthogonal groupK=SO(n)and dot denotes the natural action n of this group onRaverage of. This uover the sphere with centerpand radius n x=kXk(the Euclidean norm ofR) only depends onpand this radius; it may be x writtenM u(p)as well. n ForX; Y; p2Rtheiterated spherical meanis Z X Y X+kY (1.2)M M u(p) =M u(p)dk, K as easily checked. Takingz=kX+kYkas the new variable this transforms into Z x+y x y z n1 (1.3)M M u(p) =M u(p)a(x; y; z)z dz, jxyj         (n3)=2 Cnx+y+z x+yz xy+zx+y+z a(x; y; z) = n2 (xyz) 2 2 2 2       n3n n1 1 forx; y >0, a formula rst proved by John; hereCn= 2 = . 2 2 2 A nice proof is given in Chapter VI of Helgasons book [6]. Is there a similar result for symmetric spaces?The purpose of this note is to prove an analog of Johns formula for general Riemannian symmetric spaces and,

2000Mathematics Subject Classication.Primary 43A85, 53C35; Secondary 33C80, 43A90. Key words and phrases.Symmetric space, mean value. 1

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