Invariant operators on function spaces on homogeneous trees

Abstract

A homogeneous tree X of degree q + 1 is a connected graph with no loops in which each vertex is adjacent to q + 1 others.We assume that q ≥ 2. The tree X has a natural measure, counting measure, and a natural distance d, viz.d(x, y) is the number of edges between vertices x and y.Let o be a fixed but arbitrary reference point in X, and let G o be the stabiliser of o in the isometry group G of X.We write |x| for d(x, o).The mapWe endow the totally disconnected group G with the Haar measure such that the mass of the open subgroup G o is 1.The reader may find much more on the group G in the book of Figà-Talamanca and Nebbia [FTN].We denote by |E| the measure of a subset E of a measure space.We write S n for {x ∈ X : |x| = n}.Clearly, |S 0 | = 1, and |S n | = (q + 1)q n-1 when n ∈ Z + .We pick points w 0 , w 1 , w 2 , . . . in X such that |w d | = d.A radial function f on X is determined by its restriction to these points.It is well known that G-invariant linear operators from L p (X) to L r (X) correspond to linear operators from L p (G/G o ) to L r (G/G o ) given by convolution on the right by G o -bi-invariant kernels.We denote by Cv r p (X) the space of radial functions on X associated to these G o -bi-invariant kernels.The norm of an element k of Cv r p (X) is then defined as the norm of the corresponding operator from L p (X) to L r (X), and denoted by |||k||| p;r .Equipped

Locations

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