ANNALES UNIVERSITATISMARIAECURIE-SKŁODOWSKA LUBLIN–POLONIA
VOL. LX, 2006 SECTIO A 65–73
ANDRZEJ MIERNOWSKI and WITOLD RZYMOWSKI
On homogeneous distributions
Abstract. Any homogeneous function is determined by its values on the unit sphere. We shall prove that an analogous fact is true for homogeneous distributions.
n 1. Test functions on the unit sphere. For x, y ∈ R we will write n X x · y = xiyi i=1 and v √ u n uX 2 |x| = x · x = t xi . i=1 n−1 n By S we denote the unit sphere in R , i.e. n−1 n S = {x ∈ R : |x| = 1} . n−1 Let X be a linear space and f : S → X. For any α ∈ R we define the extension of f, of degree α, by the formula x (E f)(x) = |x|α f , x ∈ n \{0} . α |x| R
2000 Mathematics Subject Classification. 60E05. Key words and phrases. Homogeneous distribution, spherical derivative, distribution on the sphere. 66 A. Miernowski and W. Rzymowski
In the case of α = 0 we have x (E f)(x) = f , x ∈ n \{0} . 0 |x| R Definition 1. Let (X, k·k) be a normed space and f : Sn−1 → X. We n−1 say that f is differentiable at x0 ∈ S if there exists a linear operation n A : R → X such that f (x) − f (x0) − Ax Ax0 = 0 and lim = 0. n−1 S 3x→x0 |x − x0| It is not very hard to check that such an operation is unique, so that we call it the spherical derivative of f at the point x0. The spherical derivative of S n−1 f at the point x0 will be denoted by ∂ f (x0) . The mapping f : S → X is called differentiable if ∂Sf (x) exists for all x ∈ Sn−1. The notion of the spherical derivative agrees with the usual derivative in the following sense. If f : U → X, where U is an open neighbourhood n−1 n−1 of S , then f is differentiable at x0 ∈ S if and only if there exists S n ∂ f (x0) . Moreover, for any ξ ∈ R with ξ · x0 = 0, we have then S 0 0 ∂ f (x0) ξ = f (x0) ξ = (E0f) (x0) ξ. The symbol Ck Sn−1,X will stand for the space of all f : Sn−1 → X having continuous spherical derivatives ∂Sf, ∂S(2) f, . . . , ∂S(k) f up to degree k. For f ∈ Ck Sn−1,X we define