ANNALES UNIVERSITATISMARIAECURIE-SKŁODOWSKA LUBLIN–POLONIA

VOL. LX, 2006 SECTIO A 65–73

ANDRZEJ MIERNOWSKI and WITOLD RZYMOWSKI

On homogeneous distributions

Abstract. Any homogeneous 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 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

S(j) kfkCk = max max ∂ f (x) , j=1,2,...,k x∈Sn−1

S(j) S(j) where ∂ f (x) denotes the of linear operation ∂ f (x) . In the case of k = 0 the symbol C0 Sn−1,X denotes the space of all continuous f : Sn−1 → X with the norm

kfkC0 = max kf (x)k . x∈Sn−1 In the sequel we will consider the space ∞ def \ C∞ Sn−1,X = Ck Sn−1,X , k=0 being the space of test functions for distributions on the sphere Sn−1, equipped with the sequence of semi-norms k·kCk , k = 0, 1,... . Clearly, the space C∞ Sn−1,X is locally convex and complete. It can be shown that any distribution on the sphere Sn−1 in the sense of [2], see Section 6.3, is a distribution in the following sense. ∞ n−1  Definition 2. Any linear continuous functional u : C S , R → R we call the distribution on the sphere. The space of all distributions on the 0 n−1  sphere we denote by D S , R . On homogeneous distributions 67

∞ n−1  Since the topology in C S , R is given by the sequence of semi- ∞ n−1  norms k·kCk , k ∈ N0, a linear functional u : C S , R → R is continuous if and only if there exist k ∈ N0 and C ≥ 0 such that ∞ n−1  |hu, ϕi| ≤ C kϕkCk , ϕ ∈ C S , R . Each distribution on the sphere is thus of finite degree. n−1 Any f : S → R is a regular distribution {f (x)} given by Z h{f (x)} , ϕi = f (x) ϕ (x) Hn−1 (dx) , Sn−1 n−1 n where H denotes the (n − 1)-dimensional Hausdorff measure in R . 0 n−1  0 n−1  Let um ∈ D S , R , m ∈ N, and u ∈ D S , R be given. We say that

u = lim um m→∞ ∞ n−1  if, for each ϕ ∈ C S , R ,

hu, ϕi = lim hum, ϕi . m→∞ 0 n Let us recall that u ∈ D (R \{0} , R) is homogeneous of degree α if α+n u (ψ) = r u (ψr) , ∞ n for all ψ ∈ C0 (R \{0} , R) and r > 0, where n ψr (x) = ψ (rx) , x ∈ R \{0} . 0 n We will denote by Dα (R \{0} , R) the space of all distributions u ∈ 0 n 0 n−1  D (R \{0} , R) homogeneous of degree α. For any u ∈ D S , R and 0 n any α ∈ R we define Eαu ∈ D (R \{0} , R) , being the extension of order α of u, by the formula, see formula (3) of [1], p. 387, Z ∞ α+n−1 ∞ n (1) hEαu, ψi = r hu, ψri dr, ψ ∈ C0 (R \{0} , R) . 0

It is easy to prove that Eαu is homogeneous of degree α and 0 n−1  0 n Eα : D S , R → Dα (R \{0} , R) is a linear continuous and univalent mapping.

2. Main result. We are going to prove in this section that for any ho- 0 n mogeneous u ∈ D (R \{0} , R) , of degree α, there exists a unique Rαu ∈ 0 n−1  D S , R such that EαRαu = u. 0 n−1  In other words Eα is a continuous linear between D S , R 0 n and Dα (R \{0} , R) . 68 A. Miernowski and W. Rzymowski

0 n Theorem 1. For any u ∈ Dα (R \{0} , R) we have

u = EαRαu, 0 n 0 n−1  where Rα : Dα (R \{0} , R) → D S , R is a linear continuous mapping given by the formula    x   hR u, ϕi = u, ϕ ψ (|x|) , u ∈ D0 ( n \{0} , ) , α |x| 0 α R R ∞ with a fixed ψ0 ∈ C0 ((0, ∞) , R) such that Z ∞ n+α−1 ψ0 ≥ 0, r ψ0 (r) dr = 1. 0 The proof will be divided into a few steps.

∞ n Claim 1. If f ∈ C0 (R \{0} , R) and a ∈ R then the equation 0 n aΦ(x) + x · Φ (x) = f (x) , x ∈ R \{0} , ∞ n has exactly one solution Φ ∈ C (R \{0} , R) given by the formula Z 1 a−1 n Φ(x) = t f (tx) dt, x ∈ R \{0} . 0 ∞ n n Moreover, if f ∈ C0 (R \{0} , R) and, for each x ∈ R \{0} , Z ∞ ta−1f (tx) dt = 0 0 ∞ n then Φ ∈ C0 (R \{0} , R). Proof of Claim 1. Let us define Z 1 a−1 n Φ(x) = t f (tx) dt, x ∈ R \{0} . 0 ∞ n n Clearly Φ ∈ C (R \{0} , R) . For any x ∈ R \{0} we have Z 1 Z 1 d x · Φ0 (x) = tax · f 0 (tx) dt = ta f (tx) dt 0 0 dt Z 1 a t=1 a−1 = [t f (tx)]t=0 − a t f (tx) dt 0 = f (x) − aΦ(x) , so that Φ satisfies the equation. ∞ n Let us suppose that ψ ∈ C (R \{0} , R) satisfies the equation. Let n x ∈ R \{0} be fixed arbitrarily. Define v (t) = ψ (tx) , w (t) = f (tx) , t ∈ (0, ∞) . On homogeneous distributions 69

For all t > 0 we have d (tav (t)) = ata−1v (t) + tav0 (t) = ata−1ψ (tx) + tax · ψ0 (tx) dt = ata−1ψ (tx) + ta−1tx · ψ0 (tx) = ta−1 · aψ (tx) + tx · ψ0 (tx) = ta−1 · f (tx) = ta−1 · w (t) , thus Z 1 Z 1 ψ (x) = v (1) = ta−1w (t) dt = ta−1f (tx) dt = Φ (x) . 0 0 ∞ n n Suppose now that f ∈ C0 (R \{0} , R) and, for each x ∈ R \{0} , Z ∞ ta−1f (tx) dt = 0. 0 n Since supp (f) ⊂ R \{0} there exist a, b ∈ R such that 0 < a < b and |x| ∈/ (a, b) ⇒ f (x) = 0. n Let us fix arbitrarily an x ∈ R \{0} . If |x| ≤ a then Z 1 Φ(x) = ta−1f (tx) dt = 0. 0 If |x| ≥ b then Z 1 Z ∞ Φ(x) = ta−1f (tx) dt = ta−1f (tx) dt = 0. 0 0  0 n Claim 2. If u ∈ Dα (R \{0} , R) then    x   u, ϕ ψ (|x|) = 0 |x| ∞ n−1  ∞ for all ϕ ∈ C S , R and ψ ∈ C0 ((0, ∞) , R) such that Z ∞ tn+α−1ψ (t) dt = 0. 0 Proof of Claim 2. Let us define a = n + α. Using the Euler’s identity n X ∂ x u = au, i ∂x i=1 i ∞ n for any Φ ∈ C0 (R \{0} , R) we obtain * n + X ∂ u, aΦ + x Φ = 0. i ∂x i=1 i 70 A. Miernowski and W. Rzymowski

∞ n By Claim 1, there exists a Φ ∈ C0 (R \{0} , R) such that n  x  X ∂ ϕ ψ (|x|) = aΦ(x) + x Φ(x) . |x| i ∂x i=1 i  0 n Claim 3. If u ∈ Dα (R \{0} , R) then there exists a distribution Rαu ∈ 0 n−1  D S , R such that     Z ∞ x n+α−1 u, ϕ ψ (|x|) = hRαu, ϕi · r ψ (r) dr, |x| 0 ∞ n−1  ∞ for all ϕ ∈ C S , R and ψ ∈ C0 ((0, ∞) , R) . Moreover, 0 n 0 n−1  Rα : Dα (R \{0} , R) → D S , R is a linear continuous mapping. ∞ Proof of Claim 3. Let us fix a ψ0 ∈ C0 ((0, ∞) , R) such that Z ∞ n+α−1 ψ0 ≥ 0, r ψ0 (r) dr = 1. 0 ∞ n−1  Define, for all ϕ ∈ C S , R ,    x   hR u, ϕi = u, ϕ ψ (|x|) . α |x| 0 0 n−1  ∞ Clearly Rαu ∈ D S , R . For each ψ ∈ C0 ((0, ∞) , R) and each r > 0 define Z ∞  n+α−1 ψ1 (r) = ψ (r) − % ψ (%) d% · ψ0 (r) . 0 ∞ Since ψ1 ∈ C0 ((0, ∞) , R) and Z ∞ n+α−1 r ψ1 (r) dr = 0, 0 we have    x   u, ϕ ψ (|x|) = 0. |x| 1 ∞ n−1  ∞ Consequently, for all ϕ ∈ C S , R and ψ ∈ C0 ((0, ∞) , R) , we obtain      Z ∞      x n+α−1 x u, ϕ ψ (|x|) = r ψ (r) dr · u, ϕ ψ0 (|x|) |x| 0 |x| Z ∞ n+α−1 = hRαu, ϕi · r ψ (r) dr. 0 The and continuity of the mapping 0 n 0 n−1  Rα : Dα (R \{0} , R) → D S , R are obvious.  On homogeneous distributions 71

It is easy to check that in the case of regular homogeneous distribution 0 n u = {f (x)} ∈ Dα (R \{0} , R) , n−1 the restriction Rαu coincides with f restricted to S . 0 n Claim 4. Given a homogeneous u ∈ Dα (R \{0} , R) . Then, for all ϕ ∈ ∞ n−1  ∞ C S , R and ψ ∈ C0 ((0, ∞) , R) we have    x      x   E R u, ϕ ψ (|x|) = u, ϕ ψ (|x|) . α α |x| |x| ∞ n−1  Proof of Claim 4. Let us fix arbitrarily ϕ ∈ C S , R and ψ ∈ ∞ C0 ((0, ∞) , R) . According to the extension formula (1), by Claim 3, we obtain      Z ∞ x n+α−1 EαRαu, ϕ ψ (|x|) = r hRαu, {ϕ (ω) ψ (r)}i dr |x| 0 Z ∞ n+α−1 = hRαu, ϕi · r ψ (r) dr 0    x   = u, ϕ ψ (|x|) . |x|  ∞ ∞ n−1  Let us define, for f ∈ C0 ((0, ∞) , R) and g ∈ C0 S , R ,  x  (f ⊗ g)(x) = f (|x|) · g , x ∈ n \{0} . |x| R ∞ n Claim 5. For each ϕ ∈ C0 (R \{0} , R) there exists a sequence k Xm ϕm = tm,kfm,k ⊗ gm,k, m ∈ N, k=1 such that tm,k ∈ R, ∞ ∞ n−1  fm,k ∈ C0 ((0, ∞) , R) , gm,k ∈ C0 S , R , k = 1, 2, . . . , km and k Xm ϕ = lim tm,kfm,k ⊗ gm,k m→∞ k=1 ∞ n (in the space C0 (R \{0} , R)). n Proof of Claim 5. Since supp (ϕ) ⊂ R \{0}, there exist 0 < a < b < ∞ such that |x| ∈/ (a, b) ⇒ ϕ (x) = 0. Let us define F (r, ω) = ϕ (rω) , t ∈ (0, ∞) , ω ∈ Sn−1. 72 A. Miernowski and W. Rzymowski

∞ n There exists an Fe ∈ C0 ((0, ∞) × (R \{0}) , R) such that (   F r, w if 2 ≤ kwk ≤ 4 , F (w) = kwk 3 3 e 1 5 0 if kwk < 3 or kwk > 3 . Let us define, for 0 < α < β < ∞, n Rα,β = {x ∈ R : α < |x| < β} . By Lemma 1 of [3], p. 48, one can find a sequence

k Xm tm,kfm,k · gm,k k=1 such that tm,k ∈ R, 1  f ∈ C∞ ((0, ∞) , ) , supp (f ) ⊂ , 2b , m,k 0 R m,k 2 ∞ n gm,k ∈ C0 (R \{0} , R) , supp (gm,k) ⊂ R 2 4 , k = 1, 2, . . . , km 3 , 3 and k Xm Fe = lim tm,kfm,k · gm,k m→∞ k=1 ∞  1   (in the space C0 , 2b × R 2 4 , R ). Since 2 3 , 3  x  Fe |x| , = ϕ (x) , x ∈ n \{0} , |x| R we obtain k Xm ϕ = lim tm,kfm,k ⊗ gm,k m→∞ k=1 ∞ n (in the space C0 (R \{0} , R)). 

Proof of Theorem 1. By Claim 4, u = EαRαu in the set Z being the linear hull of the set ∞ ∞ n−1  C0 ((0, ∞) , R) ⊗ C S , R ∞ ∞ n−1  of all f ⊗ g where f ∈ C0 ((0, ∞) , R) and g ∈ C0 S , R . Since, by ∞ n Claim 5, the set Z is dense in the space C0 (R \{0} , R), we obtain

u = EαRαu.  0 n Corollary 1. Any homogeneous distribution u ∈ D (R \{0} , R) is of fi- nite order. On homogeneous distributions 73

References [1] Gelfand, I. M., Shilov, G. E., Generalized Functions and Operations on Them, Second Edition, Gos. Izd. Fiz.-Mat. Lit., Moscow, 1959 (Russian). [2] H¨ormander, L., The Analysis of Linear Partial Differential Operators I, Springer- Verlag, Berlin–Heidelberg–New York–Tokyo, 1983. [3] Shilov, G. E., Mathematical Analysis, Second Special Lecture, MGU, Moscow, 1984 (Russian).

Andrzej Miernowski Witold Rzymowski Institute of Mathematics Department of Quantitative Methods M. Curie-Skłodowska University in Management pl. Marii Curie-Skłodowskiej 1 Lublin University of Technology 20-031 Lublin, Poland ul. Nadbystrzycka 38 e-mail: [email protected] 20-618 Lublin, Poland e-mail: [email protected]

Department of Applied Mathematics The John Paul II Catholic University of Lublin ul. Konstantynów 1 H 20-708 Lublin, Poland

Received July 24, 2006