HADAMARD OPERATORS ON D (R d )

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1 HADAMARD OPERATORS ON D (R d DIETMAR VOGT BERGISCHE UNIVERSITÄT WUPPERTAL FB MATH.-NAT., GAUSS-STR. 20, D WUPPERTAL, GERMANY DVOGT@MATH.UNI-WUPPERTAL.DE Dedicated to the memory of Paweª Doma«ski Abstract. We study continuous linear operators on D (R d which admit all monomials as eigenvectors, that is, operators of Hadamard type. Such operators on C (R d and on the space A (R d of real analytic functions on R d have been investigated by Doma«ski, Langenbruch and the author. The situation in the present case, however, is quite dierent and also the characterization. An operator L on D (R d is of Hadamard type if there is a distribution T, the support of which has positive distance to all coordinate hyperplanes and which has a certain behaviour at innity, such that L(S = S T for all S D (R d. Here (S T ϕ = S y (T x ϕ(xy for all ϕ D(R d. To describe the behaviour at innity we introduce a class O H (Rd of distributions dened by the same conditions like in the description of class O C (Rd of Laurent Schwartz, but derivatives replaced with Euler derivatives. In the present note we study Hadamard operators on D (R d, that is, continuous linear operators on D (R d which admit all monomials as eigenvectors and we give a complete characterization. Such operators on C (R d have been studied and characterized in [9, 10], on A (R in [1, 2, 3] and on A (R d in [4]. There you nd also references to the long history of such problems. Since it can be shown that Hadamard operators commute with dilations our problem is, by duality, closely related to the study of continuous linear operators in D(R d which commute with dilations. They have the form ϕ T x ϕ(xy where T is a distribution. The class D H (Rd of distributions T such that T x ϕ(xy D(R d for every ϕ D(R d is studied. These are distributions with positive distance to the coordinate hyperplanes and certain behaviour at innity, similar to the class O C of L. Schwartz of rapidly decreasing distributions. We dene a class O H (Rd of distributions by the same conditions like in the description of class O C in [7], but derivatives replaced with Euler derivatives. We denote by M(R d the class of Hadamard 2010 Mathematics Subject Classication. Primary 46F10; Secondary 47B38, 46F05. Key words and phrases. Operators on distributions, operators on test functions, monomials as eigenvectors, spaces of distributions. 1

2 2 D. VOGT operators in D (R d and for x R d = (R \ {0} d we set σ(x = x j j, that x j is, the signum of x which, of course, is constant on each `quadrant'. Then our main result is (see Corollary 1.6, Theorem 2.10 and Theorem 4.2: Main Theorem L M(R d if and only if there is a distribution T O H (Rd, the support of which has positive distance to all coordinate hyperplanes, such that L(S = S T for all S D (R d. Here ( (S T ϕ = S y (T x ϕ(xy for all ϕ D(R d. The eigenvalues are m α = T σ(x x. x α+1 Moreover, we show that every Hadamard operator on D (R d maps C (R d to C (R d, that is, denes a Hadamard operator on C (R d. On the other hand not every Hadamard operator on C (R d can be extended to an operator on D (R d. We follow the notation in [9, 10]. The class of Hadamard operators in C (R d, that is, of continuous linear operators which admit all monomials as eigenvectors is denoted by M(R d. The Hadamard operators are given by distributions T E (R d by means of the formula (M T ϕ(y = T x ϕ(xy. The dual E (R d is an algebra with respect to -convolution given by the formula (T Sϕ = T x S y ϕ(xy where xy = (x 1 y 1,..., x d y d. T E (R d denes a -convolution operator N T : S S T and N T = MT, that is, the dual operator of M T. Dierential operators of the form P (θ where P is a polynomial and θ j = x j j or, equivalently, of the form α c αx α α are called Euler operators and θ j is called an Euler derivative. On C these are the Hadamard operators M T with supp T = {1} where 1 = (1,..., 1. We use standard notation of Functional Analysis, in particular, of distribution theory. For unexplained notation we refer to [5], [6], [7], [8]. The author thanks Paweª Doma«ski, who passed away too early, for useful discussions on the topic of this paper. 1. Basic properties Denition 1. A map L L(D (R d is called a Hadamard operator if it admits all monomials as eigenvectors. The set of Hadamard operators we denote by M(R d. Since the condition means that L(x α span{x α } for all α N d 0 the set M(R d is a closed subalgebra in L σ (D (R d and therefore also in L b (D (R d. Here σ denotes the topology of pointwise convergence and b the topology of uniform convergence on bounded sets. We dene m α by L(x α = m α x α. Since the polynomials are dense in D (R d the operator L M(R d is uniquely determined by the family m α,

3 HADAMARD OPERATORS ON D (R d 3 α N d 0, of eigenvalues. Clearly the set Λ(R d of eigenvalue families is an algebra and L m α is an algebra isomorphism. To study and characterize the Hadamard operators in D (R d we rst need some preparations. We set R := R \ {0}. For a R d we dene the dilation operator D a L(D (R d by (D a T ϕ := T x ( σ(a a 1... a d ϕ( x a for T D (R d and ϕ D(R d. By direct verication we see that D a ξ α = a α ξ α. Lemma 1.1. For L M(R d and a R d we have L D a = D a L. Proof. For any α we have (L D a ξ α = a α m α ξ α = (D a Lξ α. So the claim is shown for all polynomials and these are dense in D (R d. Notation. If L D a = D a L for all a we say: L commutes with dilations. By denition of D a we obtain for the dual map D a L(D(R d of D a : Lemma 1.2. For a R d and ϕ D(R d we have (Daϕ(ξ = σ(a ( ξ ϕ. a 1 a d a If L commutes with dilations then Da L = L Da for all a R d. For ϕ D(R d we set ψ = L ϕ and obtain σ(a ( x ( ( σ(a ξ ψ = L ξ ϕ [x]. a 1 a d a a 1 a d a For η R d we set a = 1/η and obtain ψ(ηx = L ξ (ϕ(ηξ[x]. We have shown: Lemma 1.3. If M = L L(D(R d and L commutes with dilations, then for all ϕ D(R d and η R d. For ϕ D(R d we dene now M ξ (ϕ(ηξ[x] = (Mϕ(ηx T ϕ = (Mϕ(1 = (Lδ 1 (ϕ. Then T D (R d and for all η R d we have (1 (Mϕ(η = T ξ ϕ(ηξ. The problem with the right hand side of equation (1 is that for η j = 0, in general, T cannot be applied to the non compact support function ξ ϕ(ηξ. For T as above, however, the function η T ξ ϕ(ηξ, ξ R d, is the restriction of a function in D(R d.

4 4 D. VOGT Denition 2. By D H (Rd we denote the set of distributions T D (R d such that for every ϕ D(R d the function y T ξ ϕ(ξy, y R d, is the restriction of a function in D(R d. This means that T must have the following properties: (* For every ϕ D(R d there is r > 0 such that T ξ ϕ(ξy = 0 for y R d with y > r, (** For every ϕ D(R d the map y T ξ ϕ(ξy, y R d, extends to a function in C (R d. For T D H (Rd we denote by M T the map which assigns to ϕ D(R d the continuous extension of y T ξ ϕ(ξy. From the closed graph theorem it follows easily: Lemma 1.4. M T L(D(R d for every T D H (Rd. We obtain the following representation theorem: Theorem 1.5. L L(D (R d commutes with dilations if and only if there is T D H (Rd such that L(S = S T for all S D (R d, here (S T ϕ = S y (T x ϕ(xy for ϕ D(R d. In this case T = L(δ 1. Proof. The assertion can be written as L = MT. If L commutes with translations then, by the above, L = M T for T = L(δ 1 and T D H (Rd by formula (1. This formula also implies the result. Notation. For T D H (Rd we set L T (S = S T for all S D (R d. Corollary 1.6. If L M(R d then there is T D H (Rd such that L = L T. 2. Properties of D H (Rd First we will exploit the fact that M T L(D(R d. For ε > 0 we set W ε = {x R d : min j x j ε} and we will use the following notations: For r = (r 1,..., r d, where all r j 0, we set B r = {x R d : x j r j for all j} and for s 0 we set B s := B s1 = {x R d : x s}. For r = (r 1,..., r d R d and s R we set r+s = r+s1 = (r 1 +s,..., r d +s. Lemma If T D H (Rd then there is ε > 0 such that supp T W ε. 2. If T D (R d and there is ε > 0 such that supp T W ε then T satises (*.

5 HADAMARD OPERATORS ON D (R d 5 Proof. 1. If T D H (Rd there is r > 0 such that supp M T ϕ B r for any ϕ D(B 1 and this implies, that T ξ ϕ(ηξ = 0 for any ϕ D(B 1, η R d and η > r. We set ε = 1/r and assume that supp ϕ W ε =. That is, supp ϕ j {x R d : x j < ε}. Then we can write ϕ = j ϕ j with ϕ j D({x R d : x j < ε}. We x j and choose η R d such that sup{ x ν η ν : x supp ϕ j } = 1 for all ν. We set ψ(ξ = ϕ j (ξ/η. Then supp ψ B 1 and η > r and therefore we have T ϕ j = T ξ ψ(ηξ = 0. Since this holds for every j, the proof of 1. is complete, the proof of 2. is obvious. A special case is that of distributions with compact support. Corollary 2.2. E (R d D H (Rd = E (R d. Proof. This follows from Lemma 2.1 and the fact that (** is fullled for distributions with compact support. This will be used in Section 4 to handle the case of T with compact support. After having settled property (* of T D H (Rd we turn to property (**. It is quite restrictive. Lemma 2.3. D H (Rd S (R d. Proof. Let T D H (Rd, we may assume that supp T W 2. For k N d 0 we set ϕ k = ϕ (k L1 and remark that for every D(B r these norms are a fundamental system of seminorms. On D(B 1 the family of distributions T (y ϕ := T x ϕ(xy, y R d B 1, is weakly bounded hence equicontinuous. This means that there is k N d 0 and C > 0 such that T (y ϕ C ϕ k, ϕ D(B 1, y 1. For r with r j 1 for all j and ϕ D(B r we set ψ(x = ϕ(rx. Then ψ D(B 1 and therefore T ϕ = T (1/r ψ C ψ k = Cr k ϕ k. For every r there is t r L (B r, t r Cr k such that T ϕ = t r (xϕ (k (xdx for all ϕ D(B r. We restrict us now to r N d and set U r = [r 1, r 1 + 1[ [r d, r d + 1[. Then r N d U r = W 1. We choose χ D(B 1/2 with χ = 1 and set χ r (x =

6 6 D. VOGT U r χ(x ξdξ. We obtain: T ϕ = T (χ r ϕ = t r+2 (x(χ r ϕ (k (xdx r r = ( ( k t r+2 (x χ (k ν r (xϕ (ν (x dx ν r ν = ( ( k ϕ (ν (x t r+2 (xχ (k ν r (x dx ν ν r = ( k ϕ (ν (xτ ν (xdx. ν ν We have to estimate the functions τ ν. We set γ(x = {r : x j [r j 1 2, r j [} τ ν (x r γ(x t r+2 (x χ (k ν r (x. For all 2 d elements r γ(x we have t r+2 (x C(r + 2 k and r x (here x = ( x 1,..., x d and therefore t r+2 (x C( x + 3 k. With a new constant C ν we have τ ν (x C ν x k. This shows the result. The necessary conditions we have found are far from being sucient as the following example shows Example 2.4. Let d = 1, we set T ϕ = ϕ(xdx, then T 1 x ϕ(xy = 1 ϕ(xdx for all y > 0 which, in general, is unbounded near 0. The y y distribution is T S (R and has support in W 1. O C The following denition is in analogy to a characterization of the space of rapidly decreasing distributions in L. Schwartz [7], Ÿ5, p Denition 3. T O H (Rd if for any k there are nitely many functions t β such that (1 + x 2 k/2 t β L (R d and such that T = β θβ t β. Proposition 2.5. If T O H (Rd then for every ϕ D(R d the function y T x (ϕ(xy, y R d, extends to a function in C (R d. Proof. We have to show that for every α the function y α y T x (ϕ(xy, y R d, extends to a continuous function on R d. By denition of O H (Rd we have to show our proposition only for T = θ β t where (1 + x 2 k/2 t L (R d and k is suitably chosen. Since the formal

7 HADAMARD OPERATORS ON D (R d 7 adjoint (θ β of the Euler-operator θ β is again an Euler-operator we have (θ β t(xϕ(xdx = t(x((θ β ϕ(xdx = c ν t(xx ν ϕ (ν (xdx ν where the sum is nite with suitable c ν. Therefore it is enough to study the function (2 F (y := t(x(xy ν ϕ (ν (xydx, y R d. We have to show that all limits lim y y0 F (α (y, y 0 R d, α N d 0 exist. (3 F (α (y = t(xx α (x ν ϕ (ν (x (α [xy]dx = c γ t(xx α (xy ν α+γ ϕ (ν+γ (xydx. α ν γ α If k was chosen so large that t(xx α requested. Therefore we have shown: L 1 (R d then the limits exist as Theorem 2.6. If T O H (Rd and supp T W ε for some ε > 0, then T D H (Rd. To show the inverse we give a description of D H (Rd in terms of the regularizations of a distribution. Theorem 2.7. T satises (** if and only if T χ satises (** for all χ L 1 (B 1. In this case the set of maps {ϕ (T χ x ϕ(x : χ L 1 (B 1, χ L1 1} is equicontinuous in L(D(R d, C (R d. Proof. We assume that T satises (** and want to show that the same holds for T χ. We need some preparation. We set: ψ(ξ, y 2 = χ(x 2 ϕ(ξ + x 2 y 2 dx 2 and remark that for ϕ D(B r and χ L 1 (B 1 we have ψ(, y 2 D(B r+ y2. This implies that the map which assigns to every ϕ D(R d the function y 2 ψ(, y 2 is a continuous linear map Φ : D(R d C (R d, D(R d. We set F (y 1, y 2 = T x1 ψ(x 1 y 1, y 2 C (R d R d. By assumption it extends to a function ˆF on R d R d such that ˆF (, y 2 is in C (R d for every y 2 R d.

8 8 D. VOGT The map which assigns to every g D(R d the continuous extension of T ξ g(ξy 1 denes a continuous linear map Ψ : D(R d C (R d. We obtain ˆF (y 1, y 2 = Ψ{Φ(ϕ[y 2 ]}[y 1 ] C (R d, C (R d = C (R d R d. Therefore ˆF (y, y C (R d and it is the extension of F (y, y = T x1 χ(x 2 ϕ((x 1 + x 2 ydx 2 = (T χ x ϕ(xy. This proves one direction of the Theorem, it remains to show the other implication. We will use the idea of proof in [7, Ÿ7, Théorème XX]. First we show the additional assertion of the theorem. We assume that T χ satises (** for all χ L 1 (B 1. We consider the map L 1 (B 1 L(D(R d, C (R d dened by χ [ϕ (T χ x ϕ(x ] (cf. Lemma 1.4. If χ 0 and [ϕ (T χ x ϕ(x ] A in L(D(R d, C (R d then for xed y R d and all ϕ we have (T χ x ϕ(xy 0 and therefore (Aχϕ = 0 on R d, hence on R d. So the map L 1 (B 1 L(D(R d, C (R d has closed graph and, by de Wilde's Theorem, is continuous. This shows the assertion. We x χ D(R d with χ = 1. Then for every ϕ D(R d the function (T χ x ϕ(xy, y R d, extends to a function in C (R d and the set {F χ : ϕ (T χ x ϕ(xy : χ D(R d, χ 0, χ = 1} is equicontinuous. Therefore it is relatively compact in L(D(B R, C (R d. We x χ and set χ ε (x = ε d χ(x/ε for ε > 0. Then there is a sequence ε n 0 such that F χεn converges to some F L(D(B R, C (R d. Since ( = T ξ χ εn (xϕ((x + ξydx T ξ ϕ(ξy F χεn for every y R d we see, that T ξ ϕ(ξy extends to a function in C (R d. Now we can show the inverse of Theorem 2.6. Theorem 2.8. If T D H (Rd then for every β > 0 there is a function t β such that x β t β is bounded and an Euler operator P (θ such that T = P (θt β. In particular T O H (Rd. Proof. Let T D H (Rd with supp T W 2ε0. Let χ D(R d with χ 0, χ = 1 and supp χ B ε0. Then for every ϕ D(B 1 the function (T χ x ϕ(xy, y R d, extends to a function in D(R d. We set τ = T χ. We set for y R d and ϕ D(R d : F (y = τ(xϕ(xydx. Then F C (R d, F (y = 0 for y > 1/ε 0. For β N 0, we have F (β (y = τ(xx β ϕ (β (xydx.

9 HADAMARD OPERATORS ON D (R d 9 By Theorem 2.7 on D(B 1 the set of distributions ϕ F (β (y, y R d, χ as above, is equicontinuous. Hence there is p such that all these distributions extend to D β +p (B 1 and the set of these distributions is bounded in D β +p (B 1. For α N 0 we choose ϕ α D α [0, +1] such that ϕ α C (R \ {0, 1} and α+1 (x 1α+1 ϕ α (x = x for 0 x 1. (2α + 2! We dene for α N d 0 such that D α (B 1 D β +p (B 1 d ϕ α (x = ϕ αj (x j j=1 and consider F (y = τ(xϕ α (xydx. Then we have for y (0, + d, setting α + 1 = (α 1 + 1,..., α d + 1, etc. F (2α+2 (y = τ(xx 2α+2 dx [0,1/y] d and therefore F (2α+3 (y = 1 ( 1 y τ. 2α+4 y With an analogous argument for the other `quadrants' we get for general y R d F (2α+3 (y = σ(y ( 1 y τ. 2α+4 y We set G(x = σ(xf (β ( 1. Then G is a bounded function on x ( Rd with a bound independent of χ and σ(x F (β 1 (x = G. We calculate the x derivatives. With certain coecients c ν we have: σ(x F (β+q (x = 1 ( 1 cν x ν+q G(ν x ν where ν runs over 1 ν j q. We choose q = 2α + 3 β. Replacing x with 1/x and c ν with c ν we obtain: (4 x β+1 τ(x = ν cν x ν G (ν (x. We analyze now functions of type x m+ν G (ν (x and claim Lemma 2.9. Any such function can be written as a linear combination of the functions (x m+j G(x (j, 0 j i ν i for all i. Proof. For any m Z there are constants c m,ν,j such that ν (x m+ν G(x (ν = c m,ν,j y m+ν j G (ν j (x j=0

10 10 D. VOGT and therefore,since c m,ν,0 = 1, x m+ν G (ν (x = (x m+ν G(x (ν j cm,ν,j x m+ν j G (ν j (x. where j runs over all j with j 0 and 0 j i ν i for all i. Induction over ν yields the result. End of the proof of Theorem 2.8: From Lemma 2.9 and equation (4 we obtain with new constants c j : τ(x = c j (x j (x β 1 G(x (j. j p+2 We x now χ D(R d with χ 0 and χ = 1. For ε > 0 we set χ ε = ε d χ(x/ε, τ ε = T χ ε and G ε the corresponding function. Then we have (5 (T χ ε ϕ = τ ε (xϕ(xdx = c j x β 1 G ε (xx j ϕ (j (xdx. j p+2 We have lim ε 0 (T χ ε ϕ = T ϕ. On the other hand {G ε : ε > 0} is bounded in L (R d = L 1 (R d. Therefore there is G L (R d and a sequence G εn which converges to G in the weak -topology with respect to L 1 (R d. From equation (5 then follows: T ϕ = c j x β 1 G(xx j ϕ (j (xdx = (x β 1 G(x(P (θϕ(xdx = j p+2 (P (θ (x β 1 G(xϕ(xdx. Putting t β := x β 1 G(x we have shown: For every β there is a function t β such that x β t β L (R d and an Euler operator Q(θ := P (θ such that T = Q(θt β. This completes the proof. From Theorems 2.6 and 2.8 we obtain one of the main results of this paper: Theorem D H (Rd = {T O H (Rd : supp T W ε for some ε > 0}. 3. The space O H (Rd We recall the denition of the space O H (Rd : T O H (Rd if for any k there are nitely many functions t β such that (1 + x 2 k/2 t β L (R d and such that T = β θβ t β. The space O C (Rd of L. Schwartz may be dened by any of the following equivalent properties (see [7, Ÿ5, Théorème IX]. Let T D (R d, then T O C (Rd if and only if 1. or 2.:

11 HADAMARD OPERATORS ON D (R d 11 (1 For any k there are nitely many functions t β such that (1+ x 2 k/2 t β L (R d and such that T = β β t β. (2 For any χ D(R d, T χ is a rapidly decreasing continuous function. Without proof we admit. Lemma 3.1. E (R d O C (Rd O H (Rd. And this implies: Lemma 3.2. If T R d \B R = S R d \B R for some R > 0 and S O H (Rd or S O C (Rd then T O H (Rd or T O C (Rd, respectively. Proof. T = S + (T S and T S E (R d. Proposition O C (R O H (R. 2. If supp T W ε for some ε > 0 and T O C (Rd then T O H (Rd. Proof. 2. It is enough to show the claim for T = t (β β, (1+ x 2 k/2 t β bounded. Set τ β = 1 t x β β. Then τ β is decreasing even faster and (x β τ β (β = t (β β. Since f (x β f (β is an Euler operator the proof of 2. is complete. 1. Choose χ D[ 1, +1], χ 1 in a neighborhood of 0. For T O C (R set S = (1 χt. Then supp S W ε for some ε > 0 and, due to Lemma 3.2, S O C (R. By part 2. of this Proposition we have S O H (R and therefore, again by Lemma 3.2, T O H (R. The space O C (R is a proper subspace of O H (R, as the following example shows. Example 3.4. If T = e ix, that is, T ϕ = e ix ϕ(xdx, then 1. T O C (R, 2. T O H (R. Proof. 1. Let χ D(R, then (T χ(x = e iξ χ(x ξdξ = ˆχ( 1 e ix. By the second denition of O C (R (see above T O C (R. 2. To show that T O H (R we choose χ D[ 1, +1], χ 1 in a neighborhood of 0. For given k we set t k (x = ik (1 χ(xe ix. Then (1+x 2 k/2 t x k k (x is bounded and (x k t k (x (k = (i k (1 χ(xe ix (k = e ix +g(x where g has compact support. Hence T = (x k t k (x (k g where g has compact support. This shows the result like above. By Proposition 2.5 we know now that for T as in Example 3.4 the function T x ϕ(xy, y R, extends to a C -function on R. For this example we can make it explicit, even for higher dimensions, setting e ix = e i(x 1+ +x d.

12 12 D. VOGT For ϕ D(R d and y R d we set F (y = T x ϕ(xy. We obtain for y R d α F (y = e ix x α ϕ (α (xydx = σ(y e i(x/y x α ϕ (α (xdx = σ(y y α+1 xα ϕ (α ( 1 y. y α+1 Since x α ϕ (α S (R d we obtain lim y y0 F (α (y = 0 for every y 0 R d \ R d. That means, if we denote the extended function again by F, that F (α (y = 0 on all coordinate hyperplanes and for all α. Returning to the one-dimensional case we present another example which we take from [7, Ÿ5, p. 100, (VII,5;1]. Example 3.5. If T = e iπx2 then T O C (R and therefore T O H (R. The function e iπx2 is bounded, but its derivatives are not. 4. Eigenvalues In this section we study Hadamard operators in terms of the representing distribution in D H (Rd. A special case are the distributions in D H (Rd with compact support. In consequence of Lemma 1.1, Theorem 1.5 and Corollary 2.2 we obtain: Theorem 4.1. For T D (R d the following are equivalent: 1. T E (R d and the -homomorphism N T can be extended to a map in M(R d. 2. T D H (Rd and L T E (R d E (R d. 3. T E (R d. In this case m α = T x ( σ(x x α+1. Proof : By assumption there is L M(R d such that L(S = N T (S = S T for all S E (R d. Then T = N T (δ 1 = L(δ 1 D H (Rd. By denition N T (S = S T = L T (S for S E (R d : By assumption T = L T (δ 1 E (R d. Hence T E (R d D H (Rd = E (R d, by Corollary By Corollary 2.2 T E (R d D H (Rd and we have L T (S = S T = N T (S E (R d for S E (R d. Moreover ( 1 L T (ξ α [ϕ] = ξ α (T x ϕ(xξdξ = T x ( σ(x = T x η α ϕ(ηdη x α+1 = (m α η α ϕ(ηdη. x α (xξ α ϕ(xξdξ

13 HADAMARD OPERATORS ON D (R d 13 This shows that L T (ξ α = m α ξ α with m α = T x ( σ(x x α+1. Notice that the interchange of T and the integral is clear since T E (R d. For T with non-compact support this is more complicated. We further study operators represented by distributions in D H (Rd. By Theorems 1.5 and 2.10 we know that they dene operators L T L(D (R d commuting with dilations. We show that these are, in fact, Hadamard operators. Theorem 4.2. If T D H (Rd, then L T is a Hadamard operator with eigenvalues m α = T x ( σ(x x α+1. Proof. It remains to show that L T admits all monomials as eigenvectors. It is sucient to assume that T = ( 1 k (x k τ (k, where τ L 1 (R d and k N d 0. Then the function f(x, ξ = ξ α τ(xξ k ϕ (k (xξ is in L 1 (R d R d. To see this let supp ϕ B R. Then f(x, ξ 0 only if x j ε and x j ξ j R for all j, hence only for ξ j R/ε for all j. Therefore ξ α T x ϕ(xξdξ = = = = ( ξ α τ(x(xξ k ϕ (k (xξdx dξ τ(x 1 x α ( τ(x σ(x x α+1 ( (xξ α+k ϕ (k (xξdξ dx η α+k ϕ (k (ηdη dx τ(x σ(x (α + k! ( 1 k xα+1 α! ξ α T x ϕ(xξdξ = (m α η α ϕ(ηdη ( η α ϕ(ηdη dx. where k (α + k! m α = ( 1 α! τ(x σ(x dx. xα+1 Taking into account that for x with min j x j > 0 we nally obtain which proves the result. ( σ(x (k x k = ( 1 k (α + k! x α+1 α! ( σ(x m α = T x α+1 σ(x x α+1

14 14 D. VOGT Remark: We needed in the proof only very weak assumptions on T. Under these ϕ T x ϕ(xy might not send D(R d into D(R d hence its adjoint sends something much smaller to D (R d, but it sends ξ α to m α ξ α for all α N d Hadamard operators in D (R d and in C (R d We study now the problem when a Hadamard operator M on C (R d extends to an operator on D (R d and, on the other side, when an operator L M(R d leaves C (R d invariant, that is, LC (R d C (R d. We start with the latter question, which has a rather straightforward answer. Theorem 5.1. If T D H (Rd then L T C (R d. M(R d and L T (C (R d Proof. We may assume that T = P (θt where t L 1 (R d with supp t W ε for some ε > 0 and P (θ is an Euler operator. Let P (θ denote the formal adjoint of P (θ which is again an Euler operator. Then for f C (R d, ϕ D(R d the function f(yt(xp (θ x ϕ(xy, x, y R d is in L 1 (R d R d. We obtain,using that Euler operators commute with dilations: (L T fϕ = f(y( t(xp (θ x ϕ(xydx dy = f(yt(x(p (θϕ(xydxdy. We apply the substitution xy = η, y = ξη, its Jacobian determinant is ( 1 d ξ 1 ξ d. ( (L T fϕ = ( 1 σ(ξ f(ξη t dξ (P (θϕ(ηdη. ξ ξ 1 ξ d We have shown that on C (R d we have L T = P (θ M T # where ( 1 σ(ξ T # = t. ξ ξ 1 ξ d ( Notice that t σ(ξ is an L 1 (R d -function with compact support, hence T # E (R d. 1 ξ ξ 1 ξ d Remark: Like in Theorem 4.2 we needed much weaker assumptions than T O H (Rd, cf. the remark at the end of Section 4. This corresponds to the fact that not all Hadamard operators in C (R d extend to operators in M(R d as the following proposition shows:

15 HADAMARD OPERATORS ON D (R d 15 Proposition 5.2. If supp T = {0} and T 0 then M T cannot be extended to a map in M(R d Proof. If T = β c βδ (β then M T (f[x] = β c β( 1 β f (β (0x β, hence R(M T E = span{x β : β e} where e is a nite set. Assume L M(R d and L C (R d = M T. Since E is closed in D (R d and C (R d is dense in D (R d we have R(L E and this implies that L(S = β S(ϕ βx β for all S D (R d where ϕ β D(R d for all β e. This implies c β ( 1 β f (β (0 = f(ξϕβ (ξdξ for all β e and f C (R d which is possible only if c β = 0 for all β e, that is, T = 0. To study the problem which Hadamard operators on C (R d extend to operators in M(R d we consider an operator M = M T M(R d, T E (R d. We may assume that T = ( 1 β t (β where t L 1 (R d with compact support. Here we assume that f C (R d and ϕ D(R d. Then the function t(xϕ(yy β f (β (xy, x, y R d is in L 1 (R d R d. We use the same coordinate transformation as above. Then the function σ(ξ 1 t( ϕ(ξη(ξη β f (β (η, ξ, η R d ξ 1 ξ d ξ is again in L 1 (R d R d. By Fubini's theorem and the change of variables we obtain: { } ϕ(y t(xy β f (β (xydx dy = t(xϕ(yy β f (β (xydxdy σ(ξ 1 = t( ϕ(ξη(ξη β f (β (ηdξdη ξ 1 ξ d ξ ( σ(ξ 1 = t( ξ β ϕ(ξηdξ (P (θf(ηdη ξ 1 ξ d ξ = (P (θf(ηs ξ ϕ(ξηdη where P (θf(η = ξ β f (β (η and the distribution S is dened by S = σ(ξ 1 ξ t( β. ξ 1 ξ d ξ Our problem now comes down to the question: when is S D H (Rd? Then we have M T = L S f(θ. Since clearly supp S W ε for some ε > 0, the question is: when does η S ξ ϕ(ξη, η R d extend to a function in C (R d?

16 16 D. VOGT Sucient for that is if equivalent to We have shown: ξ γ ξ 1 ξ d t ( 1 ξ x γ t(x L 1 (R d for all γ N d 0. L 1 (R d for all γ N d 0 and this is Theorem 5.3. If T E (R d and there are nitely many functions t β L 1 (R d with compact support such that T = β t(β β and x γ t β (x L 1 (R d for all γ N d 0 then M T : C (R d C (R d extends to a map in M(R d. A condition about the behaviour of T at 0 is necessary as we have seen on Proposition 5.2. That in the assumptions for Theorem 5.3 we need a strong vanishing condition at zero and that a condition in the spirit of the O H (Rd -condition is not sucient is shown by the following easy example. Example 5.4. For any p N 0 we dene a function t p as follows: t p (x = 0 for x < 0, t(x = χ(xx p /p! for x 0 where χ D(R d and χ 1 in a neighborhood of 0. Then δ = t (p p + t 0 where t 0 D(]0, + [. So for any P the distribution δ is a sum of derivatives of functions with zeroes of order p in 0, but M δ does not extend to an operator in M(R d (see Proposition 5.2. References [1] P. Doma«ski, M. Langenbruch, Representation of multipliers on spaces of real analytic functions, Analysis 32 (2012, [2] P. Doma«ski, M. Langenbruch, Algebra of multipliers on the space of real analytic functions of one variable, Studia Math. 212 (2012, [3] P. Doma«ski, M. Langenbruch, Hadamard multipliers on spaces of real analytic functions, Adv. Math. 240 (2013, [4] P. Doma«ski, M. Langenbruch, D. Vogt, Hadamard type operators on spaces of real analytic functions in several variables, J. Funct. Anal. 269 (2015, [5] J. J. Duistermaat, J. A. C. Kolk, Distributions. Theory and Applications, Birkhäuser, Boston [6] L. Schwartz, Théorie des distributions I, Hermann, Paris [7] L. Schwartz, Théorie des distributions II, Hermann, Paris [8] R. Meise, D. Vogt: Introduction to functional analysis, Clarendon Press, Oxford, (1997.

17 HADAMARD OPERATORS ON D (R d 17 [9] D. Vogt, Operators of Hadamard type on spaces of smooth functions, Math. Nachr., 288 (2015, [10] D. Vogt, E as an algebra by multiplicative convolution, arxiv: (2015.