HOMOGENEOUS CONES AND ABELIAN THEOREMS

This paper deals with analysis on homogeneous cones in ℝn. This subject has its origins in 
one-dimensional topics that are connected, often implicitely, with some group properties. The homogeneous 
cones are open convex cones in ℝn that are at the same time homogeneous spaces, and they are more general 
than the classical, or symmetric cones. As an example of application of the theory of homogeneous cones we 
study the asymptotic behaviour of some integral transforms.

In sections 2 and 3 we review some properties of the homogeneous cones following mostly Vinberg [4].As an example of an application of this theory to analysis we study integral transforms and their asymptotic behaviour.Analysis on homogeneous cones was developed, and to a much larger extent, by Gindikin [5].We shall eventually reprove some of his results in the framework of Vinberg's theory.In this form these results look more similar to the ones for classical cones.In section 4 we consider the power functions which are the n-dimensional functions t-.-} ta; they also give the invariant measure and take part in the definition of homogeneous kernels.In sections 5 and 6 we deal with a class of integral transforms with homogeneous kernels and prove a theorem of Abelian type.Examples of these integral operators are the Laplace transform, the Riemann-Liouville operator and the Stieltjes transform.
2. THE CONE, THE GROUP, THE LIE ALGEBRA AND THE VINBERG ALGEBRA In this section we review some notions we shall need from the theory of homogeneous cones.In particular, we shall see how the group of automorphisms of the cone and its Lie algebra determine a Vinberg algebra in the ambient space DEFINITION 2.1.Let V be an open convex cone in n, which doesn't contain any straight line.
The cone V is said to be homogeneous if there is a group G of linear automorphisms (a subgroup of GL(n,]) which leaves V invariant) which is transitive on V; i.e., such that for every u, ve V there is an element g of G such that v gu.
A most important property of homogeneous cones is that they always have a simply transitive group of automorphisms, i.e. a group G such that for every u,v V there is a unique g G such that v gu.In other words, there is a bijection 1"I: G V which, if we fix an element c V once for all, assigns to every v V a unique gv G such that v gv c.Thus the group operation induces an operation for the elements of the cone v u gvguC. (2.1) We shall call this simply transitive group G the group of the cone.This group is triangular (real solvable) and if is the Lie algebra of G, then the exponential mapping is a bijection from the Lie algebra onto G.By taking the derivative of the mapping 1"I above, we see that the mapping n: IRn defined by X Xc is a vector space isomorphism.Thus we have bijective mappings between the four sets: V, G, and IRn.In particular, if for x A n we write L(x) for the unique element of such that x L(x)c, we can define the following operation x y L(x)y L(x)L(y)c.(2.2) This bilinear operation introduces into IR n the structure of a Vinberg algebra.This is the nonassoeiative algebra described in the next definition.
Equation (2.2) gives an action of the Lie algebra on n.Also, by the bijectivity the exponential mapping, every g G is of the form g expL(x), with some L(x) and the corresponding action of the Lie group G is given by gy expL(x)y. (2. 3) The orbit of the point c is the cone V.The Vinberg algebra has an identity c if moreover (B.3) xac cr x.
A Vinberg algebra 15 with identity c has a Pierce decomposition.First of all, there is a complete system of orthogonal idempotents Cl C,n, with cl+...+c,n c; the number m is called the rank of 15.Then 15 is decomposed into a direct sum of subspaces 15 i<j 15ij, i,j ,m (2.4) where 'Sii IRci and the subspace Eij is characterized by the fact that the operators L(ci) and L(cj) have the eigenvalue 1/2 on it, and the operator R(cj) has the eigenvalue 1; all the other L(ct) and R(ck) are zero on Eij.The subspaces Sij have dimensions nij.
We shall write xij for an element of Bij, and in particular, xii xici, with xi ]R.Thus by (2.4), every xe is of the form (2.5) Then for L(x) we have an analogous decomposition L(x) -'i xiL(ci) + i<j L(xij). (2.6) Consider the Lie algebra 9, which, as every triangular Lie algebra, is decomposed into the sum 9 + % of the abelian subalgebra and the niipotent subalgebra 'll, [9,].The operators L(ci) constitute a basis of the they commute since ci are orthogonal idempotents; this can be easily seen if we note that in terms of the operators of left multiplication condition (B) takes the form [L(x), L(y)] L(xay yax).The elements L(xij) belong to the nilpotent subaigebra 'II,.The trace of L(x) equals where we have put di TrL(ci) (since TrL(xij) 0).By the remark above about the eigenvalues of L(ci) We shall write d (dl dm), v (Vl Vm) and I.t (lXl lSm) for these three multi-indices, which are characteristic for the cone.We shall need them in sections 5, 7 and 8 and the values of these indices for different examples of cones are given in section 9.
In a Vinberg algebra we can introduce an inner product by (x ly) TrL(xay).
(2.9) Condition (B2) shows that this bilinear form is positive definite.In particular, by (2.7) we see that for the identity c we have (c x) ,xiai.
(2.0) i=1 3. THE MATRIX CALCULUS In this section we write an element x of a Vinberg algebra, given as in (2.5), in the form of a symmetric "matrix" in the following way.We write the real numbers xi as diagonal entries and the nij- dimensional elements xij ij in the ij and ji position.We thus define a mapping m: 5 --% from into the space of generalized symmetric matrices .T he matrix product for two such matrices is defined by making use of the operation (2.2) (see Vinberg [4] for details).It is a nonassociative operation.
Let be the subspace of upper triangular matrices in .I f A %, let , e W be the "upper trangular half' of A, i.e. a matrix , such that A =/] + ,r, where r is the transposition in this matrix algebra.For A, Be put A B= A B + Bt]r.
(3.1) THEOREM 3.1.(Vinberg [4]).The space % with the operation (3.1) is a Vinberg algebra and the mapping m: --% is an isomorphism of Vinberg algebras, i.e. m(x y) re(x) re(y).Isomorphic Vinberg algebras have equal inner products (2.9).Thus for the algebras and % we have (x ly) (re(x)Ira(y)) and in particular, since m(c) I, the identity matrix, we deduce from (2.10) m (IIA) aiid (3.2) i=1 where A re(x) and aii (= xi) are the diagonal elements of the matrix A. Now, consider the triangular subalgebra 7. It turns out that the restriction of the matrix product to is associative.Then the commutator [X, Y] XY-YX makes into a Lie algebra, which is isomorphic with .T his isomorphism is given by the mapping V: L(x) X which writes the element L(x) from (2.6) in the form of an upper triangular matrix by putting 1/2xi in the diagonal position and xij in the ij position, for < j.
(The factor 1/2 is introduced into the diagonal entries so that for A m(x) and X (L(x)) we have A X + xr.) Formula (3.1) actually defines the action of the Lie algebra 9" on %: ifXe 9" and Be % the mapping p(X): B XB + BX (3.3) acts on % and by Theorem 3.1 it is isomorphic with the action of the element L(x) of the Lie algebra defined in (2.2).Now let T expg" be the Lie group with Lie algebra 9".It is easily seen, by standard arguments, that the exponential mapping is given by expX I + X + X212! + X3/3! + where X 2 XX, etc. is the matrix product.Then it follows that T is the subset of 9" consisting of matrices with positive diagonal elements.
The groups T and G are isomorphic and the isomorphism P is the exponential of , i.e.F: g expL(x) expl/(L(x)).
(3.4) The exponential of the action p(X) of (3.3) gives the following representation P of T in P(t): B -* t(Btr) (3.5) which is isomorphic to the action of g expL0c) in (2.3).LEMMA 3.1.Let te T be the image by F of g expL(x).Then the diagonal elements of are equal to tii ex/2.Indeed, we are looking for the i-th diagonal element of expw(L(x)).Since is a triangular matrix, which is equal to the exponential of a triangular matrix X w(L(x)), we have that tii is equal to the exponential of the i-th diagonal element of X, which is equal to x//2.
The orbit by T of the point I re(c) is a cone W, isomorphic to V. Indeed, it is readilly seen that W re(V). Thus every we Wis in P(T)L in other words, there is a te T such that w ttr.
(3.6) Equation (3.6) is called the Gauss decomposition of the symmetric matrix w.Matrices having a Gauss decomposition will be also called "positive definite" (because the diagonal elements of are positive).Thus W is the cone of positive definite matrices in %.This is analogous to the case of the symmetric cones consisting of positive definite elements of the corresponding Jordan algebra.Note how here the Gauss decomposition tt r is obtained immediately, whereas in Jordan algebras it requires some work (see Faraut and Koranyi [6]).
Let us close this section with a formula in which an instance of the matrix product is computed and which we shall need in section 7.For an upper triangular matrix whose matrix components are tij the diagonal components of the product tt r are equal to m j=i+l where It/j[] 2   (tijltij) (see Vinberg [4]).And now by <3.2) we have (I ttr) t + - (3.7)   i--1 i<j 4. THE POWER FUNCTIONS The power functions are generalizations of the function x x a in one variable.They are in fact the group characters.For the classical cones the power functions are the principal minors of the matrix.This is generalized to any homogeneous cone, the main tool being the Gauss decomposition; if in w tt r we take the entries of the matrix as coordinates for w, then the power function has a particularly simple form it is a product of diagonal coordinates (see (4.5) below).
Note first that the mapping I'I: G ---> V gives a bijeetion between the functions F: V ---> IR+ and f: G ---> IR+ in the following way  It follows that (gv) lff,g)(v) and dp(v.v')dO(v) qp(v') (4.2) for every g G and v,v' V (where. is defined as in (2.1)).It is easy to determine the exact form of all power functions.
LEMMA 4.1.Let be a power function and let g G be written as g expL(x), with L(x) as in (2.6).
Then there exist real numbers i such that (g) eqX,+--.+,x,.( If o (1 or,n) is a multi-index, we shall write a for the power function in (4.3) and Oa for the power function related to o as in (4.2).Denote by i and I the elementary power functions Oi(g) eX', i(v) e x, ( Thus every a is equal to a , ...amm" Now let T be the triangular matrix group from section 3, isomorphic with G.We shall write 6afor the power function on T and iii a for the corresponding power function on the cone W m(V). Then we have for W(g) as in (3.4)   ....,,- Ov) ilia(re(v)) Indeed, it is obvious that isomorphic groups have equal characters.Now by Lemma 3.1 we have tii eX/2, and the second equality in (4.5) follows by (4.3).
An important example of a power function is the norm N: V --IR+ defined as N(gv) Detg N(v).
As another example, take the multi-index 11 (1 and put and call this function the delta function of the cone. REMARK 4.1.By applying (4.5) we see that when v is an ordinary matrix (written as v ttr), then the delta function A(v) is the ordinary determinant of v. Indeed, A(v) l(t) (t l...tmm) 2 according to (4.5) and this equals (det 0 2 dew.When in (4.8) the multi-index d is a constant di d, and this is the case only for symmetric cones, we obtain a formula N(v) (A(v))d, well-known in Jordan algebras (see Faraut and Koranyi [6]); in the theory of Jordan algebras A(v) equals detv, the determinant of an element in the Jordan algebra.We write capital letters when Det and Tr are the determinant and the trace of a linear operator (in (4.8) g is identified with the linear operator, as in (2.3)).

AN INTEGRAL FORMULA
In section 6 we shall deal with integral transforms of functions defined on V.The formula in Theorem 5.1 below will be used to compute some integrals.The Gauss decomposition (3.6) provides a good change of variables in an integral containing a power function, since the Jacobian for this change of variables is also a power function.We shall apply this formula in section 7 to compute the gamma function of the cone.For the classical cones this formula can be found in Garding [7].It was generalized to the symmetric cones in Faraut and Koranyi [6].Gindikin [5] has evaluated the integral for the gamma function by a different parametrization.
First consider the invariant measure on V.Here a measure dm is called invariant if it is invariant under the action of G: dm(gv) din(v), for every g G, v V. Then by (4.8) we find easily the following lemma.
LEMMA 5.1.The invariant measure on V is equal to din(v) dv/N(v).
Here N is the norm of V and, for v written as in (2.5), dv dvl...dvmdvl2...dv,n.lm is the Lebesgue measure on IRn, where dvij is the nij-dimensional Lebesgue measure on the space IRnij ij.
We shall take the cone in its W form (and write again ot in the place of ili a).The change of variables in the following theorem is given by I'I !: T--o W with w= Ill(t) ttr. (5.1) Recall that the multi-indices d, v and Ix were defined in (2.8).
THEOREM 5.1.Let F: W-+ IR+ be integrable with respect to the invariant measure din.Then Iw F(w)dm(w) 2m IT F(ttr-'(t)dt where , is the multi-index , 1/2 + IX/2.PROOF.By Lemma 5.1 the invariant measure is given by the power function -d(w) -d(t).Thus we only have to show that the Jacobian of (5.1) is equal to 2mdpd-l(t).This is done in the following three lemmas.
LEMMA 5.2.Letlt: T --Tbe the left translation in the group: It(s) ts and let Ill be as in (5.1).
(5.2) PROOF.Since Ill(t) tt r P(t)l (by (3.5)) we have for the left-hand side of (5.2) Ill It(s)   1-11(ts) P(ts)l and for the fight-hand side of (5.2) we have P(t)* Ill(s) P(t)P(s)l.Since P(t) is a group action, the lemma follows.
LEMMA 5.4.If It is the left translation in T, then Dt It tV(t).PROOF.Since It Itl we have that .(t)Det It is indeed a power function.Thus we only have to determine the exponent ),.By Lemma 4.1 a power function depends only on the abelian part.This means that if a is the abelian part of t T, then k(t) ,(a).Thus to compute .(a)consider la T T. By Lemma 3.1 the elements of the diagonal matrix a are d2 so that la multiplies the i-th row of s T by a factor ex/2.
Then in the Jacobian we have this factor ?i times, where ?i is the sum of the dimensions in the i-th row, and this is equal to + ti (see (2.8)).
Now to finish the proof of the theorem, we have to substitute in Lemma 5.3 the values of Det/t '(t) and DetP(t).Since P(t) (3.5) is an action isomorphic with (2.3), we have Det P(t) Detg and this equals t(g) t(t).Finally the Jacobian is equal to 2-'t(t).This proves the theorem.

INTEGRAL TRANSFORMS WTI'H HOMOGENEOUS KERNELS
Let vRn be a homogeneous cone and let F: V -lq.+.We shall consider operators of the form o:T(v) vk(V,u)F(u)dm(u) (6.1)where k: V V-+ 1;I + is a given function, the kernel of and dm is the invariant measure.The kernel is said to be homogeneous of order T if k(gv, gu) #:'t(g) k(v, u) (6.2) for every g G and v,u V, where ' is the power function.It is easily seen that if ."has a homogeneous kernel, it transforms power functions into power functions.LEMMA 6.1.Let an operator with homogeneous kernel of order ?.If the integral rx Jv k(c,u)da(u)dm(u) (6.3) is convergent, then the integral defining ..Xe'v) is convergent for every v V and .v):a t(v). (6.4) PROOF.Equation (6.2) can be written as k(v,u) (v(C,gv-lu).Now by putting this into the definition of ..eF(v) and changing the variable y gv'lu we obtain .'F(v)t(v) Iv k(c, y)F(v y)dm(y).(6.5) Finally put F D ot in this formula and use (4.2).
Postponing until the next section the computation of r, for some examples of operators (6.1), we are now going to prove that if a function F is asymptotically equivalent to a power function, then its integral transform is asymptotically equivalent to a power function.To define limits we consider translation invariant filters.DEFINITION 6.1.A filter ' of open convex subsets of V is called translation invariant if it has a countable basis and if, for every y V, we have that S 'implies S.y .
We shall write v-->, for the convergence with respect to this filter.Thus we have that v-->, implies v.y-->=o, for every y V.
Two functions F, G: V --> -,+ are said to be equivalent at infinity with respect to , if lim F(v)/G(v) 1.This will be denoted by F ,x, G. REMARK 6.1.A different kind of asymptotic relations for functions defined on cones in IR n were considered in Vladimirov et al. [8] and Yakymiv [9].The notion of limit in Definition 6.1 is more general (it is taken after Bajsanski and Karamata [10]).Indeed, by the identification (4.1), it is equivalent to consider a filter of subsets in G and then, for example, lim F(v.y) is the same as lim F(gy).In our notation, the convergence in Vladimirov et al. [8] becomes convergence with respect to one parameter subgroups g(t) (with t-->,) in G.The asymptotic in Yakymiv [9] is even more special, with g(t) the group of dilations.(Compare also with Stadtmiiller [11], where g(t) is an arbitrary curve, but only in the cone lRn+.)Both in Vladimirov et al. [8] and Yakymiv [9] the cone is only supposed to be (the closure of) an open, convex cone, and it has less structure than a homogeneous cone.THEOREM 6.1.Let .,r'bean integral operator with homogeneous kernel of order ?. Let F: V --> IR+ be such that F is bounded.Then F ,x, tl implies ,x, ctl+' for those t for which the integral r c) is convergent.
We shall first consider the case when t 0. Then F(v) ---> 1, v --> .Since we are only interested in the asymptotic behaviour, we may modify the functions so that they become bounded on the complements of the elements of the filter basis (without changing its behaviour at o). Together with F(v) --> 1, this modification yields that F is bounded everywhere.PROPOSITION 6.1.Let be an integral operator with homogeneous kernel of order [ and such that :0 < ".Let F: V--+ R+ be bounded.Then F(v) --> 1, when v --> implies to0 . ( (Here gO is defined in (6.3), with 0 .)PROOF.By applying (6.5) we have that (6.6) is equivalent to the fact that the following expression tends to zero q-l(v)..rF(v) to0 [.,k(c,y)(F(v y)-1)din(y).( Now for > 0 take a (large) compact set D c_ V such that v_Dk(C, y)dm(y) < (6.8) which is possible by the assumption c0 < oo, and write the integral in (6.7) as To estimate ;l, note that, by the translation invariancc of the filter, v --> implies v.y--> oo, for every y, so that F(v.y) tends to l, when v --> oo, and in fact uniformly in y D. Thus we can find an Se .9'so that for v $ we have [11 < .ok(c,y)dm(y) < .;Vk(c, y)dm(y) For 12, since F is bounded (by C, say), we have 12 < (C+I) g_Dk(C,y)dm(y) < (C+1) by (6.8).This proves the proposition.
PROOF OF THEOREM 6.1.Put FI(y) F(y)-'y) and kl(c,y) qa(y)k(c,y).Then we have ..fffFI (v) .,ff"F(v)and we are going to apply Proposition 6.1 to the operator .,fir.By the assumptions of the theorem, F1--> and is bounded, so that it satisfies the conditions of the proposition.The kernel kl is homogeneous of order + ?and satisfies the conditions of the proposition, since the constant r,0 for this operator equals ..'O0(c) ."OC(c)ga and is finite.Now an application of the propostion gives .,ffi'F1,x, rO13 and since [3 + y, we finally find ..Z,'F .,i'F1rq+, which proves the theorem.
7. THE LAPLACE TRANSFORM In this and the following two sections we consider some examples of the operators (6.1).They are n- dimensional analogues of some classical operators and were considered by Gindikin [5].Here we shall apply Vinberg's theory of homogeneous cones to compute the integrals as in (6.3).
To define the Laplace transform, consider first the norm N like in (4.7) and the inner product defined in (2.9), and for v Vlet v* R n be such that for every y 1 n we have (v*ly) dlogN(v)y.(See Koecher [2] or Rothaus [3]; actually v* belongs to the dual cone of V.) The mapping * has the property ((gv)*l gu) (v*lu) (7.1) for every g G and v,u V.
Here the kernel is k(v,u) e-(V*lu), and by (7.1) this is a homogeneous kernel of order 0. In order to apply Theorem 6. to this operator we need to know the integral "Z'dhCt(c) Jv e'(c lu)diX(u)dm(u) FV(o:) vhich is called the "gamma function of the cone" (see Koecher [2] or Gindildn [5]).
PROPOSITION 7.1.Let a be a multi-index such that o > l.t/2.Then the integral def'ming Fv(ot) is convergent.
PROOF.First since W m(V), we have Fv(c) w e'(! tw)(w)dm(w)" where now a is the power function of W, which was denoted before by ili a (see (4.6)), dm is the invariant measure on W and the inner products on two isomorphic cones are exlual.Now in this integral we put w tt r (see (5.1)).By Theorem 5. we have FIAt) 2m[r e'(llttr)O?ct(t)O-(t)dt.
The second type of integrals in (7.3) is I O" [ e'I/2(t)dt where t/ are the coordinates of tij in some basis in ij.By a well-known formula, I O. (2t) I/2.This proves the proposition.Now an application of Theorem 6. gives the following.THEOREM 7.1.(Abelian Theorem for the Laplace Transform).Let ot > 0/2.Let F: V --IR+ be such that F -t is bounded.Then F ,x, Oa implies .Z'F Fv(a) o.

THE RIEMMAN-LIOUVILLE OPERATOR AND THE STIELTJF TRANSFORM
In this section we define two more operators.Again we shall have to prove that the integrals (6.3) for these operators are convergent.But now there is no need to evaluate these integrals again, since they are generalizations of the beta function and can be obtained by reducing to the gamma functions, by formulas completely analogous to the classical ones.This was done in Gindikin [5]; for completeness we reproduce briefly these formulas.
The cone V defines a partial order in IR n in the following way: v < u iff u ve V. Write (a,b) for the "interval" with respect to this order, i.e. for the set of all elements v V such that a < v < b.
Let [ be a multi-index, then the Riemann-Liouville operator is defined by where the kernel k(v,u) %(o,v)(u)p[(v-u) is homogeneous of order [.Here : is the characteristic function; and since g G preserves the cone, it also preserves the order: u < v implies gu < gv.LEMMA 8.1.Let c > 2 and 3'> p/2.Then the integral is convergent (0,c) :(u)P'r-d(c-u)dm(u) Bv(fx,3').This is easily sen if we consider the double integral defining I'v()Fv(3'), which by Proposition 7.1 is convergent for x > IX/2 and 3'> la/2.By changing the order of integration this integral is equal to rl/(x)rv(3") IV e-(c Iv) I(0,v) Pu)OY-d(v-u)dm(u)dv and by Fubinrs theorem the inner integral is convergent for every v V and for and 3' as above.THEOREM 8.1.(Abelian Theorem for the Riemann-Liouville Operator).Let [ > -1 v/2 and let for c > la/2 the function F: V IR+ be such that F is bounded.Then F ,',., I implies .F Bv(ot,[+d) Oo+l.
Let p be a multi-index.Then the Stieltjes transform is defined by F(v) v -P(v+u)F(u)dm(u) and its kernel k(v,u) -P(v+u) is homogeneous of order -p.LEMMA 8.2.Let p > IX and /2 < x < p IH2. Then the following integral is convergent v O-P(c+u)O(u)dm(u).This is sen by introducing the change of variables u c + y, and after that the change x u "1 gu-lc (the Jacobian is equal to Det(g-2) -2d(g), se Gindikin [5]).Then we obtain Bv(p-o,c).THEOREM 8.2.(Abelian Theorem for the Stieltjes Transform).Let p > Ix and let for IH2 < x < p p/2 and let the function F: V -+ IR+ be such that F -c is bounded.Then F ,' c implies .pF ,',., Bv(p-c,c)I--P.

EXAMPLES OF CONES
In this section we consider some examples of homogeneous cones.By Vinberg's matrix calculus we find the explicit form of the power function in much the same way as for the cones of ordinary matrices in example 1) below.We also write down the exact values of the constants Ixi, v and di appearing in the formulation of the Abelian Theorems.
1) The Classical Cones.Let IR n m(F) be the space of symmetric mm matrices with coefficients in F R, C or I-I.Let V be the cone of positive definite matrices m(F).Here x IR n is already given in its matrix form.We have nq 1, 2 or 4, for P IR, : or l[-I, respectively, and if we put b 1, 2 or 4, then vi b(i 1) and Ixi b(m i).Then di d is constant, and equals (m+1)/2 in 1,; equals m in ; and equals 2m-in If v V, then v n r is the well-known representation of a positive definite matrix.We know by (4.5) that (v) tii .T hen the delta function equals A(v) ql...tmm; i.e. it is equal to the determinant ofv and the norm is N(v) ld(v)...fYPmd(v) (A(v)) d (det v)d.
2) The Light Cone.This is the only symmetric cone not having a classical representation as a matrix cone.
It is the set V_ 1, n of all elements v such that VlV2 v32 Vn 2 > 0 and v2 > 0.Here m 2. The matrices from section 3 are 22 matrices A with al Vl, a22 v2 and a12 (v3 Vn).Thus we have n12 n-2 and nl 0, n2 n-2, n n-2 and n 2 0. It follows that dl d2 n/2.Now A tt r with an upper triangular matrix of the same type.In this product the entries of the form tii t12 are SCalar by vector products and t12 t12 is the inner product of two (n-2)-vectors.If we write down this product we find l(V) tl 12 al a12a12/a22 Vl (v32 +...+ Vn2)/v2 2(v) t222 a22 v 2. The delta function is A(v) bl(v)cl2(v) VlV2 v32 vn 2, and the norm N(v) (A(v))n/2.