SEVERAL PRODUCTS OF DISTRIBUTIONS ON MANIFOLDS

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1 Novi Sad J. Math. Vol. 39, No., 2009, 3-46 SEVERAL PRODUCTS OF DISTRIBUTIONS ON MANIFOLDS C. K. Li Abstract. The problem of defining products of distributions on manifolds, particularly un the ones of lower dimension, has been a serious challenge since Gel fand introduced special types of generalized functions, which are needed in quantum field. In this paper, we start with Pizetti s formula and an introduction on differential forms and distributions defined on manifolds, and then apply Pizetti s formula and a recursive structure of j (X l φ(x)) to compute the asymptotic product X l δ(r ). Furthermore, we study the product f(p,, P ) α δ(p,, P ) P α P α on smooth manifolds of lower dimension, which extends a few results obtained earlier. Several generalized functions, such as δ(qp,, QP ) and δ(q P,, Q P ), are derived based on the transformation of differential form ω. AMS Mathematics Subject Classification (2000): 46F0 Key words and phrases: Distribution, product, manifold, differential form, transformation and Pizetti s formula. Pizetti s formula and differential forms The simplest example of a generalized function concentrated on a manifold of dimension less than n is one defined by (f, φ) = f(x)φ(x)dσ, S where S is the given manifold, dσ is the induced measure on S, f(x) is a fixed function, and φ D(R n ). As an example, let us consider the distribution δ(r a), where r 2 = n i= x2 i and a > 0. The equation r a = 0 defines the sphere O a of radius a. We have (δ(r a), φ) = φdo a. O a where do a is the Euclidean element on the sphere r a = 0. Department of Mathematics and Computer Science, Brandon University, Brandon, Manitoba, Canada R7A 6A9, lic@brandonu.ca

2 32 C. K. Li To mae this paper as self-contained as possible, we begin to state Pizetti s formula and briefly introduce differential forms in the following, which are extremely helpful in defining distributions on manifolds in an invariant way. Please refer to reference [] for detail. Assume dσ is the Euclidean area on the unit sphere Ω (= O )in R n, and S φ (r) is the mean value of φ(x) D(R n ) on the sphere of radius r, defined by S φ (r) = φ(rσ)dσ Ω n Ω where Ω n = 2π n 2 /Γ( n 2 ) is the hypersurface area of Ω. asymptotic expression for S φ (r)(see [] ), namely We can write out an S φ (r) φ(0) + 2! S φ(0)r = =0 (2)! S(2) φ(0)r 2 2! n(n + 2) (n + 2 2) φ (0)r 2 + ( is the Laplacian) which is the well-nown Pizetti s formula and it plays an important role in the wor of Li, Aguirre and Fisher [2-0]. Remar: Pizetti s formula is not a convergent series for φ D(R n ) from the counterexample below. φ(x) = { exp{ r 2 ( r 2 )} if 0 < r <, 0 otherwise. Clearly, φ(x) D(R n ) and S φ (r) 0 for 0 < r <, but the series in the formula is identically equal to zero. Obviously, S φ (r) 0 as r 0. However, it converges in spaces of analytic functions from the reference []. A differential form of th degree on an n-dimensional manifold with coordinates x, x 2,, x n is an expression of the form aii 2 i (x)dx i dx i2 dx i, where the sum is taen over all possible combinations of indices. The coefficients a i i 2 i (x) are assumed to be infinitely differentiable functions of the coordinates. Two forms of degree are considered equal if they are transformed into each other when products of differentials are transposed according to the anti-commutation rule dx i dx j = dx j dx i and all similar terms are collected. This rule implies that if a term in a differential form has two differentials with the same index, it must be zero. It can be used to write any differential form into canonical form, in which the indices in each term appear in increasing

3 Several products of distributions on manifolds 33 order. Clearly, the anti-commutation rule holds for any differential forms of first degree. Indeed, let α = a j (x)dx j and β = b (x)dx ; then αβ = j, a j (x)b (x)dx j dx = j, a j (x)b (x)dx dx j = βα. Let us find how differential forms transform under an infinitely differentiable change of coordinates given by x i = x i (x, x 2,, x n). We have and i < <i a i i dx i dx i = dx i = n j= i < <i x i x dx j j j x i a i i x x i j x dx j dx j. j In the sum we have obtained, the terms in which the same differential occurs twice will vanish. Different terms containing the same combination of differentials can be combined using the anti-commutation rule, which holds also for the dx j. Then it follows that for j < j 2 < < j, the coefficient of dx j dx j is multiplied by the Jacobian ( ) xi x D i2 x i x j x j 2 x. j We thus arrive at where i < <i a i i dx i dx i = a j j = ( xi x D i2 x i i < <i The exterior derivative of a differential form j < <j a j j dx j dx j, x j x j 2 x j ) a i i. α = a i i dx i dx i is defined as the ( + )st degree differential form dα = ( ) a i i dx i dx i dx i, x i i i i which, of course, can be simplified by using the anti-commutation rule. Let a(x) be a scalar function. Then n a(x) da(x) = dx i. x i i=

4 34 C. K. Li It is easily shown that according to the anti-commutation rule, any differential form α satisfies the equation ddα = 0. Let us assume that and the claim holds since and the anti-commutation rule α = a i i dx i dx i 2 a i i (x) x i x j = 2 a i i (x) x j x i dx i dx j = dx j dx i. Let α be a differential form of degree n defined on some bounded n- dimensional region G with a piecewise smooth boundary Γ. We assume an orientation of G corresponding to the positive direction of the normal to Γ. Then dα = α which is called the Gauss-Ostrogradsii formula. G As an example, consider a second degree form α given below in three dimensions α = a dx 2 dx 3 + a 2 dx 3 dx + a 3 dx dx 2 and its exterior derivative is dα = ( a x + a 2 x 2 + a 3 x 3 )dx dx 2 dx 3, so that the Gauss-Ostrogradsii formula turns to be a dx 2 dx 3 + a 2 dx 3 dx + a 3 dx dx 2 = ( a + a 2 + a 3 )dx dx 2 dx 3 x x 2 x 3 Γ which is seen in calculus. We consider a manifold S given by P (x, x 2,, x n ) = 0, where P is an infinitely differentiable function such that { P gradp =, P,, P } 0 x x 2 x n on S, which therefore has no singular points. The differential form ω is defined by Γ G dp ω = dv

5 Several products of distributions on manifolds 35 where dv = dx dx n, and dp is the differential form of P. Note that if P (x) is the Euclidean distance of x from the P = 0 surface, the differential form ω on S coincides with the Euclidean element of area dσ on S. Since gradp 0 on S, there exists j ( j n) such that P/ x j 0. We may introduce a local coordinate system u, u 2,, u n to be () u = x,, u j = P (x),, u n = x n. Then and thus we may set ( ) x D = u [ D ( )] u = x P/ x j, ω = ( ) j dx dx j dx j+ dx n P/ x j. We naturally define the characteristic function θ(p ) for the region P 0 as (θ(p ), φ(x)) = φ(x)dx P 0 where φ D(R n ), and the generalized function δ(p ) by (δ(p ), φ(x)) = φ(x)ω. Kanwal [2] studied certain distributions defined on the surface Σ(t) and their extensions to the whole space. The basic distribution concentrated on Σ(t) is the Dirac delta function, whose action on a test function φ(x, t) is given by + δ(σ), φ) = φ(x, t)ds(x)dt, Σ(t) where ds(x) is the surface element. Observe the special treatment of time in the above integral. The integration with respect to the space variables is surface integration while that with respect to time is ordinary integration. According to Kanwal, the relation between δ(p ) and δ(σ) is given as (δ(p ), φ(x)) = φ(y)ds(y) gradp, which implies Σ δ(p ) = δ(σ) gradp. Another way of introducing the distribution δ(p ) is used by DeJager [3]; (δ(p ), φ(x)) = lim φ(x)dx. c 0 c 0 P c

6 36 C. K. Li Similarly, its higher derivatives can be defined as δ () (P ) = lim c 0 c [δ( ) (P + c) δ ( ) (P )], =, 2,. It follows from DeJager [3] that (δ(p ), φ(x)) = lim c 0 c = lim c 0 c = which coincides with the Kanwal s result. It was proven in [] that θ(p ) x j 0 P c φ(x)γds(x) φ(x) c ds(x) gradp φ(x) ds(x) gradp = P x j δ(p ). We shall first add the following identity, which has never appeared so far, according to the author s nowledge Indeed, θ(p ) P = δ(p ). ( θ(p ) P, φ(x)) = (θ(p ), P φ(x)). Since φ = φ(x, x 2,, x j (P ),, x n ) by the substitution of (), we come to (θ(p ), P φ(x)) = (θ(p ), φ(x) φ(x) ) = dx. x P j x j P 0 x P j x j On the other hand, (δ(p ), φ(x)) = φ(x)ω. Let us assume that P 0 defines a bounded region. Then we may apply the Gauss-Ostrogradsii formula to the above integral over this region and to the differential form of degree n in the integrand. We also use the fact that P increases into the interior of the region to derive φ(x)ω = d(φ(x)ω) P 0 and d(φ(x)ω) = φ(x) x j P x j dx + φ ( x j x j P )dx = φ(x) x j P x j dx,

7 Several products of distributions on manifolds 37 which implies φ(x) φ(x)ω = dx. P 0 x P j x j Hence the identity holds on any bounded region. If P 0 does not define a bounded region, we replace it by its intersection G R with a sufficiently large ball x R outside of which φ(x) is nown to vanish. Let Γ R be the boundary of G R, we have φ(x)ω = Γ R G R φ(x) x j P x j dx. Now, since φ(x) vanishes outside of x R, we arrive at φ φ(x)ω = P 0 P dx, which completes the proof. It is well nown that in one dimension every functional concentrated on a point is a linear combination of the delta function and its derivatives. For n >, we have a similar role played by generalized functions, δ(p ), δ (P ),, δ () (P ) (the derivatives of δ(p ) with respect to the argument P ), which we shall define based on the differential forms ω (φ) given by ω 0 (φ) = φ ω, dω 0 (φ) = dp ω (φ), dω (φ) = dp ω (φ), where d denotes the exterior derivative. Now we are able to define (δ () (P ), φ) = ( ) ω (φ) for = 0,, 2,, since the above integral over the P = 0 surface of any of the ω (φ) is uniquely determined by P (x). Furthermore, we define the generalized function δ(p )/ P as ( φ δ(p ), φ) = P P ω. We shall show that P δ(p ) = δ (P ).

8 38 C. K. Li In fact, On the other hand, ( φ δ(p ), φ) = P P ω = ω 0 ( φ P ). (δ (P ), φ) = ω (φ) = P ( φ )dx dx j dx j+ dx n. P/ x j Since φ = φ(x, x 2,, x j (P ),, x n ) and P/ x j is not a function of P, we imply P ( φ )dx dx j dx j+ dx n = P/ x j φ P P/ x j dx dx j dx j+ dx n = ω 0 ( φ P ), by choosing the coordinates u i = x i, and u j = P. Under these coordinates This completes the proof. ω (φ) = P ( φ )dx dx j dx j+ dx n. P/ x j Similarly, we can obtain P δ() (P ) = δ (+) (P ) for =, 2,. We now prove the following recurrence relations, identities between δ(p ) and its derivatives: P δ(p ) = 0 P δ (P ) + δ(p ) = 0 P δ (P ) + 2δ (P ) = 0 P δ () (P ) + δ ( ) (P ) = 0 The first of these is obvious, since the integral of P φ over the P = 0 surface clearly vanishes. We now tae the derivative with respect to P to get as well as the rest similarly. P δ (P ) + δ(p ) = 0

9 Several products of distributions on manifolds The product X l δ(r ) Let X = n i= x i. We shall use a recursion and Pizetti s formula to derive the asymptotic product X l δ(r ) for any integer l, which is not possible to obtain along the differential form approach, since X is clearly not a function of r. Setting ψ(x) = X l φ(x) and obviously ψ(x) D(R n ). We naturally have (X l δ(r ), φ(x)) = (δ(r ), X l φ(x)) = X l φ(x)dσ r= = ψ(x)dσ = Ω n S ψ (). r= It follows from Pizetti s formula and ψ(0) = 0φ(0) = 0 that (X l δ(r ), φ(x)) Ω n j= j ψ(0) 2 j j! n(n + 2) (n + 2j 2). In order to calculate X l δ(r ), we need to express j ψ(0) in terms of a finite combination of φ and its derivatives at x = 0. First, we claim for j 0 that (2) j+ (Xφ) = 2(j + ) j φ + X j+ φ where = / x + + / x n. We use an inductive method to prove it. It is obviously true for j = 0. Assume j =, we have 2 (x i φ) = 4 x i φ + x i 2 φ simply by calculating the left-hand side. Hence 2 (Xφ) = 4 φ + X 2 φ. By hypothesis, it holds for the case of j, that is Hence it follows that j (Xφ) = 2j j φ + X j φ. j+ (Xφ) = j (Xφ) = (2j j φ + X j φ) = 2j j φ + (X j φ) = 2(j + ) j φ + X j+ φ. Clearly, we have from equation (2) that (3) j (Xφ(x)) = 2j j φ(0) = 2j( j δ(x), φ(x)) for j.

10 40 C. K. Li Next, we are going to calculate j (X 2 φ(x)) based on j (Xφ(x)). Indeed, j (X 2 φ(x)) = j (XXφ(x)) = 2j j (Xφ(x)) + X j (Xφ(x)). By simple calculation, Hence it follows that (Xφ(x)) = nφ(x) + X φ(x). j (X 2 φ(x)) = 2nj j φ(0) + 2j j (X φ(x)). Using equation (3), we obtain Thus, j (X φ(x)) = 2(j ) 2 j 2 φ(0). j (X 2 φ(x)) = 2nj j φ(0) j(j ) 2 j 2 φ(0). In order to construct a recursion of computing j (X l φ(x)), we need to search for a pattern, and continue on j (X 3 φ(x)) = j (XX 2 φ(x)) = 2j j (X 2 φ(x)) + X j (X 2 φ(x)). Similarly, Therefore, (X 2 φ(x)) = 2nXφ(x) + X 2 φ(x). j (X 3 φ(x)) = 2j j (2nXφ(x) + X 2 φ(x)) = 2 2 nj j (Xφ(x)) + 2j j (X 2 φ(x)). Since, j (Xφ(x)) = 2(j ) j 2 φ(0) and j (X 2 φ(x)) = 2n(j ) j 2 φ(0) (j )(j 2) 3 j 3 φ(0). Finally, we arrive at j (X 3 φ(x)) = 2 3 nj(j ) j 2 φ(0) nj(j ) j 2 φ(0) j(j )(j 2) 3 j 3 φ(0). In general, j (X l φ(x)) = j (XX l φ(x)) = 2j j (X l φ(x)) + X j (X l φ(x)). Clearly, (X l φ(x)) = n(l )X l 2 φ(x) + X l φ(x).

11 Several products of distributions on manifolds 4 Hence, j (X l φ(x)) = 2j j (n(l )X l 2 φ(x) + X l φ(x)) = 2nj(l ) j (X l 2 φ(x)) + 2j j (X l φ(x)). This is obviously dependent on the two previous terms of j (X l 2 φ) and j (X l φ), and forms a recursion for computing j (X l φ(x)), although the author is unable to write out the explicit formula at this moment. In particular, we have Xδ(r ) Ω n j=0 j δ(x) 2 j j! n(n + 2) (n + 2j), X 2 δ(r ) Ω n δ(x) + Ω n δ(x) 2(n + 2) + Ω n 2 δ(x) n(n + 2) n j δ(x) + 2j 2 j δ(x) + Ω n 2 j j! n(n + 2) (n + 2j). j=2 3. The product f(p,, P ) α δ(p,,p ) P α P α We now turn our attention to new generalized functions associated with manifolds S of lower dimension defined by equations of the form P (x,, x n ) = 0, P 2 (x,, x n ) = 0,, P (x,, x n ) = 0. where is in general greater than one. Following [], we shall mae the two assumptions: (i) The P i are infinitely differentiable functions. (ii) The P i (x,, x n ) = η i hypersurfaces (i =, 2,, ) form a lattice such that in the neighborhood of every point of S there exists a local coordinate system in which u i = P i (x,, x n ) for i =, 2,, and the remaining u +,, u n can be chosen so that the Jacobian D ( x u) > 0. Consider the element of volume in R n dv = dx dx n a differential form of degree n, and let us write it as the product of the firstdegree differential forms dp dp with an additional differential form ω of degree n ; i.e. dv = dp dp ω.

12 42 C. K. Li It was proven in [] that such ω exists, but can not be unique, and dp dp = i < <i D ( ) P P 2 P dx x i x i2 x i dx i. i We define the generalized function δ(p,, P ) by the equation (δ(p,, P ), φ) = It can be easily shown that this definition is independent of the particular choice of ω. Let us denote ω 0,,0 (φ) = φω. Then we define the differential form ω,0,,0 (φ) (whose integral over S will give δ(p,, P )/ P ) as follows. We tae the exterior derivative of the differential form of degree n, dp 2 dp ω 0,,0 (φ), and write it in the form S φω. d(dp 2 dp ω 0,,0 (φ)) = dp dp ω,0,,0 (φ). We choose the local coordinate system in which the u i = P i for i =,,, and denote φ(x (u,, u n ),, x n (u,, u n )) by φ(u,, u n ) = φ(u); we then obtain ( ) ω 0,,0 (φ) = φω = φd x du + du n, u ( ) dp 2 dp ω 0,,0 (φ)ω 0,,0 (φ) = φd x du 2 du n, u d(dp 2 dp ω 0,,0 (φ)ω 0,,0 (φ)) = [ ( )] x φd du du n, u u which implies ω,0,,0 (φ) = u [ φd ( )] x du + du n. u Any of the indices of ω 0,,0 (φ) can be changed from zero to one in the same way. In general, assuming that we now ω α,,α (φ), we may raise its jth index by multiplying on the left by all the dp i with i j, taing the exterior derivative, and writing d(dp dp j dp j+ dp ω α,,α (φ)) = ( ) j dp dp ω α,,α j,α j+,α j+,,α (φ). This defines the ω α,,α (φ) for any nonnegative integral indices.

13 Several products of distributions on manifolds 43 Obviously, if ω 0,,0 (φ) is not unique, neither are the ω α,,α (φ). We now define the generalized function α δ(p,,p ), where α = α + + α, by P α P α ( α δ(p,, P ) P α P α, φ) = ( ) α which is independent of the choice of ω α,,α. S ω α,,α (φ), Theorem. Let f(u, u ) be an infinitely differentiable function of variables. Then the product f(p,, P ) α δ(p,,p ) exists and f(p,, P ) α δ(p,, P ) P α P α = ( ) ( ) α α α j j j P α P α u α j u α j α j =0 α j =0 ( ) α j f(0,, 0) j δ(p,, P ) P j P j. Before going into the proof, we would lie to give the following products, if f(p,, P ) = P i, by Theorem. P i δ P i (P,, P ) + δ(p,, P ) = 0, P i δ (m) P i,,p i (P,, P ) + mδ (m ) P i,,p i (P,, P ) = 0, which were obtained in []. Proof. Maing the substitution (without loss of generality) u i = P i for i =,,, and denoting we come to φ(u) = φ(u,, u n ) = φ(x (u,, u n ),, x n (u,, u n )), (f(p,, P ) α δ(p,, P ) P α P α, φ) = ( α δ(p,, P ) P α P α, f(p,, P )φ) = ( ) α α [ ( )] S u α f(u uα,, u ) φd x du + du n u α α ( ) ( ) = ( ) α α α α j j j u α j u α j f(u,, u ) j u j u j α = ( ) α S S j =0 j =0 j [ φd u j u j α j =0 j =0 ( )] x du + du n u ( ) ( ) α α j j [ ( )] x φd du + du n u α j u α j u α j f(0,, 0)

14 44 C. K. Li Using the identity S j u j u j [ ( )] x φd we complete the proof of Theorem. u du + du n = ( ) j ( j δ(p,, P ) P j P j, φ), To end this section, we would lie to mention that Aguirre studied the following product P l P l α δ(p,, P ) P α P α, which is a special case of Theorem if f(p,, P ) = P l P l. 4. The generalized function δ(q P,, Q P ) Assuming that Q is a nonvanishing function and P is a manifold of dimension n, we have for any m 0 that (4) δ (m) (QP ) = Q (m+) δ (m) (P ). This is a powerful formula which can be used to derive some products, such as X l δ(r 2 ), since δ(r 2 ) = δ(r ). 2 We are interested in extending equation (4) to smooth manifolds of lower dimension. First of all, we would lie to see how the differential form ω and functional δ(p,, P ) change while maing the substitution W j (x) = α ij (x)p i (x). i= Here the α ij (x) are assumed to be infinitely differentiable functions and the matrix they form is assumed nonsingular. The defining equations for the initial differential form ω and for the new one ω are dp dp ω = dv = dw dw ω = ( α i dp i ) ( α i dp i ) ω. By expanding the terms in parentheses and using the anti-commutation rule dp i dp j = dp j dp i, we write det α ij dp dp ω = dv, which implies ω = det α ij ω. Hence (δ(w,, W ), φ) = (δ(p,, P ), φ det α ij ).

15 Several products of distributions on manifolds 45 Let us find the generalized function δ(qp,, QP ), where Q 0. By the substitution W = QP,, W = QP, we arrive at det α ij = Q (x). This indicates (5) δ(qp,, QP ) = Q (x)δ(p,, P ). In particular, we obtain for = that δ(qp ) = Q δ(p ), which coincides with equation (4) for m = 0. It follows that δ (m) (QP,, QP ) = Q (+m) (x)δ (m) (P,, P ) by differentiating both sides of equation (5) m times with respect to some P i. Similarly, δ(q P,, Q P ) = Q Q δ(p,, P ) where the Q i are nonzero and infinitely differentiable functions. Let α = α + + α, then δ α (Q P,, Q P ) = Acnowledgement Q +α Q +α δ α (P,, P ). The author would lie to deeply than the referee s very constructive suggestions and corrections in several places, which have improved the quality of the paper. This wor is partially supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). References [] Gel fand, I.M., Shilov, G.E., Generalized functions. Vol. I, Academic Press 964. [2] Li, C.K., Fisher, B., Examples of the neutrix product of distributions on R m. Rad. Mat. 6 (990), [3] Cheng, L.Z., Li, C.K., A commutative neutrix product of distributions on R m. Math. Nachr. 5 (99), [4] Li, C.K., The product of r and δ. Int. J. Math. Math. Sci. 24 (2000), [5] Li, C.K., An approach for distributional products on R m. Integral Transform. Spec. Func. 6 (2005), [6] Li, C.K., The products on the unit sphere and even-dimension spaces. J. Math. Anal. Appl. 305 (2005), [7] Li, C.K., A review on the products of distributions. (Mathematical Methods in Engineering) Springer (2007), 7-96.

16 46 C. K. Li [8] Aguirre, M.A., Li, C.K., The distributional products of particular distributions. Appl. Math. Comput. 87 (2007), [9] Aguirre, M.A., A convolution product of (2j)-th derivative of Diracs delta in r and multiplicative distributional product between r and ( j δ). Int. J. Math. Math. Sci. 3 (2003), [0] Aguirre, M.A., The series expansion of δ () (r c). Math. Notae 35 (99), [] Courant, R., Hilbert, D., Methods of mathematical physics. Vol. II, Interscience New Yor 962. [2] Kanwal, R., Generalized functions: theory and applications. Boston: Birhäuser [3] Roubine, É., Mathematics applied to physics. New Yor: Springer-Verlag Inc Received by the editors January 4, 2008