CHANGES OF VARIABLES IN HYPERSINGULAR INTEGRALS RICARDO ESTRADA Abstract. We prove that if A is a non singular n × n matrix and φ is smooth for x 6= 0, integrable of a ball, and at the origin it has an asymptotic expansion of the type Poutside ∞ φ (x) ∼ j=0 aj (x/ |x|) rαj as r = |x| → 0, where aj ∈ D (S) and αj % ∞, αi = −n, R then the hypersingular integral F.p. Rn φ (Ax) dx is given as Z F.p. φ (Ax) dx = Rn Z Z 1 1 F.p. ai (w) ln A−1 w dσ (w) . φ (x) dx+ |det A| |det A| S Rn We apply this and similar formulas to obtain the transformation rules for linear β changes of variables in pseudofunctions, Pf (|Ax| ), if A is a non singular n × n matrix.
1. Introduction The change of variables formula Z φ (Ax) dx = (1.1) Rn
1 |det A|
Z φ (x) dx , Rn
valid if A is a non singular n × n matrix and φ ∈ L1 (Rn ) is very well known. Our aim is to study the corresponding transformation formulas when the integral is not an ordinary one P but it is hypersingular at the origin. When φ has singular behavior of the αj as r = |x| → 0, where aj ∈ D (S) and αj % ∞, but is type φ (x) ∼ ∞ j=0 aj (x/ |x|) r integrable outside a ball, then in Section 4 we establish the change of variables formula Z Z Z 1 1 (1.2) F.p. φ (Ax) dx = F.p. φ (x) dx+ ai (w) ln A−1 w dσ (w) , |det A| |det A| S Rn Rn R if αi = −n for some i, F.p. Rn φ (x) dx being the notation we use for the hypersingular integrals, as explained in Section 2. We then employ (1.2) to study transformation formulas for pseudofunctions under linear changes of variables. Indeed, it is well known [2] that in D0 (Rn ) the distribution Pf (|x|β ) is homogeneous of degree β, Pf (|λx|β ) = λβ Pf (|x|β ), λ > 0, if β is not of the form −n − 2m for some m ∈ N, but 1 1 1 ln λ cm,n (1.3) Pf = n+2m Pf + ∇2m δ (x) , n+2m n+2m n+2m λ (2m)!λ |λx| |x| R 2m where cm,n = S ωj dσ (ω) = 2Γ (m + 1/2) π (n−1)/2 /Γ (m + n/2) . More generally, in the space of thick distributions D∗0 (Rn ) , introduced in [9] and considered in Section 2, we 2010 Mathematics Subject Classification. 46F10. Key words and phrases. Hypersingular integrals, changes of variables, pseudofunctions. 1
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have Pf (|λx|β ) = λβ Pf (|x|β ), λ > 0, if β ∈ / Z, but if q ∈ Z, (1.4)
Pf (|λx|q ) = λq Pf (|x|q ) + Cλq log λ δ∗[−q−n] (x) ,
where C = c0,n = 2π n/2 /Γ (n/2) is the surface area of the unit sphere S of Rn . Actually (1.3) follows from (1.4) by projection [9]. In Section 5 we generalize these results and obtain the transformation formula (1.5)
Pf {|Az|q ; x} = Pf {|z|q ; Ax} − C |Aw|q ln |Aw| δ∗[−q−n] (x) ,
when A is a non singular n × n matrix. In the notation of [1], the ordinary part of Pf {|Az|q ; x} is Pf {|z|q ; Ax} , but the distributions are not equal because Pf {|Az|q ; x} [−q−n] has no delta part, but Pf {|z|q ; Ax} has a delta part, namely, C |Aw|q ln |Aw| δ∗ (x) . 2. Preliminaries Let us recall the notion of the finite part of a limit, introduced by Hadamard in his studies of the fundamental solutions of partial differential equations [4]. Here we follow [2, Section 2.4]; see also [7, 8]. Let G (ε) be a function defined for 0 < ε < ε0 that satisfies limε→0+ G (ε) = ∞. Suppose F is a family of strictly positive functions defined in the same interval such that all of them tend to infinity at 0 and such that given two different elements f1 , f2 ∈ F then limε→0+ f1 (ε) /f2 (ε) is either 0 or ∞. Then the finite part of the limit of G (ε) as ε → 0+ with respect to F exists and equals A if we can write G (ε) = G1 (ε) + G2 (ε) , where G1 , the infinite part, is a linear combination of the basic functions and where G2 , the finite part, has the property that the limit A = limε→0+ G2 (ε) exists. Such a decomposition is unique if it exists since any finite number of elements of F has to be linearly independent. We then employ the notation F.p.F limε→0+ G (ε) = A. In the standard Hadamard finite part limit one takes F as the family of functions ε−α |ln α|β , where α > 0 and β ≥ 0 or where α = 0 and β > 0 and uses the simpler notation F.p. limε→0+ G (ε) . Consider now a function f defined in Rn that is possibly not integrable over the whole space but which is integrable in the region |x| > ε for any ε > 0. Then the radial 1 finite part integral is defined as Z Z (2.1) F.p. f (x) dx = F.p. lim+ f (x) dx , Rn
ε→0
|x|>ε
if the finite part limit exists. Other, non-radial finite part integrals can be considered, of course, but the results would be different, in general , an important point in many situations [5], especially in the numerical solution of integral equations [3]. Finite part integralsR are referred as hypersingular integrals, while those for which the ordinary limit limε→0+ |x|>ε f (x) dx exists are called singular integrals. Interestingly, already Hadamard worried about the behavior of hypersingular integrals under changes of variables [4, §83]. We follow the standard notation on distributions [6, 8]. We shall also employ the recently developed thick distributional calculus [9]. The space of test functions with a thick point at x = 0 is the space D∗ (Rn ) of all smooth functions φ defined in Rn \{0} , with support of the form K \ {0} , where K is compact in Rn , that admit a strong asymptotic 1Other,
non-radial finite part integrals can be considered, of course, but the results would be different, in general , an important point in many situations [5], especially in the numerical solution of integral equations [3].
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3
P∞
expansion of the form φ (x) = φ (rw) ∼ j=m aj (w) rj , as x → 0, where m ∈ Z, and where aj ∈ D (S) . We call D∗ (Rn ) the space of test functions on Rn with a thick point located at x = 0. The space D∗ (Rn ) has a natural topology, which makes it a locally convex topological vector space [9]. The space of distributions on Rn with a thick point at x = 0 is the dual space of 0 D∗ (Rn ) , denoted as D∗ (Rn ) . Observe that D (Rn ) , the space of standard test functions, is a closed subspace of D∗ (Rn ) ; if i : D (Rn ) → D∗ (Rn ) is the inclusion map then the dual of the inclusion is the projection operator Π : D∗0 (Rn ) → D0 (Rn ) . One can easily introduce other spaces of thick test functions and distributions [10], such as E∗ (Rn ) and E∗0 (Rn ) or S∗ (Rn ) and S∗0 (Rn ) . The usual operations on thick distributions, such as derivatives are defined in much the same way as with standard distributions, that is, by duality, namely, if f ∈ D∗0 (Rn ) then its thick distributional derivative ∂ ∗ f /∂xj is defined as h∂ ∗ f /∂xj , φi = − hf, ∂ ∗ φ/∂xj i if φ ∈ D∗ (Rn ) . The same applies to linear changes of variables, our main concern in this article: hf (Ax) , φ (x)i = (1/ |det A|) hf (x) , φ (A−1 x)i . Let g (w) be a distribution on S. The thick delta function of degree q, denoted either [q] [q] [q] as gδ∗ , gP (w) δ∗ , or as g (w) δ∗ (x) acts on a thick test function φ (x) with expansion ∞ φ (rw) ∼ j=m aj (w) rj , as r → 0+ , as
(2.2)
1 gδ∗[q] , φ D0 (Rn )×D∗ (Rn ) = hg (w) , aq (w)iD0 (S)×D(S) . ∗ C 3. Identities on the sphere
If ψ : S −→ S is a smooth diffeomorphism of the unit sphere S of Rn , then for any smooth function a defined on S we have a formula of the type Z Z a (ψ (w)) dσ (w) = a (v) J (v) dσ (v) , (3.1) S
S
where dσ is the Lebesgue measure on the sphere and where J is the corresponding Jacobian of the transformation. Naturally, J may or may not have a simple formula. We would like to consider a case when such a formula is very simple, the case when ψ (v) = Av/ |Av| , where A is a non singular n × n matrix. Proposition 3.1. Let A be a non singular n × n matrix and let a be a Lebesgue integrable function on S, and let a0 (x) = a (x/ |x|) be the extension of a to Rn \ {0} that is homogeneous of degree 0. Then for any α ∈ C, Z Z 1 a (v) α (3.2) a0 (Aw) |Aw| dσ (w) = dσ (v) . |det A| S |A−1 v|α+n S Proof. Let F (x) R= a0 (x) |x|α , which will be defined in Rn if 0, and let us compute the integral I = |x|≤1 F (Ax) dx in two different ways. On the one hand, Z Z
1 α
a0 (Aw) |Aw| r
I= S
0
α+n−1
1 dr dσ (w) = α+n
Z S
a0 (Aw) |Aw|α dσ (w) ,
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while on the other, if B is the unit ball in Rn , Z 1/|A−1 v| Z Z 1 1 rα+n−1 dr dσ (v) I= F (u) du = a (v) |det A| A(B) |det A| S 0 Z 1 a (v) = dσ (v) . −1 (α + n) |det A| S |A v|α+n Formula (3.2) follows when 0, and thus for any α ∈ C by analytic continuation. The formula remains true for distributions on the sphere if the integrals are understood as evaluations at the constant function 1 : D −1 −α−n E α −1 ,1 . ha0 (Aw) |Aw| , 1i = |det A| a (v) A v S
S
The cases α = 0, Z (3.3)
1 a0 (Aw) dσ (w) = |det A| S
Z S
a (v) dσ (v) , |A−1 v|n
and α = −n, Z
Z a0 (Aw) 1 (3.4) a (v) dσ (v) , n dσ (w) = |det A| S S |Aw| will be particularly useful in our analysis. It should be clear that one can actually derive (3.2) from (3.3) or from (3.4) by replacing the function a in an appropriate fashion. Another useful identity is the ensuing one. Proposition 3.2. Let A be a non singular n × n matrix and let a be a Lebesgue integrable function on S, and let a0 (x) = a (x/ |x|) be the extension of a to Rn \ {0} that is homogeneous of degree 0. Then for any α ∈ C, Z Z −1 a (v) ln |A−1 w| α a0 (Aw) |Aw| ln |Aw| dσ (w) = (3.5) dσ (v) . |det A| S |A−1 v|α+n S Proof. Formula (3.5) follows by differentiation of (3.2) with respect to α. It can also be obtained directly from (3.2) by employing an appropriate new function a. These identities allow us to complete a formula for the change of variables in thick [q] distributions. Indeed, the change of variables (gδ∗ ) (Ax) was computed when A is a rotation or when A is a multiple of the identity, A = tI, in [9, Example 5.1]. We can now give a formula for a general matrix A. Proposition 3.3. Let A be a non singular n × n matrix and let g ∈ D0 (S) . Then (3.6) gδ∗[q] (Ax) = g (Aw/ |Aw|) |Aw|−q−n δ∗[q] (x) . [q]
[q]
Proof. Indeed, [9, (5.7)] gives (gδ∗ ) (Ax) = gA,q δ∗ (x) , where −1 q 1 A (w) −1 (3.7) hgA,q (w) , a (w)i = g (w) , a A (w) . |det A| |A−1 (w)| If we now employ (3.2), we see that also A (w) −q−n (3.8) hgA,q (w) , a (w)i = g |Aw| , a (w) , |A (w)|
CHANGES OF VARIABLES
5
and (3.6) follows immediately.
[q]
[q]
The cases [9, Example 5.1] (gδ∗ ) (Ax) = g (Aw) δ∗ (x) when A is a rotation and [q] [q] (gδ∗ ) (tx) = t−n−q g (w) δ∗ (x) are recovered. An interesting example of (3.6), which will play a role in our analysis, is the change of variables identity (3.9) ln A−1 (w) δ∗[q] (Ax) = − |A (w)|−q−n ln |A (w)| δ∗[q] (x) . 4. Linear changes in finite part integrals 1 n R If φ ∈ L (R ) and A is a non R singular n × n matrix then we of course have the formula φ (Ax) dx = (1/ |det A|) φ (x) dx. Our aim in this section is to find a transformaRn Rn tion formula for the hypersingular integral Z (4.1) I = F.p. φ (Ax) dx , Rn
when φ (x) is smooth for x 6= 0, it is integrablePoutside of a ball, and at the origin it has αj as r = |x| → 0, where an asymptotic expansion of the type φ (x) ∼ ∞ j=0 aj (x/ |x|) r aj ∈ D (S) and αj % ∞. In order to do so we need to find the Hadamard limit of the convergent integrals Z φ (Ax) dx , (4.2) I (ε) = |x|≥ε
that is, I = F.p. limε→0 I (ε) . We shall start with the case when φ (x) = a (w) rα ρ (r) , x = rw being polar coordinates, where a is homogeneous of degree 0 and where ρ ∈ D (R) is such that ρ (r) = 1 in a neighborhood of the origin. In this case we have Z Z ∞ I (ε) = a (Aw) |Aw|α ρ (|Aw| r) rα+n−1 drdσ (w) ZS Zε ∞ a (Aw) |Aw|−n ρ (s) sα+n−1 dsdσ (w) , = ε|Aw|
S
but Z (4.3)
∞
F.p. lim
ε→0
ρ (s) s
α+n−1
Z ds = F.p.
ε|Aw|
∞
ρ (s) sα+n−1 ds ,
if α 6= −n ,
0
while Z (4.4)
∞
F.p. lim
ε→0
ρ (s) s
α+n−1
Z ds = F.p.
ε|Aw|
∞
ρ (s) s−1 ds − ln |Aw| ,
if α = −n .
0
Hence, if α 6= −n we obtain Z
(4.5)
Z
a (Aw) ρ (s) s ds n dσ (w) 0 S |Aw| Z ∞ Z 1 α+n−1 = F.p. ρ (s) s ds a (w) dσ (w) |det A| 0 S Z 1 = F.p. φ (x) dx . |det A| Rn
F.p. lim I (ε) = F.p. ε→0
∞
α+n−1
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When α = −n on the other hand, I = F.p. limε→0 I (ε) would be given as Z Z ∞ Z a (Aw) ln |Aw| a (Aw) −1 dσ (w) I = F.p. ρ (s) s ds n dσ (w) − |Aw|n S 0 S |Aw| Z Z 1 1 (4.6) F.p. φ (x) dx+ a (w) ln A−1 w dσ (w) . = |det A| |det A| S Rn These results immediately give the following formula for the change of variables in hypersingular integrals. Theorem 4.1. Let A be a non singular n×n matrix and suppose φ is smooth for x 6= 0, it is integrable and at the origin it has an asymptotic expansion of the type P outside of a ball, αj φ (x) ∼ ∞ a (x/ |x|) r as r = |x| → 0, where aj ∈ D (S) and αj % ∞. If αi = −n j=0 j for some exponent αi , then Z Z Z 1 1 F.p. φ (x) dx+ ai (w) ln A−1 w dσ (w) . (4.7) F.p. φ (Ax) dx = |det A| |det A| S Rn Rn Proof. Let ρ ∈ D (R) be such that ρ (r) = 1 in a neighborhood of the origin. Then if J is large enough, we can write (4.8)
φ (x) =
J X
aj (x/ |x|) |x|αj ρ (|x|) + φJ (x) ,
j=0
R
R where φJ ∈ L1 (Rn ) , so that Rn φJ (Ax) dx = (1/ |det A|) Rn φJ (x) dx. The transformation formula for each of the terms of the sum in (4.8) can be obtained from (4.5) and (4.6), an extra term being obtained only when αi = −n, the extra term being precisely that in (4.7). Notice that no extra term is obtained if A is an orthogonal matrix. Notice also that when A = tI, t > 0, then we recover the formula [10, Prop. 6.2] Z Z Z 1 ln t φ (tx) dx = n F.p. (4.9) F.p. φ (x) dx− n ai (w) dσ (w) , t t Rn Rn S where αi = −n. 5. Transformation of pseudofunctions In this section we shall study the transformation formulas for pseudofunctions under linear changes of variables. We only consider the case of thick distributions, the case of ordinary distributions being obtained by projection. Definition 5.1. If g is a locally integrable function in Rn \ {0} such that the radial finite part integral of gφ exists for each φ belonging to a space of thick test functions A∗ (Rn ) , then we can define a thick distribution Pf {g (z) ; x} = Pf (g) ∈ A0∗ (Rn ) as Z (5.1) hPf {g (z) ; x} , φ (x)i = hPf (g) , φi = F.p. g (x) φ (x) dx . Rn
The same notation is employed for standard distributions.
CHANGES OF VARIABLES
7
Following Schwartz [8] the distribution Pf (g) is called a pseudofunction. One may usually employ the notation Pf (g (x)) when the variable of the pseudofunction is important, but when considering the transformation properties of the association g 7−→ Pf (g) it is convenient to employ the more detailed notation Pf {g (z) ; x} . Indeed, since g and Pf (g) are not exactly the same object, then applying an operation to g does not give the same result as applying the same operation to Pf (g) ; our aim is to find the relationship between the distributions Pf {g (Az) ; x} and Pf {g (z) ; Ax} when A is a non singular n × n matrix. If g is a regular distribution, that is, if the (5.1) are convergent integrals, then g (Ax) is uniquely defined, since Pf {g (Az) ; x} = Pf {g (z) ; Ax} . Let us start with the case of homogeneous functions. Proposition 5.2. Let A be a non singular n × n matrix. Let b ∈ D (S) and let Bα (z) = b (z/ |z|) |z|a be its extension to D0 (Rn \ {0}) that is homogeneous of degree α. Then in D∗0 (Rn ) , (5.2) Pf Bα (Az) ; A−1 x = Pf {Bα (z) ; x} , if α ∈ / Z, while if α = k ∈ Z, (5.3) Pf Bk (Az) ; A−1 x = Pf {Bk (z) ; x} + Cb (w) ln A−1 w δ∗[−k−n] (x) . P j Proof. Indeed, if φ ∈ D∗ (Rn ) , with expansion φ (x) ∼ ∞ j=−m aj (x/ |x|) r as r = |x| → 0, then Z
1 −1 (5.4) Pf Bα (Az) ; A x , φ (x) = Bα (Ax) φ (Ax) dx . F.p. |det A−1 | Rn R If we now apply the formula of the Theorem 4.1 we obtain F.p. Rn Bα (x) φ (x) dx if α∈ / Z, and Z Z Bk (x) φ (x) dx + b (w) ln A−1 w a−k−n (w) dσ (w) , (5.5) F.p. Rn
S
if α = k ∈ Z, so that (5.3) follows.
We can rewrite these formulas in a slightly different way by replacing A−1 x by x. Proposition 5.3. Let A be a non singular n × n matrix. Let b ∈ D (S) and let Bα (z) = b (z/ |z|) |z|a be its extension to D0 (Rn \ {0}) that is homogeneous of degree α. Then in D∗0 (Rn ) we have Pf {Bα (Az) ; x} = Pf {Bα (z) ; Ax} if α ∈ / Z, while if α = k ∈ Z, (5.6)
Pf {Bk (Az) ; x} = Pf {Bk (z) ; Ax} − Cb (w) |Aw|k ln |Aw| δ∗[−k−n] (x) .
Proof. It follows immediately from the Proposition 3.3, (3.9) in particular.
We therefore obtain the ensuing transformation formulas for more general pseudofunctions. Proposition 5.4. Let A be a non singular n × n matrix. Let ψ ∈ E∗ (Rn ) , with ψ (x) ∼ P∞ j j=p bj (w) r , as x → 0. Then −n−p
(5.7)
Pf {ψ (z) ; Ax} = Pf {ψ (Az) ; x} + C
X log |Aw| [j] n+j b−n−j (w) δ∗ (x) . |Aw| j=−∞
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RICARDO ESTRADA
P [j] 0 n Observe that series of the type M j=−∞ gj (w) δ∗ (x) always converge in D∗ (R ) since DP E [j] M n 0 j=−∞ gj (w) δ∗ (x) , φ (x) becomes a finite sum for any φ ∈ D∗ (R ) . We finish by indicating how one can obtain transformation formulas for standard disP∞ n tributions. Suppose, for example, that ψ ∈ E∗ (R ) , with ψ (x) ∼ j=−n bj (w) rj , as x → 0, that is with p = −n. If we apply the projection operator Π to (5.7), and observe [9] that the projection of a thick delta of order j vanishes unless j ≥ 0, we obtain that only the term j = 0 remains. Hence, in D0 (Rn ) Pf {ψ (z) ; Ax} = Pf {ψ (Az) ; x} + Kδ (x) ,
(5.8) where
Z (5.9)
K= S
ln |Aw| b−n (w) dσ (w) , |Aw|n
since [9, Prop. 4.7] (5.10)
Π Cg
(w) δ∗[0]
(x) = Kδ (x) ,
Z K=
g (w) dσ (w) . S
That is, if φ ∈ D (Rn ) then Z Z 1 −1 ψ (Ax) φ (x) dx + Kφ (0) . ψ (x) φ A x dx =F.p. F.p. |det A| Rn Rn References [1] Estrada, R., Regularization and derivatives of multipole potentials, J. Math. Anal. Appls. 446 (2017), 770-785. [2] Estrada, R. and Kanwal, R.P., A distributional approach to Asymptotics. Theory and Applications, second edition, Birkh¨ auser, Boston, 2002. [3] Farassat, F., Introduction to generalized functions with applications in aerodynamics and aeroacoustics, NASA Technical Paper 3248 (Hampton, VA: NASA Langley Research Center) (1996); http://ntrs.nasa.gov. [4] Hadamard, J., Lectures on Cauchy’s Problem in Linear Differential Equations, Dover, New York, 1952 (reprint of the 1923 edition by Yale University Press). [5] Hnizdo, V., Generalized second-order partial derivatives of 1/r, Eur. J. Phys. 32 (2011), 287–297. [6] Kanwal, R.P., Generalized Functions: Theory and Technique, Third Edition, Birkh¨ auser, Boston, 2004. [7] Paycha, S., Regularised Integrals, Sums and Traces, A.M.S., Providence, 2012. [8] Schwartz, L., Th`eorie des Distributions, vol. 1, Hermann, Paris, 1957. [9] Yang, Y. and Estrada, R., Distributions in spaces with thick points, J. Math. Anal. Appls. 401 (2013), 821-835. [10] Yang, Y. and Estrada, R., Asymptotic expansion of thick distributions, Asymptotic Analysis 95 (2015), 1-19. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803 USA E-mail address:
[email protected].
