CC BY
8
Том 24, № 128 2019
© Molchanov V.F., 2019
DOI 10.20310/2686-9667-2019-24-128-432-449
УДК 517.98
Radon problems for hyperboloids
Vladimir F. MOLCHANOV
Derzhavin Tambov State University 33 Internatsionalnaya St., Tambov 392000, Russian Federation
Задачи Радона для гиперболоидов
Владимир Федорович МОЛЧАНОВ
ФГБОУ ВО «Тамбовский государственный университет им. Г.Р. Державина» 392000, Российская Федерация, г. Тамбов, ул. Интернациональная, 33
Abstract. We offer a variant of Radon transforms for a pair X and Y of hyperboloids in R3 defined by [x, x] = 1 and [y, y] = — 1,yy > 1, respectively, here [x,y] = —xyyy +x2y2 +x3y3 . For a kernel of these transforms we take 5([x,y]), S(t) being the Dirac delta function. We obtain two Radon transforms D(X) ^ CTO(Y) and D(Y) ^ CTO(X) . We describe kernels and images of these transforms. For that we decompose a sesqui-linear form with the kernel 5([x,y\) into inner products of Fourier components.
Keywords: hyperboloids; Radon transform; distributions; representations; Poisson and Fourier transforms
Acknowledgements: The work is partially supported by the Ministry of Education and Science of the Russian Federation (project no. 3.8515.2017/8.9).
For citation: Molchanov V.F. Zadachi Radona dlya giperboloidov [Radon problems for hyperboloids]. Vestnik rossiyskikh universitetov. Matematika - Russian Universities Reports. Mathematics, 2019, vol. 24, no. 128, pp. 432-449. DOI 10.20310/2686-9667-2019-24-128-432449.
Аннотация. Мы предлагаем некоторый вариант преобразований Радона для пары X и Y гиперболоидов в R3 , определенных уравнениями [x, x] = 1 and [y,y] = — 1, yy ^ 1, соответственно, здесь [x, y] = —xyyy + x2y2 + x3y3 . В качестве ядра этих преобразований мы берем 5([x,y]) , где 5(t) - дельта-функция Дирака. Мы получаем два преобразования Радона D(X) ^ CTO(Y) и D(Y) ^ CTO(X) . Мы описываем ядра и образы этих преобразований. Для этого мы разлагаем полуторалинейную форму с ядром 5([x,y]) по скалярным произведениям компонент Фурье.
Ключевые слова: гиперболоиды; преобразование Радона; обобщенные функции; представления; преобразования Пуассона и Фурье
Благодарности: Работа выполнена при поддержке Министерства образования и науки РФ (проект № 3.8515.2017/8.9).
Для цитирования: Молчанов В.Ф. Задачи Радона для гиперболоидов // Вестник российских университетов. Математика. 2019. Т. 24. № 128. С. 432-449. DOI 10.20310/26869667-2019-24-128-432-449. (In Engl., Abstr. in Russian)
In this paper we offer a variant of Radon transforms for a pair of dual hyperboloids in R3 : the one-sheeted hyperboloid X : [x, x] = 1 ([x, y] = — xiyi +x2y2 +x3y3 ) and the upper sheet of the two-sheeted hyperboloid Y : [y, y] = — 1, y1 ^ 1 (the Lobachevsky plane). For a kernel of these transforms we take #([x,y]), x G X, y G Y, £(t) being the Dirac delta function. This kernel gives two Radon transforms R : D(X) M C~(y) and R* : D(Y) M C~(X). We describe kernels and images of these transforms. For that we consider a sesqui-linear form with the kernel $([x,y]) and write the decomposition of this form into inner products of Fourier components. Results of this paper were announced in [4].
1. Hyperboloids
Let G be the group SOo(1, 2), it is a connected group of linear transformations of R3, preserving the form
[x, y] = —xiyi + x2y2 + x3y3.
We consider that G acts on R3 from the right. In accordance with this we write vectors in the row form.
Let us take the following basis of the Lie algebra 0 of the group G:
00 0 \ / 010 \ /001 Lo = I 0 0 —1 I , Li = I 1 0 0 I , L2 = I 0 0 0 | . (1.1)
0 1 0 0 0 0 1 0 0
The Casimir element in the universal enveloping algebra of 0 is (1/2)A0, where
Ag = —L2 + Li + L2. (1.2)
Consider subgroups K, H, A of the group G generating by elements L0, Li, L2, respectively. The subgroup K is a maximal compact subgroup of G.
Denote by X the one-sheeted hyperboloid [x, x] = 1, and by Y the upper sheet of the two-sheeted hyperboloid [y,y] = —1, yi ^ 1 (we consider that the xi -axis goes up). These hyperboloids X and Y are homogeneous spaces of the group G with respect to translations x M xg, namely, X = G/H and Y = G/K. The subgroups H and K are stabilizers of points x0 = (0, 0,1) G X and y0 = (1, 0,0) G Y respectively.
These hyperboloids have a G-invariant metric. It gives rise to the measures dx and dy and the Laplace-Beltrami operators AX and Ay respectively (all are G-invariant).
As local coordinates on the hyperboloids we can take any two variables from xi, x2, x3. For Y it is natural to take y2,y3. Then we have
dx = |xi|-idx2dx3, dy = y-idy2dy3,
d2 d2 . ^ ^ d d
Ax = — + D + Di, Di = xi---+ x^^,
dx2 dx2 dxi dx2
d2 d2 . ^ ^ d d Ay = 7TT — 7^2 + D\ + Di, Di = y^ + y3^-. dy2 d 32 dy2 dy3
If M is a manifold, then D(M) denotes the space of compactly supported infinitely differentiable C-valued functions on M, with the usual topology, and D'(M) denotes
the space of distributions on M — of antilinear continuous functionals on D(M). For a differentiable representation of a Lie group, we retain the same symbol for the corresponding representations of its Lie algebra and of the universal enveloping algebra.
Let us denote by UX and Uy representations of our group G by translations on functions on X and X respectively (quasiregular representations):
(Ux(g)f)(x) = f(xg), (Ux(g)f)(y) = f(yg).
The representations UX and Uy on the spaces L2(X,dx) and L2(Y,dy) are unitary with respect to the inner products
(F,f )x = / F (x)f(X)dx, (F, f )y = / F (y)fy)dy. (1.3)
J X Jy
We have
Ux(Aa) = Ax, Uy(Afl) = Ay. (1.4)
2. Representations of the group SO0(1, 2)
Recall some material about the principal non-unitary series of representations of the group G = SO0(1, 2), see, for example, [5]. Let C + be the cone [x,x] = 0, xi > 0. The group G acts transitively on it. For a E C, let Dr(C+) be the space of C^ functions p on C + homogeneous of degree a :
p(tx) = tatp(x), t > 0. Let Ta be the representation of G acting on this space by translations:
(Ta (g)p)(x) = p(xg).
Take the section S of the cone C+ by the plane x1 = 1, it is a circle consisting of points s = (1, sin a, cos a). The Euclidean measure on S is ds = da. For a function p on S, sometimes we write p(a) instead of p(s). The representation Ta can be realized on the space D(S) as follows (index 1 indicates the first coordinate of a vector):
(Tr(g)p)(s) = p(-S^) (sg)i.
The element Ag, see (1.2), goes to a scalar operator:
Tr (Afl) = a(a + 1)E. (2.1)
The Hermitian form
(i>,P)s =i ^(s)p(s)ds (2.2)
Js
is invariant with respect to the pair (Tr,T-r-1), i. e.
(Tr (g)^,p)s = (^,T-r-i(g-1)p)s. (2.3)
This formula follows from ds = (sg)-1 ds, where Is = (sg)/(sg)1.
Define an operator ACT in D(S):
(ACT ^)(s)= f (—[s,u])-<7-V(u)du. Js
The integral converges absolutely for Rea < —1/2 and can be continued as a meromorphic function to the whole a -plane. It has simple poles at a G —1/2 + N. Here and further N = {0,1, 2,...}. For ACT we have
(ACT = (^,A^)s . (2.4)
The operator ACT intertwines TCT and T-CT-i, i. e.
T-CT-i(g)ACT = ACTTCT(g), g G G.
A sesqui-linear form (ACT<^)S is invariant with respect to the pair (TCT, T^). In particular, for a G R, this form is an invariant Hermitian form for TCT.
Take a basis = eima, m G Z, in D(S). It consists of eigenfunctions of ACT :
ACT = a(a, (2.5)
where
a(a,m) = 2^+2n(—1)m r+—^T— 1)-^. (2.6)
i(—a + m)I(—a — m)
The composition ACTA-CT-i is a scalar operator:
ActA-ct-1 = 3-7 r ■ E
where w(a) is a "Plancherel measure" (see (5.2)):
w(a) = -^ (2a + 1) cot an. (2.7)
32n2
The representation TCT can be extended to the space D'(S) of distributions on S by formula (2.3) where is a distribution and is the value of the distribution ^ at a
test function It is an extension in fact, since D(S) can be embedded into D'(S) by means of the form (2.2), namely, we assign to a function ^ G D(S) the functional ^ M in
D'(S).
Similarly the operator ACT can be extended to the space D'(S) by means of formula (2.4).
If a is not integer, then TCT is irreducible and TCT is equivalent to T-CT-i (by ACT or A^). Let a G Z, n G N. Subspaces V-,+ and V^,+ spanned by for which m ^ —a and m ^ a respectively are invariant. For a < 0 they are irreducible and orthogonal to each other. For a ^ 0 their intersection E is irreducible and has dimension 2a + 1.
Let Vrad = D(S)/En and V-n-i = V-n-i,+ + V-n-i_. Let us denote by Tf, a G Z, the representation on V^ generated by TCT. The operator An vanishes on and gives rise to the equivalence T^ ~ Tfra_i.
There are four series of unitarizable irreducible representations: the continuous series consisting of representations Tr with a = -1/2 + ip, p E R, the inner product is (2.2); the complementary series consisting of Tr with —1 < a < 1, the inner product is (Ar^, p)s with a factor; the holomorphic and antiholomorphic series. We need only their sum . We shall call Td the representations of discrete series. For p E D(S), denote by p the coset of p modulo En. Then the invariant inner product (•, •)„ for Td is
(ij,p)n = On(An^,p)s, Cn = a(n,n + 1)-1. (2.8)
3. Poisson and Fourier transforms
First we determine distributions 9 in V(S) invariant with respect to the subgroup H under the representations T . We shall use the following notation for a character of the group R*:
tx'm = |t|A(sgn t)m,
where t E R* = R\{0}, A E C, m E Z. In fact this character depends only on m modulo 2. Here and further the sign " = " means the congruence modulo 2. It is easy to check that the distribution
9 r>e = s!'£ =[x0,s] r'£,
where a E C, e = 0,1, is H -invariant. Sometimes we write an integer instead of e with the same parity as e. As a function of a, 9r,s is a meromorphic function — with simple poles at points a E —1 — e — 2N. Its residue at a = —n — 1, n = e, is the distribution const • i(n)(s3) concentrated at two points s = (1, ±1, 0). Here 5(t) is the Dirac delta function on the real line (a linear continuous functional on D(R)). The space of H -invariants has dimension 2 for a = —n — 1, n E N, and dimension 3 for a = —n — 1. Every irreducible subfactor for Tr, a E Z, contains, up to a factor, precisely one H-invariant. In particular, 9-n-1n+1 and 9n,n+1 have non-zero projections into V-n-1± and V(SW^ respectively. The operator Ar carries 9r>£ to 9-r-1>£ with a factor:
Aa9a>£ = j (a,e)9-r-M, (3.1)
where
j (a, e) = 2-rn-1/2rl —a —
j (a, e) = 2-rn-1/2r ^—a — 0 r(a + 1) [1 — (—1)£ cos an]. (3.2)
It is easy to check that
j (a,e)j (—a — 1,e) = (8nu(a))-1,
where u(a) is given by (2.7) or (5.2). The factor j(a,e) has simple poles at a E —1/2 + N.
By a general scheme [3], the H -invariant 9r,e gives rise to the Poisson kernel Pr,e (x, s) = [x, s]a'e, x E X, s E S. This kernel gives rise to two transforms. The first of them, the Poisson transform Pr,e : D(S) ^ C^(X) is a linear continuous operator defined as follows:
(Pr >ep)(x)= / [x,s] r,£p(s)ds. s
It intertwines T-CT-i with UX, therefore, its image consists of eigenfunctions of the Laplace-Beltrami operator:
Ax ◦ P,,£ = a(a + 1)P,,£
of parity e (see (2.1) and (1.4)). As a function in a, the Poisson transform behaves like 0,,£: it depends on a meromorphically and has simple poles at a G — 1 — e — 2N. Formula (3.1) gives
P^A, = j (a, e)P_,_i,£. (3.3)
Consider a G Z. The transform P_n_i,n+i vanishes on En, it generates an operator on D(S)/En which intertwines T^ with UX. The Poisson transform Pn,n+i considered on V-n-i intertwines T-n-i with UX. By (3.3), Pn,n+i has the same image as P-n-i,n+i.
The second transform generated by the Poisson kernel is the Fourier transform F,,£ : D(X) m D(S) defined by
(F,,£f) (s) = /[x,s],,£f(x)dx.
It is meromorphic in a with simple poles at points a G —1 — e — 2N. It intertwines UX with T,. It follows from (3.1) that
A, P,,£ = j (a,e)F_,_M. (3.4)
For a function f G D(X), let us call two functions F,£f, e = 0,1, the Fourier components of f corresponding to the representation T,.
The Fourier and Poisson transforms are conjugate to each other with respect to forms (1.3) and (2.2):
(P,,£^,f )x = )s .
This relation allows to extend the Poisson transform to distributions on S.
Consider the reducible case. The Fourier transform Fn corresponding to T^ is defined as the map of D(X) to D(S)/En which assigns to f G D(X) the corresponding coset of the function Pn,ra+if. By (2.8) and (3.4) we have
(Fnf, Fnh)ra = dra(F_ra_i,ra+if, Fra,ra+ih)s, dra = 2n!2/n(2n + 1)!.
The Fourier transform corresponding to T_ra-i is F_ra_i,ra+i.
The representation T, has one up to a factor K-invariant, it is the function t, equal to 1 identically on S :
t, (s) = [y°,sr = 1.
The representations of the discrete series have no K -invariants.
The corresponding Poisson transform Q, : D(S) M C^(Y) and Fourier transform D(Y) M D(S) are defined by
(Q,^)(y) = / [—y,s]CT^(s)ds,
Js
(G,h)(s) = i [—y,s]CTh(y)dy. y
Notice that [—y, s] > 0 for all y E Y and s E S.
The Poisson transform Qr intertwines T-r-1 with Uy, therefore, its image consists of eigenfunctions of the Laplace-Beltrami operator:
Ay ◦ Qa,£ = a(a + 1)Q^ (3.5)
4. Spherical functions
Let a E C, e = 0,1. Let us define a spherical function *r,e on the hyperboloid Y as follows
*r,e(y) = T(g)9a,e, T-r-1)s (4.1)
= (9a,£,T-r- 1(g-1) T-r-1) s
= f 9a,e[—y, s]-r-1ds, (4.2)
s
where g E G is such that y0g = y. As the distribution 9re does, the spherical function *r,e is given by an integral absolutely convergent for Rea > —1 and can be continued analytically in a to a meromorphic function. It has poles where 9r,e has and of the same (the first) order.
The function *r,e(y) is a function of class C™ on Y invariant with respect to H:
*r,e(yh) = *r,e(y), h E H. Therefore, it depends on y3 = [x0,y] only:
*r,e(y) = $r;e (y3), (4.3)
where $r,e(c) is a function in C^(R).
Lemma 4.1. The function $r>e has the following integral representation:
"2n ^ --N r,e
c + V C2 + 1 • cos ^ da. (4.4)
$r,e(c) = J (c + Vc2 + 1 • cos oOj
Proof. Let us take in (4.1) as g the matrix a = exp tL2, see (1.1), in A:
cosh t 0 sinh t 010 sinh t 0 cosh t
We have
(Tr(a) 9r,e) (s) = [x0, sa]r'e = (sinht + s3 cosh t)r'e .
By (4.1), the value of *r,e at the point y0a = (cosht, 0, sinht) is just (4.4) with c = sinh t. □
It follows from (4.4) that the function $r>e has parity e:
$r,e( —c) = (—1)e$r,e(c).
Equality (4.2) shows that the spherical function is the Poisson transform of the
H -invariant:
*,,£ = Q_,_ A,£. (4.5)
Consider as a distribution on Y :
(*,,£, f )y = ^,,£(y)f (y)dy, (4.6)
y
where f G D(Y). The right hand side in (4.6) can be rewritten as an iterated integral, then we obtain:
/ro
$,,£(c)(Mf )(c)dc, (4.7)
■ro
where
(Mf)(c)= / i(y3 — c)f (y)dy,
y
The map M assigns to a function f its integrals over H-orbits. It is a continuous operator from D(Y) onto D(R).
Lemma 4.2. The value (4.6) is expressed in terms of Fourier components:
(*,,£, f )y = (0„,£,G_,_if )s. (4.8)
Proof. Let h(y) be a majorant of the function f (y), depending on yi only. Then for Rea > —1 the right hand side in (4.8) is majorized by the integral
«2n
-T-1
/ | cos a|Tda / [y,s] h(y)dy, (4.9)
Jo Jy
where t = Rea. In fact, the integral over Y here does not depend on s. Therefore, integral (4.9) converges absolutely and the order of integration can be inverted. So we get equality (4.8) for Rea > —1. To other a this equality is extended by analycity. □
Let $ be a distribution on Y invariant with respect to H. Assign to it two things: a convolution with $ of functions f in D(Y) and a sesqui-linear functional K on the pair (D(X), D(Y)). The convolution $ * f is the following function on X :
($ *f )(x) = J$,Uy(g)f)y
= / $(y)f(yg)dy, y
the functional is:
K ($|h,f) = Jh, $ *f )x
= / h(x)($,Uy(g)f)y dx Jx
XxY
where h E D(X), f E D(Y) and g is an arbitrary element in G carrying x0 to x. Since $ is H-invariant, these formulae do not depend on the choice of g for given x. The convolution is a linear map D(Y) ^ C^(X), intertwining Uy and UX :
$ * (Uy (g)f ) = Ux(g) ($ *f).
For the spherical function *r,e, the convolution and the functional are expressed in terms of the Poisson and Fourier transforms:
(*r,e *f )(x) = (Pa,eG-r-1 f )(x),
K(*r,e|h,f) = {Freh, G-r-1f )s. (4.10)
The kernel Kr,e(x,y) of the functional (4.10) is
Kr,e(x, y) = f [x,s]r,e[—y,s]-r-1 ds. s
Lemma 4.3. The function *re has the following property of symmetry in a :
1 + (—1)e cos an T /A
*-r-1,e =--—--*r,e. (4.11)
sin an
Proof. By Lemma 4.2, (3.1), (2.4), (2.5) and Lemma 4.2 again we have:
{*-r-1,e,f )y = {d-r-^Grf )s
= j (a,e)-1{Ar 0a,e,Grf )s
= j(a,e)-1 {da,e, ArGrf)s
= a(a, 0)j(a,e)-1{da!e,G-r-1f)s
= a(a, 0)j(a,e)-1{*a,e,f)y.
Substituting here values of a(a, 0) and j(a,e) from (2.6) and (3.2), we get (4.11). □
Lemma 4.4. The spherical function *r,e is an eigenfunction of the Laplace-Beltrami operator:
Ay *r,e = a(a + 1)*r,e. (4.12)
Proof. The function *r,e is the Poisson transform of the function 0a,e, see (4.5). It remains to remember (3.5). □
On functions depending on y3 = c only, the operator Ay becomes to the following differential operator (the H -radial part of Ay ):
d2 d
L = <c2 + 1> a? + 2cac- <4.13»
Lemma 4.5. The function $r,e, see (4.3) and (4.4), is an eigenfunction of L :
L$r,e = a(a + 1)$r,e.
The lemma follows immediately from Lemma 4.4.
Theorem 4.1. The spherical function is expressed in terms of the Legendre
functions PCT (see [2, Ch. III]) of the imaginary argument:
= eW2 + (-Ve—/2! + (-¿ys)}. (4.14) Proof. Denote for brevity:
PCT (¿t) = B (t), (4.15) also for a function <^(t) on R we shall denote
£(t) = p(-t).
Equality (4.14) is equivalent to the following expression of the function £ :
= ^/2 + (2-ni)£e-i.n/2 (C) + (°)} . (4.16)
So we have to prove (4.16).
The Legendre function PCT(z) is analytic in the z-plane with the cut -1], satisfies
the equation:
d 2 d\
(z2 - 1)^ + 2z^Jw = a(a + 1)w (L^)
and has the integral representation
PCT(z) = T^^ (z + Vz2 - 1 cos a^ da. (4.18)
Let a be not integer. Then the functions PCT (z) and PCT (z) form a basis of solutions of equation (4.17). For z = ¿c equation (4.17) becomes the equation:
Lw = a(a + 1)w.
In virtue of Lemma 4.5 the function is a linear combination of functions B and
B-. Coefficients of this linear combination could be found out by computing values of functions and B and their derivatives at the point c = 0, using (4.6) and
explicit expressions [2, 3.4(20),(23)]. But it is more convenient for us to find them in another way.
Let z tend to ¿c, c G R, in (4.18) such that Re z > 0. We get:
Ba (c) = ¿ei<jn/2 /" (c + Vc2 + 1 cos a - ¿0^ da.
(4.19)
Denote
Z- (c) = ^c + Vc2 + 1 cos a — da.
Then
"„2
(c)= / ^c + Vc2 + 1 cos a — ¿0^ da.
Applying to (4.19) the formula:
(t - i0)_ = t°, + e-i_nt_,
we obtain
whence
B_ = ^
_ 2n
B_ = ^
_ 2n
From (4.20) and (4.21) we have
Z_
n
Z_
i sin an
n
i sin an
ei_n/2Z_ + e-i_n/2Z_
e-i_n/2Z_ + ei_n/2Z_
ei_n/2B_ - e~i_n/2B_ - e-i_n/2B_ + ei_n/2B_
(4.20)
(4.21)
(4.22)
(4.23)
Since
= Z_ + (-l)£ Z_
we obtain (4.16) by (4.22) and (4.22).
□
Let us establish some estimates for spherical functions of the continuous series (a = —1/2 + ip). They show that values of these spherical functions at f decrease rapidly when their parameter p tends to infinity.
Theorem 4.2. Let a = —1/2 + ip, p E R. For any compact set W C Y, there exists a number C > 0 such that for any f E D(Y) with the support in W the following estimate holds:
|{*r,e,f)yI ^ C • maxKA^f) (y)| (p2 + 1/4)-m, m E N. (4.24)
y
Proof. Take h E D(Y) depending on y1 only, such that h(y) ^ 0, h(y) = 1 on W. Then ^h, where ^ = max If (y)|, is a majorant for f depending on y1 only. Arguing as in the proof of Lemma 4.2, we obtain
I{*r,e,f)yI ^ C^, (4.25)
where C is the number
C = I cos aI-1/2da [—y,s]-1/2h(y) dy. 0y
Now apply the estimate (4.25) to the function A™f, m E N, transfer the operator Ay to the function *r,e, since it is self-adjoint, and use (4.12). Since Ia(a + 1)I = p2 + 1/4 for a = —1/2 + ip, we get (4.24). □
Let us write expressions of *r,e for a integer. In the notation *re sometimes it is convenient to write an integer instead of e with the same parity as e.
Let n G N. Let first a = n. For we have to evaluate an indeterminacy in (4.14).
We have
^n,n(y) = 2ni-nP„(iys),
^„,„+1 = -4i1-nQ„(iys),
where P„(z) is the Legendre polynomial and Q„(z) is the Legendre function which differs from the Legendre function of the second kind Q„(z) by the cut on the z -plane: for Q„(z) one takes the cut [-1,1], but for Q„(z) one has to take the cut (-ro,-1]U[1, ro); therefore, we have:
1 1 + z (z) = 1 P„(z)lnT^ - W„-i(z),
cf. [2, 3.6(24)], where the principal branch of the logarithm is taken and Wn-1(z) is a polynomial of degree n - 1 indicated in [2, 3.6.2].
For a = -n - 1 we use the relation (4.11). For e = n the function has a pole at a = - n - 1 because of We have
ResCT=-„-1 = (-1)n+1 (2/n)^„,„.
5. Eigenfunction decomposition of the radial part of the Laplace-Beltrami operator
In this Section we obtain the eigenfunction decomposition of the operator (see (4.13))
r / 2 ^d2 d L = (c2 + 1) a? + 2cdc
defined on the real line R. We use the function $CT,£(c), see (4.3) and (4.4). Recall that it has parity e and satisfies the equation:
Lw = a(a + 1)w.
Let us denote by the L2(R) inner product of functions :
= / ^(c)^(c) dc. J
Theorem 5.1. There is the following eigenfunction decomposition of the operator L :
^ (5.1)
CT=-1/2+ip
where
so that
w(a) = -^(2a + 1) cot an (5.2)
^ ( — 2 + ¿P j = Y6~2Ptanhpn.
oo
Proof. Let us write the resolvent R\ = (XE — L) 1 of the operator L. Let h E L2 and R\h = f, then h = (XE — L)f, so that
Lf — Xf = —h. (5.3)
Let us take X in the form X = a (a + 1). The correspondence a M X maps the half plane Rea > —1/2 onto the X -plane with the cut (—ro, —1/4] one-to-one.
Let f1, f2 be eigenfunctions of the operator L with the eigenvalue X = a(a + 1) with Rea > —1/2. They behave at infinity (±ro) as Alcf + Blcl-a-1. Let us take them such that they are square integrable at +ro and — ro respectively. Then for c M +ro:
fi(c) - Bic-a-1,
f22(c) - A2cCT + B2c-a-1,
and for c M — ro :
f1(c) - CM° + D1lcr-1,
f2(c) - D,lcl-a-1.
The wronskian W of these functions is
W0
W = -W-, Wo = (2c +1)B1A2.
c2 + 1
We have already several eigenfunctions: Pa(ic), Pa(—ic), Za(c), Za(c), $a,£(c), e = 0,1. By [2, 3.2(18)] the Legendre functions behave when c M +ro as follows:
Pa (ic) - p(a) • eian/2 • ca + p(—a — 1) • ei(-a-1)n/2 • c-a-1, Pa (—ic) - p(a) • e-ian/2 • ca + p(—a — 1) • ei(a+1)n/2 • c-a-\
where
p(a) = 2Qn-lB ( a + 2, 2
B(a,b) being the Euler beta function.
By (4.22), (4.22) it gives that when c ^ we have
2n
ZQ(c) - 2n ■ p(a) ■ cQ---p(-a — 1) ■ c-Q-1,
sin an
ZQ(c) — 2n ■ cot an ■ p(-a — 1) ■ c-Q-1.
Therefore, as a mentioned-above basis fl, f2 of solutions of the equation Lw = Xw, X = a (a + 1), we can take the pair ZQ, ZQ. Then
W0 = (2a + 1) ■ 2np(a) ■ 2n cot an ■ p(—a — 1) = 4n.
Therefore, the solution f of equation (5.3) is
f (c) = 4n{Za (c) f ' Zq (t)h(t)dt + Zq (c) f ^ Zq (t)h(t)dt}.
Thus, for ImA = 0, the resolvent Ra is an integral operator with the kernel
K (ct) , (l/4n)Zpo(c)Zo(t),c>t,
^(c,t)=S (Л,А np /.ч _ (5.4)
(l/4n)Zo (c)Zo (t),c<t,
here A = a(a + 1) and a belongs to the half plane Rea > -1/2 with the cut along the real axis.
Let p,^ G L2(R). By the Titchmarsh-Kodaira theorem [1, XIII] we have
(w, ф) = lim -
(Лл—гв^,ф) dЛ — (Дл+гв^,ф) dЛ
Let us pass to a. Then dA = (2a + 1)da and we denote So = Ra. The operator function S- is analytic in the half plane Rea > -1/2. Therefore,
1
(Р,Ф) = 2П/ (2a + l)(So ^,Ф)
dp.
o=— 1/2+ip
We can keep here only the even part in p of the integrand:
dp
1 f™
(^,Ф) = 4П (2a + l)((So — S—o—l)w, Ф)
o= — 1/2+ip
Let us compute the kernel Mo(c, t) of the operator So — S—o—l. Let c > t. By (5.4) we
have
M(c,t) = (c)Zo(t) - z^-o-l(c)Z-o-l(t)}
Let us insert here (4.22) and (4.22) and use that the Legendre function Po is unchanged under a M — a — 1. We obtain (recall notation (4.15)):
Mo(c,t) = — nCOS2an {Bo(c)Bo(t) + Bo(c)Bo(t)|. (5.5)
2 sin an I J
For c < t, we obtain the same expression.
Further, if a = —1/2 + ip, then for the Legendre function Po on the imaginary axis we
have
Po (ic) = Po (-ic) = P-o-1 ( ic) = Po (-ic),
or, in terms of Bo :
Bo (c) = Bo (c) = B—o—1 (c) = Bo (c). Therefore, equality (5.5) gives
(2a + 1) cos an
,/_„ 8 sin an l
+ (w,Bo)(Bo^)}
dp. (5.6)
o=—1/2+ip
This formula is the desired eigenfunction decomposition - in the basis Bo, B, Now let us pass in (5.6) from Bo, Bo to Ф^, e = 0,1, by
n 1 ( an . . an
Bo = — l^cos— ■ Фo,0 + i sin— ■ ?o,l
P 1 ( an . . an
Bo = — l^cos— ■ Фo,o + i sin— ■ Ф^.1
then we obtain (5.1). □
oo
ЭО
6. Decomposition of a sesqui-linear form on the pair of hyperboloids
Let us consider the following sesqui-linear form A(h, f) defined on the pair (D(X),
D(Y):
A(h,f )=/ ¿([x,y])h(x)f (y)dxdy. JXxy
The main result of our work consists of Theorem 6.1, which gives the decomposition of this form in terms of Fourier components of functions h and f. The decomposition contains Fourier components of the continuous series (a = —1/2 + ip).
Theorem 6.1. The sesqui-linear form A(h, f) decomposes into Fourier components of the continuous series Fa,0 f and Gah, a = —1/2 + ip, p E R, as follows:
a=-1/2+ip
where
A(h,f ) = / ^(a){Fa,oh,Ga f )s
Ma) = Ma)B( — 2, 2
1 , (a 1
16-2 (2a +l)cotan • B -a, 2
dp,
(6.1)
(6.2) (6.3)
the factor u(a) is given by (5.2), so that
8
— 1 + ip) = 1 n 5/2Ptanhpn • sin ( - + -p
42
n
n1 + -p
1 4 2
Proof. Let us take in (5.1) as p the characteristic function of the interval [0, a] divided by a and as ^ the function Mf, f E V(y). We can consider that a E [0,1]. We obtain
r 1 / "
Q £ (p) - / $-1/2+ip,£(c)dc dp
(6.4)
where we denoted
Q£(p) = w(a)($ff,£,Mf)
= {^a,£,f )y
a=-1/2+ip a=-1/2+ip}
see (4.7). Let a tend to 0. Then the left hand side of (6.4) goes to Mf (0). Let us prove that we can pass to the limit under the integral over p in the right hand side of (6.4). By the mean value theorem, the integral in the right hand side of (6.4) is equal to
F£(a)
tt£(p)$-1/2+ip,£(n)dp,
(6.5)
where n is a number in [0,a ] (depending on a, p and e). We have to prove that
oo
2
1
a
0
oo
F£(a) M F£(0)
(6.6)
when a —> 0, where
Fe (0)
^e{p)^-1/2+ip,s{0)dp.
(6.7)
Let us take an arbitrary number y > 0. In virtue of Theorem 4.2 both functions Q£(p), e = 0,1, decrease rapidly when I pi M 0. Therefore, there exists a number A such that
'IpI>a
fie(p)
dp < j.
It follows from formula (4.4) that the function &-1/2+ip,£(c) is bounded, i. e.
1/2+ip,e(C)
^ N,
(6.8)
(6.9)
N being some number, for all p E R, e = 0,1, and all c from some finite interval, for example, [0,1]. Indeed, formula (4.4) implies the inequality
^e(c)
r-2n
<
+ \\c2 + 1■
cos a
-1/2
da
(6.10)
since the function of c in the right hand side of (6.10) (it is the function ^-1/2>0) is continuous with respect to c.
On the other hand, since the function &-1/2+ip,£(c) is continuous with respect to p and
c, there exists a number S > 0 such that
$-1/2+ip,e(v) - $-1/2+ip,e(0)
for I pi ^ A and 0 ^ n < 8. Then for 0 < a < 8 we have
< Y
Fe(a) - Fe(0)
^e(p) ■ ®-1/2+ip,e(n) - &-1/2+ip,e(0) dp
cA
(6.11)
J-A JIpI^A < Y i ^e(p)
-A
^ (Ce + 2N )y,
dp + 2N
'IpI>a
^e(p)
dp
where
Ce
^e(p)
dp,
here we used (6.5), (6.7)-(6.9), (6.11). Inequality (6.12) proves (6.6) Now we may pass to the limit in (6.4) when a M 0. We obtain
Mf (0)
U(a) Y, $-,e(0) {^a,e,f h
a=-1/2+ip
dp.
By (4.4) we have
-2n
(cos p)a,£dp
[1 + (-1)e ]B
a + 1 1 2 , 2
(6.12)
(6.13)
oo
0
oo
oo
oo
0
We see that $o.,£(0) is equal to zero for e =1, so that only one summand in (6.13) remains - with e = 0. Since
a = -a - 1 for a = -1/2 + ip, (6.14)
equality (6.13) is
/ro
M*) (*<*),/ )y dp, (6.15)
-ro CT=-l/2+tp
where p(a) is given by (6.3), (6.4). The left hand side in (6.15) is
Mf (0) = S ([x0,y0 f (y) dy. Jy
(6.16)
Taking into account (6.16) let us apply (6.15) to a shifted function (Uy(g)/) (y) = /(yg), g G G. We get
S ([ x,y]) f (y)dy
p(a)(^CT;o,Uy (g)f )y
y
CT=-1/2+ip
dp,
(6.17)
where x = x0g.
Now multiply both sides of (6.17) by a function h(x) in D(X) and integrate over x G X. In the right hand side we may invert the order of integrations - in virtue of Lemma 6.1, see below. We obtain:
A(h,f )
M^) / (^,o,Uy(g)f)y h(x)dx
IX
CT=-1/2+ip
dp.
The integral over X is nothing but the functional K(^CT,0|h, /). Substituting its expression (4.10) in terms of Fourier components and taking into account (6.14), we get (6.1). □
Lemma 6.1. For any function /(y) in Dy the integral in the right hand side of (6.17) converges absolutely and uniformly with respect to x = x0g on any compact V C X.
Proof. The hyperboloid X can be embedded into the group G as the product AK of subgroups A and K. By continuity of Uy, the union of supports of all functions Uy(g)/, where g = ak is such that x0g G V is some compact W in Y. By Theorem 4.1 there exists C > 0 such that for any g = ak, x0g G V, the following inequality holds
(tf^Uy (g)f )y
^ C ■ max
y
A^Uy(g)f (y) ■ (P2 + 1/4)
Since Ay commutes with translations, we have
max
y
max
y
Uy (g)Amf (y)
(A^uy (g)/ (y)
so that there exist numbers Cm, m G N, such that
(^,Uy(g)/)y| ^ Cm ■ (p2 + 1/4) for all x = x0g G V and all m G N, whence the lemma.
max
y
Amf (y)
□
oo
CO
—m
— m
The quasiregular representation of G = SO0(1,2) on X contains representations of the continuous series with multiplicity two and the analytic and antianalytic series with multiplicity one, and the quasiregular representation of G on y contains representations of the continuous series with multiplicity one. Theorem 6.1 gives
Theorem 6.2. The kernel of the Radon transform R consists of functions belonging to the discrete spectrum and to the odd part of the continuous spectrum on X, its image goes in C^(y). The kernel of the Radon transform R* is {0}, its image consists of functions belonging to the even part of the continuous spectrum X.
References
[1] Н. Данфорд, J. T. Шварц, Линейные операторы. Т. II: Спектральная теория, Мир, М., 1966; англ. nep.:N. Dunford, J. T. Schwartz, Linear Operators. V. II: Spectral Theory, Wiley-Interscience, New York, 1988.
[2] Г. Бейтмен, А. Эрдейи, Высшие трансцендентные функции, М., Наука, 1965; англ. пер.:А. Erdelyi, W. Magnus, F. Oberhettinger, F. Tricomi, Higher Transcendental Functions I, McGraw-Hill, New York, 1953.
[3] В. Ф. Молчанов, "Гармонический анализ на однородных пространствах", Некоммутативный гармонический анализ - 2, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, 59, ВИНИТИ, М., 1990, 5-144; англ. пер.^. F. Molchanov, "Harmonic analysis on homogeneous spaces", Representation Theory and Noncommutative Harmonic Analysis II, Encyclopaedia of Mathematical Sciences, 59, ed. A. A. Kirillov, Springer-Verlag Berlin Heidelberg, Berlin, 1995, 1-135 pp.
[4] V. F. Molchanov, "Harmonic analysis on a pair of hyperboloids", Вестник Тамбовского университета. Серия Естественные и технические науки, 8:1 (2003), 149-150.
[5] Н.Я. Виленкин, Спектральные функции итеория представлений групп, Наука, М., 1965; англ. пер.^. J. Vilenkin, Special Functions and the Theory of Group Representations, Translations Mathematical Monographs, 22, Amer. Math. Soc., Providence, 1988.
Information about the author
Vladimir F. Molchanov, Doctor of Physics and Mathematics, Professor of the Functional Analysis Department. Derzhavin Tambov State University, Tambov, the Russian Federation. E-mail: v.molchanov@bk.ru ORCID: https://orcid.org/0000-0002-4065-2649
Received 19 September 2019 Reviewed 14 November 2019 Accepted for press 29 November 2019
Информация об авторе
Молчанов Владимир Федорович, доктор физико-математических наук, профессор кафедры функционального анализа. Тамбовский государственный университет им. Г.Р. Державина, г. Тамбов, Российская Федерация. E-mail: v.molchanov@bk.ru
ORCID: https://orcid.org/0000-0002-4065-2649
Поступила в редакцию 19 сентября 2019 г. Поступила после рецензирования 14 ноября 2019 г. Принята к публикации 29 ноября 2019 г.
