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8Математические заметки СВФУ Январь—март, 2022. Том 29, № 1
UDC 517.548+517.547.9
SYMMETRIES IN QUATERNIONIC ANALYSIS H. Orelma
Abstract. This survey-type paper deals with the symmetries related to quaternionic analysis. The main goal is to formulate an SU(2) invariant version of the theory. First, we consider the classical Lie groups related to the algebra of quaternions. After that, we recall the classical Spin(4) invariant case, that is Cauchy—Riemann operators, and recall their basic properties. We define the SU (2) invariant operators called the Coifman— Weiss operators. Then we study their relations with the classical Cauchy—Riemann operators and consider the factorization of the Laplace operator. Using SU (2) invariant harmonic polynomials, we obtain the Fourier series representations for quaternionic valued functions studying in detail the matrix coefficients.
DOI: 10.25587/SVFU.2022.72.67.007
Keywords: quaternionic analysis, Cauchy—Riemann operator, Lie group SU (2), Coif-man—Weiss operator, Fourier series, matrix element.
1. Introduction
The roots of quaternionic analysis come from the works of Moisil and Theodo-resco (see [1]), and Fueter (see [2,3]). Deavours published the first survey [4] of Fueter's works at the beginning of 70s. The idea is to found a well-defined and explicit counter-part for complex function theory on the plane. A function class in quaternionic analysis corresponding to complex holomorphic functions, called regular functions, is characterized by so-called Cauchy-Riemann operators. It seems that Sudbery's survey paper [5] was one of the starting points for modern research of the area, making the theory again visible. Around the same time in Gent, Richard Delangle and his students Fred Brackx and Frank Sommen started to study function theory on Clifford numbers, called Clifford analysis (see [6]). The quaternionic analysis is a natural special case for this theory still having its own special features. See [7] as a modern introduction to the topic.
Cauchy-Riemann operators are well-studied and known. Their counter-part in Clifford analysis is the so-called Dirac operator, whose kernel is used to define a higher dimensional analogy for complex holomorphic functions, called monogenic functions. The theory of monogenic functions is well-known and offers a nice approach for function theory in higher dimensions. The symmetry of the Dirac operator is also well-studied (see, e.g., [8,9]).
Quaternionic analysis symmetries are not so well eloborated, although the starting point is fascinating. Many classical Lie groups are associated with quaternions,
© 2022 Orelma H.
indeed SU(2), Spin(3), Spin(4), SO(3) and SO(4). In the spirit of Clifford analysis, Cauchy-Riemann operators are Spin(4) invariant under L and H actions (see Section 3). However, they are not invariant under canonical actions
R\u\f(x) = f(ux) and S[u]f(x) = f(xu), (1)
where u £ SU(2). This action is mentioned and studied in the book by Coifman and Weiss [10]. In this survey-type paper, our aim is to complete their studies and give a modern representation of their theory using quaternions instead of matrices. The fundamental tools are SU(2) invariant operators under the preceding actions. We are call them Coifman-Weiss operators. Coifman-Weiss operators give also a decomposition for the Laplacian, such as the Cauchy-Riemann operator and its conjugate. The results given in this paper offer a new way to look at quaternionic analysis via symmmetry and many possibilities to continue research in this direction.
The structure of the paper is the following. Section 2 is completely algebraic: we recall all needed tools and more as a starting point for further needs. In Section
3, we recall the Cauchy-Riemann operators and their classical actions. In Section
4, we define the Coifman-Weiss operators by SU(2) invariant differential operators and represent them by the Cauchy-Riemann operators and derive some fundamental formulas. In Section 5, we find the Fourier series representation for a quaternion valued functions using SU(2) invariant harmonic polynomials.
2. Quaternions
In this section, we recall Quanternions and classical Lie groups related to them. All theory is completely known, and we use references [11,12]. This section is the starting point for the following ones, and for this reason we want to give a detailed representation.
2.1. Algebra of Quaternions. The associative division algebra of quaternions is generated in R4 with the basis {e0, e1,e2,e3} putting
2 2 2 i
ei = e2 = e3 = eie2e3 = -1.
The algebra is denoted by H. In the above, we denote the identity by e0 = 1. We also denote basis quaternions by
i = ei, j = e2, k = e3.
A general quaternion is
where the x0 is the scalar part of x and the vector part is respectively
x = xiei + x2e2 + x3e3.
The conjugate is defined by
The conjugation is an anti-involution, that is
xy = yx
for all x, y € H. A quaternion and its conjugate satisfies the relation
3
= (2)
j=0
The real and vector part of a quaternion x may be computed by
Re(x) := xq = —(x + x), Vec(x) := x = —(x — x). The norm in H is defined by
I 12 — — 2,2,2,2 /y* - /-v» /-v» - /-v» /-v» --_L_ /y® __L_ /y® __L_ /y®
— ^^ — juJU — 0 ^^ 1 ^^ 2 ^^ 3
and it is multiplicative, that is, for all x, y € H we have
|xy| = MM-
This means, that the unit sphere
S3 = {x € H : |x| = 1}
admits the group structure. The definition of the norm gives us the explicit formulas for inverse elements _
for non-zero quaternions. For an element of the unit sphere x € S3, the inverse is just x= x.
The inner product of quaternions x, y € H may be computed by
(x, y) = Re(xy) = x0y0 + + x2y2 +
If a € H, we have
(ax, y) = (x, ay). (3)
The real part satisfies Re(xy) = Re(yx) and
Re(xyz) = Re(zxy) = Re(yzx) (4)
for all x, y, z € H.
In this paper, we often use so-called complex or polar representations for quaternions, defined by
x = Xi + X2j,
where X1 = x0 + x1i and X2 = x2 + x3i are complex numbers. Then the conjugation is
x = Xi- X2j (5)
and the multiplication
xy = (XxYi - X2Y2) + (XM + X2Y1)j. (6)
This gives
|x|2 = |Xi|2 + |X2|2. 2.2. SU(2) = S3. The special unitar group SU(2) is defined by
su^ = {(-i2 xi) ^^2x2--\Xi\2 + \x2\2 = iy
\ —X 2 X
Since
' Xi x2\(Y± Y_2\ = ( X1Y1 - XM + X2Fi
-x2 x1j\-y2 yx) + x2yx) x1y1-x2y2
we observe comparing this with the formula (6), that
SU (2) = S3.
Let us define the mapping
Pu{x) = UXU
where u £ S3 and i£R3 = Vec(H). Since ~x = —x, we have
R e(uxu) = — (uxu + uxu) = — (uxu — uxu) = 0,
and observe that for all u £ S3 we have
pu : R3 M R3.
Since \uxu\ = \x\, we obtain pu £ 0(3) for any u £ S3. Since
Pu{x x y) = iu(xy - yx)u = uxuuyu - uyuuxu) = pu(x) x pu(y)
i.e., the mapping pu preserves the orientation, so we have pu £ SO(3), or more precisely,
u M pu; S3 M SO(3) is a group homomorphims. We know, that any rotation in R3 is the composite of two plane reflections. Let u £ S3 fi R3 and decompose x = Xu + v, where v -L u. Since pu(u) = u and (v) = —v, we have
Pu (x) = Am — v
i.e., pu is the reflection of the plane Span(w)±. Then for each rotation, we may find reflections i.e., vectors ux and u2, such that « is the wanted rotation. We conclude that
u M pu; S3 M SO(3) is surjection. Since the mapping is 2 — 1, we observe that S3 is a two fold cover group of SO(3), i.e., S3 = Spin(3).
2.3. Rotations in R4. If u,v £ S3, then our first observation is that the mappings
ul : x M ux, vr : x M xv
are orthogonal mappings.
Proposition 2.1 [11,12]. The preceding mappings belongs to SO(4).
Proof. We need to prove that their determinant is one. Let now u G S3 and i£l3 such that ili. Then
ux = uqx + ux = uqx imxiçr3.
Since \ux\ = |x|, by the preceding section, there exists v G S3 such that
ux = pv (x) = vxv.
Then u = VXVX+1 and
det(iti) = det(^L) det(xL) det(i7z,) det(x^1)
= det(uL) det(xL) det(uL)_1 det^)"1 = 1. □
Let us now define the mapping pUjV{x) = uxv, where u,v G S3. Obviously Pu,v G SO(4).
Lemma 2.2 [11,12]. The mapping
P : S3 x S3 ^ SO(4)
is surjection.
Proof. For the sake of completeness, we recall the proof. Let Q G SO(4) be an arbitrary rotation and let u = Q( 1). Then |w| = 1 and x h> uQ(x) belongs to 50(4) leaving the x0-axis invariant, indeed uQ( 1) = uu = 1. Hence uQ(x) is a rotation in R3 and there exists v G S3 satisfying
uQ(x) = pv (x) = vxv.
We compute
uQ{x) = uQ(x o) + uQ(x) = xo + vxv = vxv.
Hence Q(x) = uvxv = pUVjV{x), completing the proof. □
Since the kernel of the mapping p is {(1,1), (-1, -1)}, we have that
S3 x S3 = Spin(4).
3. Cauchy—Riemann operators
In this section, we represent Cauchy-Riemann operators and recall their invariance properties.
3.1. Definition and basic properties. The Cauchy-Riemann operator and its conjugate is defined by
dx = dXo + dx
and
where = e\dXl + e2dX2 + e3dX3 is called the Dirac operator. They factorize the Laplacian by
3 3-- d-d — A — d2 4- d2 4- d2 4- d2
Vx'Jx — 'Jx'Jx — <-±x — '-'xo ' uxi ' ux2 ' ux3 ■
It is well-known (see, e.g., [13]) that the Laplacian is SO(4) invariant, that is, if we define an action by T[u,v]f(x) = f(uxv), where u,v £ SU(2), then
[Ax, T [u,v]] = 0.
As a special case of this, we have
[Ax,R[u]] = [Ax,S[u]] = 0, that is, the Laplacian is also left and right SU(2) invariant under actions (1).
3.2. On the invariance of the Cauchy—Riemann operator. Let us now
study invariance of the Cauchy-Riemann operator. For this, we need the following lemma.
Lemma 3.2. If u, v £ S3, then
duxv udxv
Proof. A proof may be found in [8, p. 222] in the Clifford analysis level. But for the sake of completeness and since the situation is a little bit different, we want to give a detailed proof. Let us first consider Q £ SO(4) and y = Q(x). Hence
3
Q(ek ) =53 Qjk ej • j=0
Consider the Cauchy-Riemann operators
33 dx = ^2 ekdxk and dy = ^ ejdyj k=0 j=0
Since x = QT (y), we find
3
xk ^ ^ Qjkyj
j=0
and = Qjk • Then we compute using the chain rule
df =^_d£dxk = V^n. —'L
dVj dxk dyj Jk dxk '
Then we obtain
3 3 3 / 3 \ 3
dy = Y1 ej dyj = ej Qjk dXk = I ^2Qjk ej I dXk =Y1 Q(ek )dXk • j=0 j,k=0 k=0 V j=0 ) k=0
Now, choosing Q(x) = uxv, we obtain the result. □
Using the preceding lemma, we deduce that the Cauchy-Riemann operator is not R or S invariant.
Proposition 3.2. For the Cauchy-Riemann operator
(a) R[u]dx = udxR[u],
(b) S[u]dx = dxuS[u].
Remark 3.3. The preceding proposition may be found in the book by Coifman and Weiss [10. p. 118], but they prove it only for differentiable functions f : H ^ R using a different technique.
Although the Cauchy-Riemann operator is not SU(2) invariant under actions R and S, we can make it invariant by defining actions
L\u\f(x) = uf{uxu)
and
H\u\f(x) = uf(uxu)u. Indeed, using Lemma 3.1, we compute
L[u]dxf(x) = uduXUf(uxu) = uudxuf(uxu) = dxL[u]f(x)
and
H[u]dxf(x) = uduXUf(uxu)u = uudxuf(uxu)u = dxL[u]f(x), since uu = 1.
Proposition 3.4. For each u € SU(2), we have
[dx,L[u}\ = [ds, L[u}\ = \8X, H[u}\ = [dm, H[u}\ = 0.
The L and H operators are defined and studied for the first time in context of the Clifford Dirac operator by Frank Sommen in his paper [14-16], but of course the formulas work also for the Cauchy-Riemann operator in quaternionic analysis.
4. Coifman—Weiss operators
The Lie algebra su(2) = H is generated by {e0, e1; e2, e3} and the exponential mapping
e(-> : su(2) ^ SU(2)
is defined via matrix exponentiation (see [10,17]). We define left SU(2) invariant derivatives by
P*.f(X) = lim^)-^.
Extending f outside of S3, we may write etej = 1 + tej + o(t2), that is,
P*f{x) = umfix + txf-fix) = (xes,dx)f(x).
Obviously dXj Ru = RudXj.
Similarly, we define left SU(2) invariant derivatives by
^ f (etejx) - f (x)
fix) = lim ■
t
Extending f outside of S3, we may write etej = 1 + tej + o(t2), that is,
fxj{X) = îim /(*+*«**)-/(*) = {ejX> dx)f{xy
Obviously dS3 S„ = S„ dS3.
We define the left and right SU(2) invariant Coifman-Weiss operators and their conjugations by
33
and ^ = j=0 j=0
and
33
= and ^ =
j=0 j=0
Let us first prove the following representation result. The first of them can be found in [10].
Proposition 4.1. Left invariant Coifman-Weiss operators satisfy
p?f(x)=xdxf(x)
and
Pif(x) = d^xf(x).
Proof. In [10], the proof is based on matrix operators. We give here a direct proof with quaternions. Let f be a differentiable function. Using (3), we have
Pxi = (xej,dx) = (ej,xdx).
Hence
33
Pxf(x) = = ^2zj(zj,xdx)f{x)=xdxf{x).
j=0 j=0
Using (4), we obtain
= (xej> dx) = Re(xëjdx) = Re(ëjxdx) = Re(dxxëj) = (ëj, &xx)
and
33
Pif(x) = = ^ëj&Mfix) = ¿hxf(x). □
j=0 j=0
Proposition 4.2. Right invariant Coifman-Weiss operators satisfy
fâf(x)=àxxf(x)
and
p§f(x) = xd3rf(x). Proof. Let f be a differentiable function. Using (4), we have
fix- = {ejxi 9x) = Re(ëjxdx) = Re(xëjdx) = Re(ëjdxx) = (ej, dxx).
Hence
33
Pxf (x) = J2ejPXlf(x) = J2ej(ej,dxx}f(x) = dxxf. j=0 j=0
In the above, we have
J>S
= R e(ejdxx) = (dxx,ej) = (ej,xdx)-
Hence
33
P§f(x) = ^2^jPxJix) = ^ëjiëj^dx )f{x) = xdxf. □ j=0 j=0 Using the preceding formulas, we can give direct proof for invariance.
Proposition 4.3. If u G SU(2), then
[p?,R[u]] = [p£,R\u]] = [psx,S\u]] = [pi,S\u]] =0.
Proof. Let u G S3. Using Lemma 3.1, we compute
s\u\{fixf(x)) = dxuxuf(xu) = dxuuxf(xu) = dxxS[u]f(x) = S[u]f(x)). Other formulas can be proved by a similar technique. □
Remark 4.4. Coifman-Weiss operators obey the quaternionic conjugation law if we allow that they act also from the right, that is, ft^f = fp§- and pxf = fft-§. Thus, Cauchy-Riemann and Coifman-Weiss operators behave algebraically similarly.
Let us next prove the following decomposition formulas for the Laplacian. One of these formulas can be found in [10]. Next, we formulate both and give a direct proof using quaternions.
Proposition 4.5. p*(p£ + 2) = (p£ + 2)p£=\x\2Ax, + 2)p§ = +
2) = |x|2Ax.
Proof. We compute
j=0 Xj
Using (2), we compute
3
Pi (Pif) = \x\2Axf + J2 ' J~' .¡'hi = M2A xf - 2x0,/ = \x\2Axf - 2pif.
j=0
Similarly, we compute
Pi(Plf) = \x\2Axf - 2p§f, p*(p£f) = \x\2Axf - 2p«f,
p£(p«f) = \x\2Axf-2p«f.
The proof follows from these. □
These formulas indicate the commutativity of Coifman-Weiss operators and their adjungated versions.
Corollary 4.6.
®Kp£ = p£pK,pspi = PiPl
x
5. Fourier series on SU (2)
In this section, we find the Fourier series expansion for quaternionic valued functions L2(SU(2)) and extend them to the whole space. This job is already mostly done, e.g., in the book by Ruzhansky and Turunen [17]. Using the Fourier series, it is easy to obtain a series representation for the quaternionic valued function on spherical domains.
5.1. Homogeneous spherical harmonic polynomials. Let us consider integrable functions f, g : S3 ^ C. We define the innerproduct
(f,g)L2(s3) '■= — f f{x)g{x)dS{x) J
S3
and in the usual way this leads us to the space L2(SU(2)), square integrable complex valued functions on S3. We define the left and right SU(2) action on L2(SU(2)) by (1). The Lie group SU(2) = S3 admits a natural biaxial nature, i.e., every point is of the form y = Y + Y2j, where Y = y0 + and Y2 = y2 + y3i
We look for irreducible representations in the usual manner. We take a space of 2l-homogeneous complex valued polynomials VI with an orthonormal basis
y l-ny i+n
ptn{v)=ptn{Yl'Y*)= ^i-n)Ki+ny:
for n G {—£, — I + 1 ,...,£— 1,1} and I € ^No- The dimension of the space Vg is 21 + 1 and I - n,l + n e Z. See all details in [10,17].
The matrix elements {Rttmn} and {St^n}, with respect to the actions (1), are defined by
RxPtn(YuY2) =Y/RtL(x)Pik(Y1,Y2) k
and
SxPin(Yi, Y2) = £ Stkn(x)pik(Yi.Yi), k
for x e SU(2). For the left action, we need to use conjugates in the second variable to complete the calculations successfully.
Proposition 5.1. Matrix elements admits to representations
2t\I (I - n)!(l + n)
and
s,i t \ 1 Í (I — m)\(l + m)\ f if) _ -iffse-nf— -id , i6\i+n i2m6 ,ü mn( ) = 2^V (f-n)!(£ + n)! 7( l 2 } ( l + 2 }
n
for x = z1 + z2j.
Proof. The right invariant version of the formulas can be found in [10] in a slightly different form.
Left invariant case. Consider the homogeneous polynomials
mniy)=mn{Yl'Y2)= ^i-n)l(i + ny.-Using (5) and (6), we have
RxPiniv) =Pin{xy) = pen(ziY1+z2Y2,z1Y2 - z2Yi) = pen(ziY1+z2Y2, z1Y2-z2Y1).§ Hence
(z1Y1+z2Y2y-(z1Y2-z2Yiy+" =srRf£ YtkY2+k
\J{i — n)\{l + n)\ Y ^-*)!(* +A)!"
To find an explicit formula for a matrix coefficient, we first substitute y = ei0j), i.e., we obtain
(zie10 + z2e-iey-n{zie-ie -z2et6y
e
-i2k8
y/(e-n)\(e + n)\ ? tkn{x) ^{i-k)\{i + k)\
k
„i2rnB
We multiply both sides by ei2m and use the fact
2n, if r.
0, other integers.
, ( 2n, if m = k,
ei2(m-k)e d(9 = I
Then we obtain
Right invariant case. Using (6), we have
Sxp£n(y) = Pin{yx) = ptn{YiZi - Y2~z2,Yiz2 + Y2zi).
Hence
(YlZl - Y2z2y-(YlZ2 + Y2z1)t+n ^ YtkYtk
^{i-nw+ny. v wii-mt+ky.'
We substitute
y = (e + e j) and we have
{Zle10 -z2e-l0y-n{z2e10 + z1e-l0y+n e e-l2k0
tkn\x)~
■\J{I — n)\{l + n)\ ^ *nV y/(£- k)\{l + k)\
Multiplying both sides by ei2m0 and integrating as in the above, we obtain
s,i t \ 1 (£ — m)l(£ + m)l f if) _ _i6g id,- -iOsi+n i2m,e ,Q r-l tmni*) = J |
n
Remark 5.2. For example, in the left invariant case, the integral can be written by standard change of variables
—n
n
= J(z+ z2y-n(zi - de
—n
2n
= 2 J {zie10 +z2f-n{zi -z2el0Y+nel{m-^0de.
Thus, we may represent the matrix elements in a coordinate independent way by
w = 1
and
sa 2 ¡{l-m)\{i + m)\ f ,..... ^ _n,^ , .....dw
'CW =--1/77-\>/p , \i' / "(zx+zaw)^
^y (I — n)!(l + n)! J w-
|w| = 1
The preceding formulas allow us to extend the matrix coefficients ^¿^«(x) and Stm«(x) to the whole space, i.e., we may assume x G H. Let us recall that if f : S3 ^ C is a restriction of a harmonic function F : H ^ C, it is called a spherical harmonics.
Proposition 5.3. Matrix coefficients are 2£-homogeneous spherical harmonics.
Proof. Let x = z1 + z2j. In a left invariant matrix coeffient, the kernel is given in the preceding remark by
kn(x) = kl{z1,z2) = (zxw + z2y-n(Zl - z2wf+n
for |w| = 1. If x = z1 + z2j, we put
dz! = dxo + idXl and 0Z2 = 0X2 + ¿d^
with their conjugations
<3zi = dXo - idXl and <9j2 = dX2 - idX3.
Hence dZpzp = dzpzp = 0 and <hpzp = dz^zp = 2 for p = 1, 2. It is easy to see, that
= dz1dzi + d^2dZ2.
It is straightforward to compute
chid^h = 4(£ - n)(£ + n)wkt-i, ck2dZ2ki = -4(£ - n)(£ + n)wkt-i,
n
that is
Ax ki = 0.
The right invariant case is similar. □
5.2. Fourier series representation. Let tlnm be either the right or left invariant matrix coefficient. Hence, any f G L2(SU(2)) admits the Fourier series representation
f (X)= (21 +1)£ (f,tnm )i2(s3) tim (x),
£eiN0 m,n
where the summation is taken over m, n satisfying —I < m, n < I and I — m, I — n G Z (see details in [10,17]). If we define x = ru, r = \x\ and u = f G S3, we may represent any f : O ^ C, where O G H is a spherical neighborhood and f |S3 G L2(SU(2)) by the series
f(x)= (21 +1)r2^(f.O^/mJu).
2(S3)'
£eiN0 m>™
This allows us to find an explicit series expansion also for quaternion valued functions.
Theorem 5.4. Let f = f1 + f2j be a quaternion valued function and f1 and f2 be complex valued functions defined on any spherical neighborhood O c H and f1|s3, f21s3 € L2(SU(2)). Then
f(x)= (21 +1)r2^ ((fl,imjL2(S3) +(— 1)m-n(f2,iim,-„)L2(s3)
££¿N0 m>™
where x = ru, r = \x\ and u = ^ £ S3.
Proof. We consider the left invariant case. Our first observation is
Rfi (Z, ) = J_ (t-my.(e + m)\ f e_ iey+n
—n
x (Zle-m + z2éef-ne-ame dd = fltlmi_n(zi, -z2). Recall the coordinate invariant form of the matrix coefficient given in Remark 5.2
Hmn{zuz2) =--J—-j ' / (ZlW + z2) n(zl-z2w)
n V (I — n)!(l + n)! J w-
|w| = 1
We assume z1z2 = 0 and make the change of variables given by Coifman and Weiss [10, p. 108]
z{z2w = t — \z2\2 + \zi\2
and we have
- M ¿-H2 H - N2-t ziw + z2 = ——- and z 1 — z2w = ——-.
¿2 Zi
n
We define T = {t = z\z2w + \z2\2 — \zi\2 : |w| = 1}. Hence the matrix coefficient takes the form
R. _ /(l-m)!(l + m)!
mn{/ U 2) 7T y (£-n)\(£ + n)\
zi\2\l-n(\z2\2-t\l+n z£1-m+V2-m+1 dt
x
Z2 J V ') (i-|z2|2+|z1|2)^™+lz1z2
r
i ¡{£-m)\{£ + m)\ n_m
z 1 zry
nV (I - n)!(l + n)!
|2\£—n^l 12 ,-\£+n_dt
x J(t -M2)l-(|z2|2 - t)
(t - |Z2|2 + |zi|2)l-m+l'
The path r is the same if we choose ±z2. Hence the integral above is invariant under substitution by —z2 and we obtain Rtm„(zi, —z2) = (—1)n-mRtm„(zi, z2). Especially
HilJzi,z2) = (-ir-nRte_mi_n(z1,z2). The proof is similar for the right invariant matrix coefficients. Hence
4„(u)j = jCJu) = (-1)m-njtimi-„(u),
and we obtain
f (x) = fi(x)+/2(x)j = £ (2i+1)r21 £(/i,tm„)i2(s3)tm„(u)
+ £ (2i+1)r2^ (/2,tm„)
fe^No
= £ (2i+1)r21 £(/i,tm„)i2(s3)tm„(u)
££¿N0 m>n
+ £ (21 +1)r21 £(/2,ti„) j(-1)m-ntm„(u)
-n,-mL2(s3)J
£ (21 + 1)r2l£((/i,<mjL2(s3) +(-1)m-n(/2,timi-n)L2(S3)j)4». □
££¿N0 m>™
Conclusions. In this paper, we recall the SU(2) invariant version of quaternionic analysis and write all proofs using modern tools. The fundamental objective is to find an SU(2) invariant Dirac type operators, which are called Coifman-Weiss operators, after the founders of the theory. In addition, we discuss the SU(2) invariant Fourier series and recall nice and explicit formulas for matrix coefficients.
This paper allows us to continue studies with Coifman-Weiss operators in different directions.
There are still open questions in the basic theory. The connection of the representation of the Coifman and Weiss matrix coefficients that we get in this paper,
and the representation of Ruzhansky and Turunen [17], should be studied. Also, the connection to Jacobi polynomials should be recorded explicitly for all cases. We leave these to the interested reader as an exercise.
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Received August 24, 2021 Revised February 17, 2022 Accepted February 28, 2022
Heikki Orelma
Tampere University,
Kalevantie 4, Tampere 33100, Finland
Heikki.Orelma@tuni.f i
