Advances in Pure Mathematics, 2012, 2, 63-103
http://dx.doi.org/10.4236/apm.2012.22013 Published Online March 2012 (http://www.SciRP.org/journal/apm)
Multidimensional Laplace Transforms over Quaternions,
Octonions and Cayley-Dickson Algebras, Their
Applications to PDE
Sergey Victor Ludkovsky
Department of Applied Mathematics, Moscow State Technical University, Moscow, Russia
Received July 8, 2011; revised November 10, 2011; accepted November 20, 2011
ABSTRACT
Multidimensional noncommutative Laplace transforms over octonions are studied. Theorems about direct and inverse
transforms and other properties of the Laplace transforms over the Cayley-Dickson algebras are proved. Applications to
partial differential equations including that of elliptic, parabolic and hyperbolic type are investigated. Moreover, partial
differential equations of higher order with real and complex coefficients and with variable coefficients with or without
boundary conditions are considered.
Keywords: Laplace Transform; Quaternion Skew Field; Octonion Algebra; Cayley-Dickson Algebra; Partial
Differential Equation; Non-Commutative Integration
1. Introduction
The Laplace transform over the complex field is already
classical and plays very important role in mathematics
including complex analysis and differential equations [1-
3]. The classical Laplace transform is used frequently for
ordinary differential equations and also for partial dif-
ferential equations sufficiently simple to be resolved, for
example, of two variables. But it meets substantial dif-
ficulties or does not work for general partial differential
equations even with constant coefficients especially for
that of hyperbolic type.
To overcome these drawbacks of the classical Laplace
transform in the present paper more general noncom-
mutative multiparameter transforms over Cayley-Dick-
son algebras are investigated. In the preceding paper a
noncommutative analog of the classical Laplace trans-
form over the Cayley-Dickson algebras was defined and
investigated [4]. This paper is devoted to its generali-
zations for several real parameters and also variables in
the Cayley-Dickson algebras. For this the preceding re-
sults of the author on holomorphic, that is (super) dif-
ferentiable functions, and meromorphic functions of the
Cayley-Dickson numbers are used [5,6]. The super-dif-
ferentiability of functions of Cayley-Dickson variables is
stronger than the Fréchet's differentiability. In those works
also a noncommutative line integration was investigated.
We remind that quaternions and operations over them
had been first defined and investigated by W. R. Ha-
milton in 1843 [7]. Several years later on Cayley and
Dickson had introduced generalizations of quaternions
known now as the Cayley-Dickson algebras [8-11]. These
algebras, especially quaternions and octonions, have found
applications in physics. They were used by Maxwell,
Yang and Mills while derivation of their equations, which
they then have rewritten in the real form because of the
insufficient development of mathematical analysis over
such algebras in their time [12-14]. This is important,
because noncommutative gauge fields are widely used in
theoretical physics [15].
Each Cayley-Dickson algebra r
A
over the real field
R has 2r generators


01 21
,, ,r ii i  such that 0=1i,
2=1
j
i

for each =1,2,,21
r
j, =
j
kkj
ii ii for every
121
r
kj

 , where 1r. The algebra 1r
A

is
formed from the preceding algebra r
A
with the help of
the so-called doubling procedure by generator 2r i. In par-
ticular, 1=
A
C coincides with the field of complex
numbers, 2=
A
H is the skew field of quaternions, 3
A
is the algebra of octonions, 4
A
is the algebra of seden-
ions. This means that a sequence of embeddings
1rr
AA

  exists.
Generators of the Cayley-Dickson algebras have a
natural physical meaning as generating operators of
fermions. The skew field of quaternions is associative,
and the algebra of octonions is alternative. The Cayley-
Dickson algebra r
A
is power associative, that is,
=
nmn m
zzz
 for each ,nm N and r
zA. It is non-
associative and non-alternative for each 4r. A

S. V. LUDKOVSKY 64
conjugation *=zz
 of Cayley-Dickson numbers
r
zA is associated with the norm 2** ==zzzzz
.
The octonion algebra has the multiplicative norm and is
the division algebra. Cayley-Dickson algebras r
A
with
4r are not division algebras and have not multi-
plicative norms. The conjugate of any Cayley-Dickson
number z is given by the formula:
(M1) **
:=zl

 .
The multiplication in 1r
A

is defined by the fol-
lowing equation:
(M2)



=ll l

   

for each

,

,

, r
A

, 1 := r zlA



 ,
1
:= r lA



 .
At the beginning of this article a multiparameter non-
commutative transform is defined. Then new types of the
direct and inverse noncommutative multiparameter trans-
forms over the general Cayley-Dickson algebras are in-
vestigated, particularly, also over the quaternion skew
field and the algebra of octonions. The transforms are
considered in r
A
spherical and r
A
Cartesian coordi-
nates. At the same time specific features of the noncom-
mutative multiparameter transforms are elucidated, for
example, related with the fact that in the Cayley-Dickson
algebra r
A
there are 21
r imaginary generators

121
,,
r
ii

 apart from one in the field of complex num-
bers such that the imaginary space in r
A
has the di-
mension 21
r. Theorems about properties of images
and originals in conjunction with the operations of linear
combinations, differentiation, integration, shift and ho-
mothety are proved. An extension of the noncommuta-
tive multiparameter transforms for generalized functions is
given. Formulas for noncommutative transforms of pro-
ducts and convolutions of functions are deduced.
Thus this solves the problem of non-commutative ma-
thematical analysis to develop the multiparameter Lap-
lace transform over the Cayley-Dickson algebras. More-
over, an application of the noncommutative integral trans-
forms for solutions of partial differential equations is
described. It can serve as an effective means (tool) to
solve partial differential equations with real or complex
coefficients with or without boundary conditions and
their systems of different types (see also [16]). An algo-
rithm is described which permits to write fundamental
solutions and functions of Green’s type. A moving bound-
ary problem and partial differential equations with dis-
continuous coefficients are also studied with the use of
the noncommutative transform.
Frequently, references within the same subsection are
given without number of the subsection, apart from
references when subsection are different.
All results of this paper are obtained for the first time.
2. Multidimensional Noncommutative
Integral Transforms
2.1. Definitions Transforms in Ar Cartesian
Coordinates
Denote by r
A
the Cayley-Dickson algebra, 0r

, which
may be, in particular, 2
=
H
A the quaternion skew field
or 3
=OA the octonion algebra. For unification of the
notation we put 0=
A
R, 1=
A
C. A function
:nr
f
RA we call a function-original, where 2r

,
nN

, if it fulfills the following conditions (1-5).
1) The function


f
t is almost everywhere conti-
nuous on n R relative to the Lebesgue measure n

on
n
R.
2) On each finite interval in R each function




1
=,,
j
jn
g
tftt by
j
t with marked all other va-
riables may have only a finite number of points of dis-
continuity of the first kind, where

1
=,,n
n
ttt R,
j
tR

, =1, ,jn. Recall that a point 0 uR is called
a point of discontinuity of the first type, if there exist
finite left and right limits




,< 0
00 =: 0
limuuuu r
g
ugu A


 and




,> 0
00 =: 0
limuuuu r
g
ugu A


.
3) Every partial function


1
=,,
j
jn
g
tftt satis-
fies the Hölder condition:




j
jj jjjjj
gt hgtAh

  for each <
j
h

,
where 0< 1
j


, =>0
j
Aconst , >0
j

are
constants for a given

1
=,,n
n
ttt R, =1, ,jn,
everywhere on n
R may be besides points of discon-
tinuity of the first type.
4) The function


f
t increases not faster, than the
exponential function, that is there exist constants
=>0
v
Cconst ,


1
=,,
n
vv v, 11 ,aa R
, where


1,1
j
v for every =1, ,jn, such that






<exp ,
vv
f
tC qt for each n tR with 0
jj
tv 
for each =1, ,jn,


11
=,,
vvnv
n
qvava; where
5)

=1
,:=n
j
j
j
x
yxy
 denotes the standard scalar pro-
duct in n
R.
Certainly for a bounded original
f
it is possible to
take 11
==0aa
.
Each Cayley-Dickson number r pA we write in the
form
6) 21
=0
=r
j
j
j
ppi

, where

01 21
,, ,r ii i  is the stand-
ard basis of generators of r
A
so that 0=1i, 2=1
j
i

and 00 ==
j
jj
iiiii for each >0j, =
j
kkj
ii ii for each
>0j and >0k with kj

, j pR for each j. If
there exists an integral
7)
 

,
:=; :=d
pt
nn n
R
F
pFpfte t



,
then


n
F
p is called the noncommutative multipara-
meter (Laplace) transform at a point r pA of the func-
tion-original


f
t, where

S. V. LUDKOVSKY 65
011 2121
=rr r iiA  

 is the parameter of an
initial phase, jR

 for each =0,1, ,21
r
j,
r
A

, =2 1
r
n,

d= d
n
tt  ,
8)


21
01 =1
21
,= r
r
j
jj
j
ptp ttpti


 

,
we also put
8.1)
 
,;=,upt pt

.
For vectors ,n vw R we shall consider a partial
ordering
9) vw if and only if
j
j
vw for each
=1,,jn and a k exists so that <
kk
vw,
1kn

.
2.2. Transforms in Ar Spherical Coordinates
Now we consider also the non-linear function


=,;uupt

taking into account non commutativity of
the Cayley-Dickson algebra r
A
. Put
1)




01 0
,:=,; :=,uptuptps Mpt

, where
2)
 

 



 
111122 2 222233322 2
22
,=, ;=cossincossin
r
Mpt Mptpsipsipspsips  

 




 
22 2
2222 222121 21 212222 222121 21
sincossin sinsin
rrrrr rrrrrrr r
pspsipspsps  
  




for the general Cayley-Dickson algebra with 2<r

.
2.1)

:;:
j
jjn
s
snt tt for each =1, ,jn,
=2 1
r
n, so that 11 =n
s
tt, =
nn
s
t. More ge-
nerally, let
3)
 
01 0
,=,; =,uptuptps wpt


, where

,wpt is a locally analytic function,
 
,=0Rewp t
for each r pA and 21
r
tR

,
 
:= 2Re zzz,
*
=zz
 denotes the conjugated number for r
zA. Then
the more general non-commutative multiparameter trans-
form over r
A
is defined by the formula:
4)



;:=exp,; d
nn
uR
F
pftuptt  

for each Cayley-Dickson numbers r pA whenever
this integral exists as the principal value of either Rie-
mann or Lebesgue integral, =2 1
r
n. This non-com-
mutative multiparameter transform is in r
A
spherical
coordinates, when

,;upt

is given by Formulas
(1,2).
At the same time the components
j
p of the number
p and
j

for

in

,;upt

we write in the p-
and

-representations respectively such that
5)
 


121 *
=1
=22 2
r
r
jjj kk
k
hhiih ihi






for each =1,2, ,21
r
j,
6)
 


121 *
0=1
=22 2
r
rkk
k
hhhihi


 


,
where 2rN

, 002121 =rr r hhihi A

 , j hR

for each j, *==
kk k
iii

 for each >0k, 0=1i,
r
hA

. Denote


;
n
u
Fp

in more details by


,; ;
n
Ffup

.
Henceforth, the functions

,;upt

given by 1(8,8.1)
or (1,2,2.1) are used, if another form (3) is not specified.
If for


,;upt

concrete formulas are not mentioned, it
will be undermined, that the function

,;upt

is given
in r
A
spherical coordinates by Expressions 1,2,2.1). If
in Formulas 1(7) or (4) the integral is not by all, but only
by (1)( )
,,
j
jk
tt variables, where 1<kn,




11<<jjkn

, then we denote a noncom-
mutative transform by

;,,
(1)( );
kt t
jjk
u
Fp

 or

;,,
(1)( ),; ;
kt t
jjk
Ffup

. If

1=1,j···, ()=jk k, then
we denote it shortly by

;
k
u
Fp

or

,; ;
k
Ffup

.
Henceforth, we take =0
m

and =0
m
t and =0
m
p
for each




11,,mjjk  if something other is
not specified.
2.3. Remark
The spherical r
A
coordinates appear naturally from the
following consideration of iterated exponents:
1)
  




 
1111 3222 1333
exp expexpips ipsips  
 

   


111122 2 2222333322 2333
= expcossincossinsin.psipsipspsi psps   
Consider 2r i the generator of the doubling procedure
of the Cayley-Dickson algebra 1r
A
 from the Cayley-
Dickson algebra r
A
, such that 22 =
jr r
j
iii  for each
=0, ,21
r
j

. We denote now the function


,;
M
pt

from Definition 2 over r
A
in more details
by r
M
. Then by induction we write:
2)
 
 






 111111
212122 2122121
exp,;= exp,,,,;
rrrrrrrrr
MptM ipiptttsii 

  
   




 


11111
21222 2121212121212121
expexp,, ,;,
rrrrrr rrrr r rr
ipsMipipt tii 
 



S. V. LUDKOVSKY 66
where

1
=,,
n
tt t,

1
=1=21
r
nnr

,
 
=1;
jj
s
snr t for each

=1, ,1jnr, since
 




(1)
2
1;=
=; 1;
mmnr
mr
snrt tt
s
nr tsnrt




for each =1,,21
r
m.
An image function can be written in the form
3)
 
21
,
=0
;:= ;
r
nn
ujuj
j
FpiF p


,
where a function
f
is decomposed in the form
3.1)
 
21
=0
=r
jj
j
f
tift

,
:n
j
f
RR for each =0,1,,21
r
j,


,;
n
uj
Fp

de-notes the image of the function-original
j
f
.
If an automorphism of the Cayley-Dickson algebra
r
A
is taken and instead of the standard generators

021
,,
r
ii

 new generators


021
,, r NN

 are used,
this provides also




,;=,;
N
M
ptM pt

relative to
new basic generators, where 2rN. In this more
general case we denote by

;
n
Nu
Fp

an image for an
original

f
t, or in more details we denote it by

,; ;
n
NFfup

.
Formulas 1(7) and 2(4) define the right multipa-
rameter transform. Symmetrically is defined a left multi-
parameter transform. They are related by conjugation and
up to a sign of basic generators. For real valued originals
they certainly coincide. Henceforward, only the right
multiparameter transform is investigated.
Particularly, if


01
=,,0,,0ppp and


1
=,0,,0tt, then the multiparameter non-commu-
tative Laplace transforms 1(7) and 2(4) reduce to the
complex case, with parameters 1 a, 1 a. Thus, the given
above definitions over quaternions, octonions and gene-
ral Cayley-Dickson algebras are justified.
2.4. Theorem
If an original


f
t satisfies Conditions 1(1-4) and
11
<aa

, then its image

,; ;
n
Ffup

is r
A
-holo-
morphic (that is locally analytic) by p in the domain




11
:< <
r
zAa Reza

, as well as by r
A

, where
1rN

, 1 221
rr
n


, the function


,;upt

is
given by 1(8,8.1) or 2(1,2,2.1).
Proof. At first consider the characteristic functions


Uvt
 , where


=1
Ut
 for each tU, while


=0
Ut
 for every \
n
tRU,


:=:0 =1,,
n
vjj
UtRvt jn is the domain in the
Euclidean space n
R for any v from § 1. Therefore,
1)


;:=
n
u
Fp 


  
=,,:,,1,1
11 exp,;d,
vv vv vU
nn vftupt t








since


=0
nv w
UU   for each vw. Each integral






exp,;d
Uvftupt t
 
 is absolutely convergent for
each r
pA

with the real part

11
<<aRepa

, since
it is majorized by the converging integral
2)
 







11 01
1 00
exp,;dexpdd
vvnvnn
n UvftupttCvw ayvw ayyy


 




1
0
=1
=,
n
vjv
j
j
Cevw a




where


=wRep , since


=exp
z
eRez for each
r
zA in view of Corollary 3.3 [6]. While an integral,
produced from the integral (1) differentiating by p
converges also uniformly:
3)
 

exp,;d
Uvftuptp ht


 


 
 

01111111 1
00
11 01
1
,,, ,
exp...dd
vnnnnnnnnnnnn
vnvn n
n
Chvy vyhvy vyhvyvyhvy
vwa yvwayyy 


 
 

 

for each r hA, since each r zA can be written in
the form

=expzz M , where 2=[0,)zzz R
,
r
M
A,


Re:=2 = 0MMM in accordance with
Proposition 3.2 [6]. In view of Equations 2(5,6):
4)
 


exp(,;)d= 0
n
Rftupttp
  
 and
5)
 




exp,;d= 0
n
Rftupt t
  
, while
6)
 

exp,;.d
Uvft uptht


 


 
 
1
0
11 01
1 00 =1
expdd =
n
vvnvnnvjv
nj
j
hCvwayvwayyyhCevwa



 
 




S. V. LUDKOVSKY 67
for each r
hA. In view of convergence of integrals
given above (1-6) the multiparameter non-commutative
transform

;
n
u
Fp

is (super)differentiable by p and

, moreover,

;=0
n
u
Fp p  
 and

;=0
n
u
Fp  
 in the considered

,p

-represen-
tation. In accordance with [5,6] a function


g
p is
locally analytic by p in an open domain U in the
Cayley-Dickson algebra r
A
, 2r, if and only if it is
(super)differentiable by p, in another words r
A
-ho-
lomorphic. Thus,

;
n
u
Fp

is r
A
-holomorphic by
r
pA with


11
<<aRepa
 and r
A

 due to Theo-
rem 2.6 [4].
Corolla ry
Let suppositions of Theorem 4 be satisfied. Then the
image

,; ;
n
Ffup

with

=,;uupt

given by 2
(1,2) has the following periodicity properties:
1) for each =1, ,jn and 2π
Z

;
2) for each =1,,1jn so that 12
00
=

and
12
=
j
j

, 12
11
=π
j
j


, 12 =
s
s

for each
s
j

and 1
s
j, while either 12 =
j
j
pp and 12 =
ll
pp for
each lj with =2  or 12 =pp and


f
t is an
even function with =2  by the

=
j
jn
s
tt
variable or an odd function by

=
j
jn
s
tt with
=1  ;
3)

1
,; ;π=,;;
nn
Ffupi Ffup

 .
Proof. In accordance with Theorem 4 the image

,; ;
n
Ffup

exists for each
 
11
:=: <<
fr
pWz AaReza

 and r
A

, where
1r. Then from the 2π periodicity of sine and cosine
functions the first statement follows. From
 
sin= sin

  ,




cos= cos

 ,
 
sin π=sin

  ,
 
cos π=cos

  we get
that
 
12
cos= cos
j
jj jjj
ps ps

 ,
 




11
11 1
22
11 1
sin cos
=sincos
jjjj jj
jjjj jj
psps
psp s


 
 

 
and
 




11
111
22
111
sin sin
=sinsin
jjjjjj
jjjjjj
psps
psp s


 
 

 
for each n tR. On the other hand, either 12 =
j
j
pp

and 12 =
ll
pp for each 1lj with =2  or
12
=pp and



11 11
11 11
,, ,,
1,, , ,,,
,,
jjjj
jjjj
j
n
jn
ftsss s tt
ftsss s tt








is an even with =2

or odd with =1

function by
the


=
j
jn
s
tt variable for each


1
=,,n
n
ttt R, where 1 =
j
jj
tss

 for
=1,,jn,


11
=;=0
nn
ssnt
 . From this and For-
mulas 2(1,2,4) the second and the third statements of this
corollary follow.
2.5. Remark
For a subset U in r
A
we put




,,
π:=:,=, =
sptvs p
vb
UuzUzwvuwswp



for each
s
pb

, where

,,
\{ ,}
:= :=
:=, ==0,,
vrsp
vb sp
rvspv
vb
twvA
zAzwvwww Rvb






where


01 21
:=,,, r bii i

 is the family of standard ge-
nerators of the Cayley-Dickson algebra r
A
. That is, geo-
metrically


,,
πspt U means the projection on the com-
plex plane ,
s
p
C of the intersection U with the plane
,,
πspt t

,


,:=: , sp
CasbpabR, since


*ˆ:=\ 1spb b. Recall that in § § 2.5-7 [6] for each
continuous function :r
f
UA it was defined the ope-
rator ˆ
f
by each variable r zA. For the non-com-
mutative integral transformations consider, for example,
the left algorithm of calculations of integrals.
A Hausdorff topological space
X
is said to be n-
connected for 0n if each continuous map
:k
f
SX from the k-dimensional real unit sphere
into
X
has a continuous extension over 1k R

for each
kn

(see also [17]). A 1-connected space is also said
to be simply connected.
It is supposed further, that a domain U in r
A
has
the property that U is


21
r-connected;


,,
πspt U
is simply connected in C for each 1 =0,1, ,2
r
k

,
2
=k
s
i, 21 =k pi

, ,,rsp tA

and ,
s
p
uC, for which
there exists =zutU

.
2.6. Theorem
If a function


f
t is an original (see Definition 1),
such that


;
n
Nu
Fp

is its imag e multiparameter non-
commutative transform, where the functions
f
and n
u
F
are written in the forms given by 3(3,3.1),


nr
f
RA
over the Cayley-Dickson algebra r
A
, where 1rN

,
1
221
rr
n


.
Then at each point t, where


f
t satisfies the Hölder
condition the equality is accomplished :
1)
    
111
11
=2π2π;exp ,; d
NN
nn
nNu
NN
n
f
tNNFapuapt p 


 
 


 

  

 
 


 





1
=:;,, ;,
nn
Nu
FFaput



S. V. LUDKOVSKY 68
where either
 
,;= ,upt pt


or
 
01 0
,; =,;
N
uptps Mpt

 (see § § 1 and 2),
the integrals are taken along the straight lines

=
j
jj r
pNA  , jR

 for each =1, ,jn;

11
<=<aRepaa
 and this integral is understood in
the sense of the principal value,

1
=,,n
n
ttt R,






112 2
=... nn dpdp NdpNdpN.
Proof. In Integral (1) an integrand

pdp  certainly
corresponds to the iterated integral as
  


11 nn
pdpN dpN  , where
11
=nn ppN pN, 1,,
n
ppR. Using Decom-
position 3(3.1) of a function
f
it is sufficient to
consider the inverse transformation of the real valued
function
j
f
, which we denote for simplicity by
f
. We
put
  
,;:=exp,; d.
nn
Nuj j
R
F
pftuptt  

If

is a holomorphic function of the Cayley-
Dickson variable, then locally in a simply connected
domain U in each ball

0
,,
r
BAz R with the center at
0
z of radius >0R contained in the interior


I
nt U
of the domain U there is accomplished the equality

0
d.
z
zaz



 






1= az  , where the inte-
gral depends only on an initial 0 z and a final z points
of a rectifiable path in

0
,,
r
BAz R, aR (see also
Theorem 2.14 [4]). Therefore, along the straight line
j
NR the restriction of the antiderivative has the form

0
d
j
jj
aN



 
, since
2)


=
=
00 0
ˆ
d= d
zN
j
j
jjj
zN
jaaNN



 
 ,
where
  
=.azaz z   
j
N for the
(super)differentiable by zU

function


z  , when
=
j
zN

, R


. For the chosen branch of the line
integral specified by the left algorithm this antiderivative
is unique up to a constant from r
A
with the given
z-representation

of the function

[4-6]. On the
other hand, for analytic functions with real expansion
coefficients in their power series non-commutative in-
tegrals specified by left or right algorithms along straight
lines coincide with usual Riemann integrals by the
corresponding variables. The functions

sin z,


cos z
and
z
e participating in the multiparameter non-com-
mutative transform are analytic with real expansion co-
efficients in their series by powers of r zA.
Using Formula 4(1) we reduce the consideration to




Uvtft
 instead of


f
t. By symmetry properties
of such domains and integrals and utilizing change of
variables it is sufficient to consider v U with


=1, ,1v. In this case n
R

for the direct multi- para-
meter non-commutative transform 1(7) and 2(4) reduces
to 00




. Therefore, we consider in this proof below
the domain 1, ,1 U only. Using Formulas 3(3,3.1) and
2(1,2,2.1) we mention that any real algebra with
generators 0=1N, k
N and
j
N with 1kj

 is
isomorphic with the quaternion skew field
H
, since


=0
jk
ReNN and =1
j
N, =1
k
N and =1
jk
NN .
Then





 
expexp= expMM M

  
for each real numbers ,,,

 and a purely imaginary
Cayley-Dickson number
M
. The octonion algebra O
is alternative, while the real field R is the center of the
Cayley-Dickson algebra r
A
. We consider the integral
3)
   

111
1,
1
:= 2π2π;exp ,; d
Nb Nb
nn
bn Nuj
Nb Nb
n
g
tNNFapuaptp 









 
 




for each positive value of the parameter 0< <b

. With
the help of generators of the Cayley-Dickson algebra r
A
and the Fubini Theorem for real valued components of
the function the integral can be written in the form:
4)
   



111
11
00
1
=[ 2πd]2dexp,;exp,; d,
Nb Nb
n
bnnN N
Nb Nb
n
g
t NNfuaptuapp  






 






 

since the integral
  
1,...,1
exp, ;d
N
Ufuap



 

for any marked

11
0< <3aa 
 is uniformly con-
verging relative to p in the domain

11
aRepa



  in r
A
(see also Proposition
2.18 [4]). If take marked k t for each kj

and
=
j
SN for some 1j in Lemma 2.17 [4] considering
the variable
j
t, then with a suitable (R-linear) auto-
morphism v of the Cayley-Dickson algebra r
A
an
expression for


,;vMpt

simplifies like in the
complex case with :=
K
CRRK

for a purely imagi-
nary Cayley-Dickson number
K
, =1K, instead of
1
:=CRRi

, where


=vx x for each real number
x
R

. But each equality =

 in r
A
is equivalent to




=vv

 . Then
5)


**
,
==
j
qjlql ql
ReNNNNReNN




for each ,ql.
If 0;
=
jll
lnl j
SN 
 

, 0; =
jll
lnl j
NN 
 
 with
1j and real numbers ,
llR

  for each l, then
6)
 
**
==
jj jj
j
jll
l
ReN SN NReSN





.

S. V. LUDKOVSKY 69
The latter identity can be applied to either




1111 11
=,,,;
kkkknnk nkknn
SMpNpNttNN 
 
  
and




111111
=,,,;
kkkknnk nkknn
NMpN pNNN 
 
  ,
or



111 1
=
kkkkknnn n
SptNpt N 
 

and



111 1
=
kkkkknnnn
NpNp N  
 
,
where
7)



1111 11
...,, ,;...
kkknnk nkknn
M
pNpNttNN 
 
 ,
 


11,12 2,12,11
sin sin
nkkk nnkk kkk n
psN psps 
  


 



8)
 
,1 ,1
=;==;
jkjkkjnk j
s
sntt tsnt
 

for each =1,,1jn;

,1 ,1
=;=
nkk nkkn
s
sntt
 . We
take the limit of


b
g
t when b tends to the infinity.
Evidently,



 
1
;;=1;=
kjk kj
snsns j




for each 1<kjn

. By our convention




1
;=;
k
s
nsn

 for <1k, while

;=0
k
sn  for
>kn. Put
9)


,0 0
,,,;
njj jnnjnj jnn
uppNpNNN     

001, 0
=,(,,);
j
jjjnnj njjnn
psM pNpNNN    
for
N
u given by 2(1,2,2.1), where
j
M
is prescribed by (7),


,,
=;
kj kj
s
sn

;
10)



,00001, =
,,,;= n
njjjnn jnjjnnjkkk k
kj
uppNpNNNpspN    

 
for =
N
uu given by 1(8,8.1). For >1j the parameter
0

for =
N
uu given by 1(8,8.1) or 2(1,2,2.1) can be
taken equal to zero.
When 111 ,, , ,,
j
jn
ttt t

 and
111
,, , ,,
j
jn
pppp

 variables are marked, we take the
parameter

 


0
001111
:=,, ,;
:=
jj
j jnnjnj jnn
j
jnnjjjj nnn
pN pNNN
NNapspsNpsN
 
 

 
 
 

for

,;up

given by Formulas 2(1,2,2.1) or

 


0
001111
:=,, ,;
:=
jj
j jnnjnj jnn
j
jnnjjjj nnn
pN pNNN
NNapspNpN
 
 

 
 
 

for

,;up

described in 1(8,8.1). Then the integral operator
 
1
0
2πdd
lim Nb
j
b
j
jjj
Nb
j
NpN 

 




(see also Formula (4) above) applied to the function


 


 
11, 00
,0 0
,,,,,exp, ,,;
exp,,,;
jjnNjjjnnjnjjnn
Njjjnn jnjjnn
fttua ppNpNttNN
uappN pNNN
 
 

 
 


S. V. LUDKOVSKY 70
with the parameter
j

instead of

treated by Theo-
rems 2.19 and 3.15 [4] gives the inversion formula cor-
responding to the real variable
j
t for

f
t and to the
Cayley-Dickson variable 00
j
j
pN pN restricted on the
complex plane =
N
j
j
CRRN

, since


=
j
j
dcd

  for each (real) constant c. After inte-
grations with =1, ,jk with the help of Formulas (6-
10) and 3(1,2) we get the following:
11)
  
11
1
11
00
1
=2πd2πd
lim NN
nk
bnn kk
NN
bnk
gt ReNN 
 



 
 

 


 
 


 

 






 


  
11,10111011
,101 11011
,,,,,exp,,,;
exp...,,,;...d.
kknNkk knnknkknn
Nkkknn knkknn
fttua pp NpNttNN
uappNpNNNp



 
 


 


 

Moreover,

=
qq
Reff for each q and in (11) the
function =q
f
f stands for some marked q in accor-
dance with Decompositions 3(3,3.1) and the beginning of
this proof.
Mention, that the algebra

,,
R
jkl
algN N N over the
real field with three generators
j
N, k
N and l N is
alternative. The product kl NN of two generators is also
the corresponding generator

(,)
1kl
m
N

 with the de-
finite number

=,mmkl and the sign multiplier

(,)
1kl

, where

,0,1kl  . On the other hand,





1212
=
kjjklkkl
NNNNNNNN



. We use decom-
positions (7-10) and take 2=kl due to Formula (11),
where Re stands on the right side of the equality, since


=0
kl
ReN N and


=0
jjkl
ReNNNN


 for each
kl

. Thus the repeated application of this procedure by
=1,2, ,jn leads to Formula (1) of this theorem.
Corolla ry
If the conditions of Theorem 6 are satisfied, then
1)
  



1
1
=2π;exp,;dd=;,,;.
nnnn
nunNu
R
ftFa pua ptppFF a put
 

 

Proof. Each algebra


,,
R
jkl
algN N N is alternative.
Therefore, in accordance with § 6 and Formulas
1(8,8.1) and 2(1-4) for each non-commutative integral
given by the left algorithm we get
2)
    


1exp, ;exp,;d
Nb
j
j
NNjj
Nb
j
Nfuaptuap pN 


 


    
    
21
=0
exp, ;exp,;d
=exp,;exp ,;d
rNb
j
jj llNNj
Nb
j
l
b
NNj
b
NNNfuaptuapp
fuaptuap p








 






 




for each =1, ,jn, since the real field is the center of
the Cayley-Dickson algebra r
A
, while the functions
sin and cos are analytic with real expansion coeffi-
cients. Thus
3)
    

1 1
00
=2πddexp,;exp,;dd
bb
n
bnNN n
bb
g
tfuaptuapp p 




  




 

hence taking the limit with b tending to the infinity im-
plies, that the non-commutative iterated (multiple) inte-
gral in Formula 6(1) reduces to the principal value of the
usual integral by real variables

1,, n

  and

1,, n
pp 6.1(1).
2.7. Theorem
An original


f
t with

nr
f
RA over the Cayley-
Dickson algebra r
A
with 1rN

 is completely de-
fined by its image

;
n
Nu
Fp

up to values at points
of discontinuity, where the function

,;upt

is given
by 1(8,8.1) or 2(1,2,2.1).
Proof. Due to Corollary 6.1 the value

f
t at each
point t of continuity of

f
t has the expression
throughout


;
n
Nu
Fp

prescribed by Formula 6.1(1).
Moreover, values of the original at points of discontinuity
do not influence on the image

;
n
Nu
Fp

, since on
each bounded interval in R by each variable
j
t a
number of points of discontinuity is finite and by our
supposition above the original function

f
t is n

—

S. V. LUDKOVSKY 71
almost everywhere on n R continuous.
2.8. Theorem
Suppose that a function

;
n
Nu
Fp

is analytic by the
variable r pA in a domain


11
:=: <<
r
WpAaRepa

, where 2rN ,
1
221
rr
n
 ,

nr
f
RA, either
 
,;= ,upt pt

 or

01 0
,; :=,;uptps Mpt

 (see § § 1 and 2).
Let

;
n
Nu
Fp

be written in the form
 
,0 ,1
;= ;;
nn n
Nu NuNu
FpFpF p

, where

,0 ; n
Nu
Fp

is holomorphic by p in the domain


1<aRep. Let also

,1 ; n
Nu
Fp

be holomorphic by
p
in the domain


1
<Re pa. Moreover, for each
1
>aa and 1 <ba

there exist constants >0
a
C,
>0
b
C and >0
a

and >0
b

such that
1)

 
,0 ;exp n
Nua a
F
pCp  
for each r pA

with


Re pa,
2)

 
,1 ;exp n
Nub b
F
pCp   for each r pA

with


Re pb

, the integral,
3)

1,
1
;d
NN
nnk
Nu
NN
n
F
wp p 

 

 converges abso-
lutely for =0k and =1k and each 11 <<awa

.
Then


;
n
Nu
Fwp

 is the image of the function,
4)
   
11
1
11
=[ 2π]2π;exp,;d
NN
nn
nNu
NN
n
f
tN NFwpuwptp 


 
















1
=( ;,,;.
nn
Nu
FFwput


Proof. For the function

,1 ; n
Nu
Fp

we consider
the substitution of the variable =pg,


1<aReg 
.
Thus the proof reduces to the consideration of

,0 ; n
Nu
Fwp

. An integration by dp in the ite-
rated integral (4) is treated as in § 6. Take marked
values of variables 111 ,, , ,,
j
jn
ppp p

 and
111
,, , ,,
j
jn
ttt t

, where

=;
kk
s
sn

for each
=1, ,kn (see § 6 also). For a given parameter


001
11 1
:=
j
j
jnn j
jj jnnn
NNwps
psN psN
 

 
 



for

,;up

prescribed by Formulas 2(1,2,2.1) or


001
11 1
:=
j
j
jnn j
jj jnnn
NNwps
pN pN
 



 



for

,;upt

given by 1(8,8.1) instead of

and any
non-zero Cayley-Dickson number r
A

 we have
=1
lim j
jj
j

  
 


  .
For any locally z-analytic function

g
z in a do-
main U satisfying conditions of § 5 the homotopy
theo-rem for a non-commutative line integral over r
A
,
2r, is satisfied (see [5,6]). In particular if U con-
tains the straight line
j
wRN and the path

:= j
j
jjj
ttN  , then

d= d
Nj
Njj
g
zz gwzz


 

,
when


ˆ0gz while z tends to the infinity, since
||j  is a finite number (see Lemma 2.23 in [4]). We
apply this to the integrand in Formula (4), since


;
n
Nu
Fwp

 is locally analytic by p in accord-
ance with Theorem 4 and Conditions (1,2) are satisfied.
Then the integral operator

1
2πNj
jNj
N




 on the
j-th step with the help of Theorems 2.22 and 3.16 [4]
gives the inversion formula corresponding to the real
parameter
j
t for


f
t and to the Cayley-Dickson va-
riable 00
j
j
pN pN

which is restricted on the com-
plex plane =
N
j
j
CRRN

(see also Formulas 6(4,11)
above). Therefore, an application of this procedure by
=1,2, ,jn as in § 6 implies Formula (4) of this
theorem. Thus there exist originals 0
f
and 1
f
for
functions


,0 ; n
Nu
Fp

and


,1 ; n
Nu
Fp

with a choice
of Rw

in the common domain


11
<<aRepa

. Then 01 =
f
ff is the original for


;
n
Nu
Fp

due to the distributivity of the multi-
plication in the Cayley-Dickson algebra r
A
leading to
the additivity of the considered integral operator in
Formula (4) .
Corolla ry
Let the conditions of Theorem 8 be satisfied, the n
1)
 




1
1
=2π;exp,;dd=;,,;.
nnnn
nNun Nu
R
ftFw puw ptppFFw put
 

 

Proof. In accordance with § § 6 and 6.1 each non-
commutative integral given by the left algorithm reduces
to the principal value of the usual integral by the cor-
responding real variable:
2)




 

1 1
2π;exp,;d=2π;exp,;d
Njn n
j
Nuj jNuj
Nj
NFwp uwptpNFwp uwptp 
 
 
 




S. V. LUDKOVSKY 72
for each =1, ,jn. Thus Formula 8(4) with the non-
commutative iterated (multiple) integral reduces to For-
mula 8.1(1) with the principal value of the usual integral
by real variables

1,, n
pp.
2.9. Note
In Theorem 8 Conditions (1,2) can be replaced on
1)

()
ˆ=0,
sup
limnpC
Rn Fp
 
where

 

()1 1
:=:=, <<
Rn r
CzAzRnaReza

 is a
sequence of intersections of spheres with a domain W,
where
 
<1Rn Rn for each n,


=
limnRn


.
Indeed, this condition leads to the accomplishment of the
r
A
analog of the Jordan Lemma for each 2r (see
also Lemma 2.23 and Remark 2.24 [4]).
Subsequent properties of quaternion, octonion and
general r
A
multiparameter non-commutative analogs of
the Laplace transform are considered below. We denote
by:
2)
 

11
=: <<
fr
WpAafRepaf

 a domain
of

;
n
Nu
Fp

by the p variable, where


11
=aaf
and


11
=aaf
 are as in § 1. For an original
3)




1, ,1
U
ft t 
 we put
 


1
=: <,
fr
WpAafRep
that is 1=a

. Cases may be, when either the left hy-
perplane


1
=Repa or the right hyperplane


1
=Re pa

is (or both are) included in
f
W. It may
also happen that a domain reduces to the hyperplane




11
=: ==
f
WpRepaa
.
2.10. Proposition
If images


;
n
Nu
Fp

and

;
n
Nu
Gp

of functions
originals


f
t and


g
t exist in domains
f
W and
g
W with values in r
A
, where the function


,;upt

is given by 1(8,8.1) or 2(1,2,2.1), then for each
,r
A



in the case 2=
A
H; as well as f and
g
with values in R and each ,r
A

  or f and
g
with values in r
A
and each ,R

  in the case of
r
A
with 3r; the function




;;
NuN u
Fp Gp



 is the image of the func-
tion




f
tgt   in a domain
f
g
WW.
Proof. Since the transforms

;
n
Nu
Fp

and


;
n
Nu
Gp

exist, then the integral
 












exp,;d=exp,;dexp, ;d
nnn
RRR
f
tgtu pttftu pttgtu ptt   

converges in the domain
 

  



1111
=: max,<<min,
fg r
WWpAafag Repafag

 .
We have n tR, 1 221
rr
n
 , while R is the
center of the Cayley-Dickson algebra r
A
. The qua-
ternion skew field
H
is associative. Thus, under the
imposed conditions the constants ,

 can be carried
out outside integrals.
2.11. Theorem
Let =>0const

, let also

;
n
Fp

be an image of
an original function


f
t with either

=,upt


or
u given by Formulas 2(1,2) over the Cayley-Dickson
algebra r
A
with 2<r, 1 221
rr
n
 . Then an
image

;
nn
Fp

 of the function

f
t

exists.
Proof. Since



==
j
jj jjjjjj
psp sps






 for each
=1,,jn, where =
j
j
s
s

,

=;
jj
s
snt,


=;
jj
s
sn

, =
j
j
t

 for each =1,,jn. Then
changing of these variables implies:






,; (,/ ;)
d= d
=;
uptupn
nn
RR
nn
ftetf e
Fp






due to the fact that the real filed R is the center Z(Ar) of
the Cayley-Dickson algebra r
A
.
2.12. Theorem
Let


f
t be a function-original on the domain 1, ,1
U
such that


k
f
tt

 also for =1kj and =kj
satisfies Conditions 1(1-4). Suppose that


,;upt

is
given by 2(1,2,2.1) or 1(8,8.1) over the Cayley-Dickson
algebra r
A
with 2<r

, 1 221
rr
n
 . Then
1)











1;
1,11,,1
,; ;=,,;; ;
j
nntjj
jU U
FftttupFfttupt p




 
，
 

 01, ,1
=1
,; ;
jn
ke U
k
k
ppSFfttup






in the r
A
spherical coordinates or

S. V. LUDKOVSKY 73
1.1)











1;
1, ,11,,1
,; ;=,,;; ;
j
nntjj
jU U
FftttupFfttupt p






 

 01, ,1,; ;
n
je U
j
ppSFfttup




 
in the r
A
Cartesian coordinates in a domain






11
=: max,<
rj
WpAafaft Rep , where


1
:=,,,,: =0
jjnj
ttttt , =
ek
k
S

  for each
1k.
Proof. Certainly,
2)
  
11
=
f
tssft t
and
2.1)
 

  



=1 =1
==
nj
j
kkj k
kk
ft tftssstftss  

for each =2,,jn, since 1 =
j
jj
tss

, 112 =tss

,
where

=;
jj
s
snt, =0
nl
s for each 1l. From
Formulas 30(6,7) [4] we have the equality in the r
A
spherical coordinates:
3)
 

 




01,
exp,;= exp,;exp,;
jj je
j
uptspuptpSupt



 ,
since







01 0
exp,;=expexp, ;uptps Mpt

,
 
01001,010
exp=exp
jj
pss pps



 ,










cossin= exp
=exp= expπ2
=cos π2sinπ2=cossin,
jj jjj jjjjjjjj
jjjj jjjjjjj
jjjjjj jjjejjjjj jj
j
psps isps is
pipsippsi
ppspsi pSpspsi
 

 







since
j
s
and k
s
are real independent variables for each kj

, where ,=0
jk

for jk

, while ,=1
jj

,
3.1)





cossin= cossin
ejjj jjjjjjj jjjjj
j
Spsps ipsps i
 

   





=cos π2sin π2
j
jjjjj j
psps i 


 


In the r
A
Cartesian coordinates we take
j
t instead of
j
s
in (3.1). If


z  is a differentiable function by
j
z for
each j, :rr
A
A

, =
j
jjj
zpt

, then
3.2)








=
exp=d expd.
j
jj
zqtq zzp



  









 

 
1
1
=1 =1=
=!
=exp=exp,
knk
n
jj
nk
jjjqe
j
qpzz zzn
qpzpSz



 







 

where either =1q or =1q, since =1
jj
z

 .
That is
3.3)




exp= 0
x
ekkk
j
Si   for each 1jk and
any positive number >0x,
3.4)




exp= expπ2
x
ejjjjjj
j
Sii x   
and






exp= expπ2
xejjjjjj
j
Sii x  

for each non-negative real number 0x, k

and
kR


, where


=
eej
jj
SS

, the zero power 0=
ej
SI
is the unit operator;
3.5)




(,;) 01 0
0,1111,11111
=cos1sincos
ps
upt q
qejjjjj jj
j
See Tipsipsps


 








  

22
11 111111 1
212121 21
=
sin cossin sin
r
kkkkr rrr
kj
ipspsi psps 

 



 




 


S. V. LUDKOVSKY 74
in the r
A
spherical coordinates, where either =1q or
=1q and
3.6)
 
:= π2
x
jj j
Tx   
for any function

j

and any real number
x
R

,
where 1j. Then in accordance with Formula (3.2) we
have:
3.7)


exp,;
qej
Supt  






1
1
=1 =1=,;
!
knk
n
j
nk upt
zqi zn










for

,;upt

given by Formulas 1(8,8.1) in the r
A
Cartesian coordinates, where either =1q or =1q

.
The integration by parts theorem (Theorem 2 in § II.2.6
on p. 228 [18]) states: if <ab and two functions f
and
g
are Riemann integrable on the segment


,ab ,
 
=d
x
a
F
xA ftt and
 
=d
x
a
GxBgt t, where
A
and B are two real constants, then
 
d= d
bb
b
a
aa
F
xgx xFxGxfxGx x

.
Therefore, the integration by parts gives
4)



 
0exp, ;d
j
j
f
ttupt t 
 

 

 


=
=0
0
=exp ,;
exp,;d.
tj
tj
j
j
ft upt
f
tupttt









Using the change of variables ts with the unit
Jacobian




11
,, ,,
nn
tt ss and applying the Fu-
bini’s theorem componentwise to
j
j
f
i we infer:
5)













.0
1, ,112
exp,;d=exp,;d
jj
Usss
n
f
tt upttftt upts 
 
  







 

 

00 1
0
00 00
=1
=exp,;dd
=exp,;d exp,;d
j
jj
sj
j
jjj
ke
k
k
ft tuptst
f
tupttppS ftuptt




 

 




 






 


in the r
A
spherical coordinates, or
5.1)





1, ,1
exp, ;d
j
U
f
ttupt t   

 


  
0
00 00
=exp,;dexp,;d
jjj
je
j
f
tupttppS ftuptt 
 


 


 
 

in the r
A
Cartesian coordinates, since

 




01 0001 0
exp= exp
j
pstpps     for each 1jn. This gives
Formula (1), where
6)
 







1;
1, ,100
,,;;;=exp ,;d
j
nt jjjjj
U
F
ftupt pftuptt  









111
0000
=d ddexp,;
jj
jj n
ttdttft upt

 
 


is the non-commutative transform by

111
=,,,0, ,,
j
j
jn
ttt tt

.
Remark
Shift operators of the form




=exp
x
ddxx   
in real variables are also frequently used in the class of
infinite differentiable functions with converging Taylor
series expansion in the corresponding domain.
It is possible to use also the following convention. One
can put

 1111221
cos=coscoscosr  

, ,


 

11
111 21
sin cos
= sincoscoscos,
kk
kk kr
 
 





where =0
j

for each 1j, 2<21
r
k, so that

11
cos=0
l
j
T   for each >1j and 1l,




11
sincos=0
l
jkk
T    for each >jk and
1l, where


1
=
ll
jjj
TTT

 is the iterated compo-
sition for >1l, lN

. Then

,;upt
l
j
Te

 gives with
such convention the same result as

,;upt
l
ej
Se

, so one
can use the symbolic notation



,; π/2
,; =upti l uptj
l
j
Te e



.
But to avoid misunderstanding we shall use e
j
S and
j
T in the sense of Formulas 12(3.1-3.7).
It is worth to mention that instead of 12(3.7) also the
formulas
1)






11
exp= cossin
nn
pi piM

   with

1/2
22
1
:= :=n pp p 





 and


11
=nn Mpi pi

 for 0

, 0=1e;
2)


11
exp nn j pipi p

 






2
=sincos
sin
j
jj
Mp
iMp







S. V. LUDKOVSKY 75
and

=1
jj jj
pt   can be used.
2.13. Theorem
Let

f
t be a function-original. Suppose that


,;upt

is given by 2(1,2,2.1) or 1(8,8.1) over the
Cayley-Dickson algebra r
A
with 2<r. Then a
(super) derivative of an image is given by the following
formula:
1)










10 11
1
,; ;=,; ;,; ;,; ;
nnn n
eenn
n
F
ftupphFftsuph SF ftsuphSF ftsuph   
in the r
A
spherical coordinates, or
1.1)
 

 



  
10 11
1
,; ;.=,; ;,; ;,; ;
nn nn
eenn
n
F
f tupphFf tsuphSFf ttuphS Ff ttuph   
in the r
A
Cartesian coordinates for each
00
=nnr hhihi A, where 0,,
n
hhR,
1
221
rr
n
 ,
f
pW.
Proof. The inequalities




11
<<afRepa f
 are
equivalent to the inequalities
  

 
11
<<a fttRepaftt
, since

|| exp= 0
lim tbtt
  for each >0b. An image




,; ;
n
Fftup

is a holomorphic function by p for






11
<<afRepa f
 by Theorem 4, also
0d<
ct n
ett


 for each >0c and =0,1,2,n.
Thus it is possible to differentiate under the sign of the
integral:
2)
 



 



{1,1}
exp,;=(exp, ;d)
nUv
RU
v
n
v
f
tuptdtphftupttph 

 


 




=exp,;d.
n
R
f
tuptpht   

Due to Formulas 12(3,3.2) we get:
3)










10 11
1
exp,;=exp,;exp, ;exp, ;
eenn
n
u ptphu ptshSu ptshSu ptsh   
in the r
A
spherical coordinates, or
4)

 







10 11
1
exp,;=exp,;exp, ;exp,;
eenn
n
u ptphu ptshSu ptthSu ptth   
in the r
A
Cartesian coordinates.
Thus from Formulas (2,3) we deduce Formula (1).
2.14. Theorem
If

f
t is a function-original, then
1)
   

,; ;=,; ;,
nn
Fft upFftupp
 
 
for either
i)




010
,; =,;uptps Mpt

 or
ii)


,;= ,upt pt


over r
A
with 2<r


in a domain
f
pW

, where n R  , 1 221
rr
n
 ,
2)

01 111
,= nnn ppspsi psi   with


=;
jj
s
sn

for each j in the first (i) and


,= ,pp

 in the second (ii) case (see also Formulas
1(8), 2(1,2,2.1)).
Proof. For p in the domain

1
>Repa the iden-
tities are satisfied:
3)




(,;)
1,...,1 1
,; ;=d
nupt
Un
F
ftupftet



 









 


,;,
1, ,1
1,...,1
=d=,;;,,
up pnU
UfteFft upp





 

due to Formulas 1(7,8) and 2(1,2,2.1,4), since

01001001
;=; ;ps ntps nps n



  and

=
j
jj jjjjj
pt pp

  and
 


;= ;;
jjjjjj jj
psntps npsn

  for each
=1, ,21
r
j, where =t



. Symmetrically we get
(2) for v U instead of 1, ,1 U. Naturally, that the mul-
tiparameter non-commutative Laplace integral for an
original
f
can be considered as the sum of 2n in-
tegrals by the sub-domains v U:
4)






exp, ;d
n
R
f
tuptt  






{1,1}
=exp,;d.
nUv
R
n
v
f
tupttt 




The summation by all possible

1,1 n v gives
Formula (1).

S. V. LUDKOVSKY 76
2.15. Note
In view of the definition of the non-commutative trans-
form n
F
and

,;upt

and Theorem 14 the term
11 2121 rr
ii


 has the natural interpretation as the
initial phase of a retardation.
2.16. Theorem
If

f
t is a function-original with values in r
A
for
2<r

, 1 221
rr
n


, bR, then
1)






1.,; ;=,;; n
bt t
nn
F eftupFftupb

  
for each


11
>>abRepab


, where u is given by
1(8,8.1) or 2(1,2).
Proof. In accordance with Expressions 1(8,8.1) and
2(1,2,2.1) one has



 
1
,;=,; .
n
uptbttup bt

  If


11
>>abRepab


, then the integral
2)

 





11
... ...
,; ;=exp,;d
nn
bt tbt t
nUvUv
F
efttup fteuptt 


 

 

=exp ,;d=,;;
nUv
Uvftupbtt Ffttupb



 

converges. Applying Decomposition 14(4) we deduce
Formula (1).
2.17. Theorem
Let a function

f
t be a real valued original,
 

;= ;;;
n
FpFftup

, where the function

,;upt

is given by 1(8,8.1) or 2(1,2,2.1). Let also

;Gp

and

qp be locally analytic functions such
that
1)


 
 
,;;;=;exp,;
n
FgtupGpuqp




for

=,upt

 or
 
01 0
=,;
n
uptt Mpt

 , then
2)


,d;;;
nn
R
Fgtf up




  
=; ;Gp Fqp

for each
g
pW and

f
qp W, where 2<r

,
1
221
rr
n
 .
Proof. If
g
pW and

f
qp W, then in view of
the Fubini’s theorem and the theorem conditions a change
of an integration order gives the equalities:





 



 
 

 

  
,dexp,;d
=,exp,;dd
=;exp,;d
=; exp(,;d
=; ;,
nn
RR
nn
RR
n
R
n
R
g
tfupt t
gtupttf
Gpuqp f
Gp fuqp
Gp Fqp
 






 









since ,n tR   and the center of the algebra r
A
is R.
2.18. Theorem
If a function


1, ,1
U
ft 
 is original together with its
derivative




1
1,. ,1
nUn
f
ttss 


 or




1
1,. ,1
nUn
f
tttt 


, where

;
n
u
Fp

is an image
function of


1, ,1
U
ft 
 over the Cayley-Dickson algebra
r
A
with 2rN

, 1 221
rr
n
 , for


010
=,;upsMpt


 given by 2(1,2,2.1), then
1)
 
1
012
12 =0
;1
lim
nm
n
eeneu
n
pm
ppSpS pSFp 









 

10,0;
()
01,
11 2
12
1<<; 1<<; ,
11
;=10,
nu
nm l
jjejeje u
jjnmj
nm
jjnllnlj
nm m
ppSpSpSFp fe
















or
1.1)
 
1
01 020
12 =0
;1
lim
nm
n
ee neu
n
pm
ppS ppSppSFp 



 
 
 






 

000
12
12
1<<; 1<<; ,
11
10,0;
;
=1 0
l
nm
jej ejeu
jjnmj
nm
jjnllnlj
nm m
nu
ppSppSppSF p
fe









 
 
 
 




for

,;upt

given by 1(8,8.1), where
 
;0
1, ,1
0=
limtU t
f
ft

,
p tends to the infinity inside the angle


<π2Arg p


for some 0< <π2

, 121
r
j

,
 
=0,
=n
l
j
j
jjl
ppi

,




1
=,,
m
lll. If the restriction

S. V. LUDKOVSKY 77

 

1
1
1
1
;0,,0; ,,
1, 1
=0, ,=0;=, ,=lim m
m
m
mtUtttkjj
jjk
tttkjj
jjk
f
tft

 

 ，
exists for all 1 1<<
m
jjn, then
2)
 
1
012
12
0=0
;1
lim
nm
n
eeneu
n
pm
ppSpSpSFp 








 


1
1
()
01,
11 2
12
1<<; 1<<; ,
11
10,0,
=0, ,=0;=, ,
=01 <...<
1
;
=1 m
m
nm l
jjejeje u
jjnmj
nm
jjnllnlj
nm m
nmu
tttkjj
jjk
mjjn
m
ppSpSpSFp
ft e









 















in the r
A
spherical coordinates or
2.1)
 
1
01 020
12
0=0
;1
lim
nm
n
ee neu
n
pm
ppS ppSppSFp 



 
 
 





 


1
1
1
()
000
12
12
1<<; 1<<; ,
1
10,0,
=0, ,=0;=,,
=01 <...<
1
;
=1
m
m
m
nml
jej ejeu
jjnmj
nm
jjnllnlj
nm
nmu
tttkjj
jjk
mjjn
m
ppSppSppSF p
ft e









 

 
 

 






in the r
A
Cartesian coordinates, where 0p inside
the same angle.
Proof. In accordance with Theorem 12 the equality
follows:
3)




 

 01,
1, ,11, ,1
,; ;=,,;,;
n n
jUjjeU
j
FftstupppSFfttuptp




 









1;
1,,1 ,,;;;
j
nt jj
U
Fftuptp



for



01 0
=,;= ,;uupt psMpt

 in the r
A
spherical coordinates, or
3.1)




 

 0
1, ,11, ,1
,; ;=,,;, ;
n n
jUje U
j
Fftttup ppSFfttuptp
 

 



 

1;
1,,1 ,,;;;
j
nt jj
U
Fft uptp



in the r
A
Cartesian coordinates, since
3.2)
   
1
=
j
jj
f
tssft tft t


for each 2j,



11
=
f
tssft t,
where 011 2121 =rr r pp pipiA

 ,
021
,,r ppR
,

021
,,
r
ii

 are the generators of the
Cayley-Dickson algebra Ar, =0
nl
s for each 1l, the
zero power 0=
ej
SI is the unit operator. For short we
write
f
instead of 1, ,1
U
f

. Thus the limit exists:
4)
 

1; ,,;;;=
j
nt jj
Fftuptp 


 

111
0000
0
ddd d
lim
exp,;.
j
jn
tj
tttt
ft upt 




 

Mention, that



2
1=0,,0,,,:=0
j
jnj
tttt



 
for every 1jn

, since 1 =
kkk
tss

 for each
1kn

. We apply these Formulas (3,4) by induction
=1,,jn, 1 221
rr
n


, to

1
nn
f
ts s,
 ,


1nj
j
n
f
tss


,,

n
f
ts
instead of


j
f
ts

.
From Note 8 [4] it follows, that in the r
A
spherical
coordinates





 11,,1
,| |<π/2
,; ;
lim
=0,
nn nU
pArgpFftssup 

 

also in the r
A
Cartesian coordinates





 11,,1
,| |<π/2
),;;
lim
=0,
nn nU
pArgpFfttt up 

 

which gives the first statement of this theorem, since






,0,=0,;= 0,0,upu tu
 
and

S. V. LUDKOVSKY 78



0,0,
0(1,1) ;=0u
u
Fpf e



，, while

;
n
u
Fp

is de-
fined for each

>0Rep.
If the limit


j
f
t exists, where


1
:=,,,,: =
jjnj
ttttt

 , then
5)
  






1;
111
000 0
ddddexp ,;=:,,;;;.
lim
j
nt jj
jjn
tj
ttttftuptFftuptp

 

 


Certainly,


2
1
11
= ,,:=,,=
j
nj
ttttt








  for each 1jn

. Therefore, the limit exists:









 


 
1
1
1
1010
0,||< π/21,,1
(0,0; )
1=0,,=0;= ,,
1,...,1 =01< <
1
012
12
0,| |<π2=0
exp,;
lim
=d=1
=;1
lim
km
m
m
nn
U
pArgp
nm
nu
ntttkjj
Ujj
mjjn
nm
n
eeneu
n
pArgp m
ft sspsMpt
ftss etft
ppSpS pSFp







 



 
 














 

11
()(0,0, )
01,
11 2
12
1<<; 1<<; ,
;10 ,
nm m
n
nm lu
jjejeje u
jjnmj
nm
jjnllnlj
ppSpSpSFp fe

















from which the second statement of this theorem follows
in the r
A
spherical coordinates and analogously in the
r
A
Cartesian coordinates using Formula (3.1).
2.19. Definitions
Let
X
and Y be two R linear normed spaces which
are also left and right r
A
modules, where 1r

. Let
Y be complete relative to its norm. We put
:=
kRR
X
XX
 is the k times ordered tensor
product over R of
X
. By

,,
k
qk
LXY
 we denote a
family of all continuous k times R poly-linear and Ar
additive operators from k
X
 into Y. Then

,,
k
qk
LXY
 is also a normed R linear and left and
right r
A
module complete relative to its norm. In
particular,

,1 , q
LXY is denoted also by


,
q
LXY.
We present
X
as the direct sum
00 2121
=rr
X
XiX i

, where 021 ,, r XX

 are pair-
wise isomorphic real normed spaces. If

,
q
A
LXY and



=
A
xbAxb or



=
A
bxb Ax
for each 0
x
X and r bA, then an operator
A
we
call right or left r
A
-linear respectively. An R linear
space of left (or right) k times r
A
poly-linear ope-
rators is denoted by

,,
k
lk
LXY
 (or


,,
k
rk
LXY

respectively).
We consider a space of test function


:= ,
n
DDRY
consisting of all infinite differentiable functions
:n
f
RY on n R with compact supports. A sequence
of functions n
f
D tends to zero, if all n
f
are zero
outside some compact subset
K
in the Euclidean space
n
R, while on it for each =0,1,2,k the sequence

()
:
k
n
f
nN converges to zero uniformly. Here as
usually

()k
f
t denotes the k-th derivative of f, which
is a k times R poly-linear symmetric operator from

k
n
R to Y, that is



 

() ()
1(1)()
.,, =.,,
kk
kk
f
thhfthhY



for each 1,, n
k
hhR and every transposition




:1, ,1, ,kk   ,

is an element of the sym-
metric group k S, n tR. For convenience one puts
(0) =
f
f. In particular,




()
11
.,, =
kk
j
jjj
kk
f
teeft tt
for all 1 1,,
k
jjn

, where


= 0,,0,1,0,,0n
j
eR with 1 on the j-th place.
Such convergence in D defines closed subsets in this
space D, their complements by the definition are open,
that gives the topology on D. The space D is R
linear and right and left r
A
module.
By a generalized function of class

:= ,
n
DDRY



is called a continuous R-linear r
A
-additive function
:r
g
DA. The set of all such functionals is denoted by
D

. That is,
g
is continuous, if for each sequence
n
f
D

, converging to zero, a sequence of numbers




=: ,
nnr
g
fgfA

converges to zero for n tending
to the infinity.
A generalized function
g
is zero on an open subset
V in n R, if


,=0gf for each
f
D equal to zero
outside V. By a support of a generalized function
g
is
called the family, denoted by

s
upp g, of all points
n
tR such that in each neighborhood of each point


tsupp g the functional
g
is different from zero.
The addition of generalized functions ,
g
h is given by
the formula:
1)






,:=, ,
g
hfgf hf.
The multiplication '
g
D

on an infinite differen-
tiable function w is given by the equality:
2)




,=,
g
wfgwf either for :nr wR A and
each test function
f
D

with a real image


n
f
RR, where R is embedded into Y; or

S. V. LUDKOVSKY 79
:n wR R and :n
f
RY.
A generalized function
g

prescribed by the equa-
tion:
3)




,:= ,
g
fgf

 is called a derivative
g

of a
generalized function
g
, where



,,
nn
q
f
DRLR Y
,


,,
nn
q
gDRLRY



.
Another space

:= ,
n
BBRY of test functions con-
sists of all infinite differentiable functions :n
f
RY
such that the limit

()
|| =0
lim mj
ttf t  exists for each
=0,1,2,m, =0,1,2,j. A sequence n
f
B

is
called converging to zero, if the sequence

()
mj
n
tf t
converges to zero uniformly on

\,0,
nn
RBR R for
each ,=0,1, 2,mj  and each 0< <R , where
  
,,:=: ,BZzRy ZyzR   denotes a ball with
center at z of radius R in a metric space
Z
with a
metric

. The family of all R-linear and r
A
-additive
functionals on B is denoted by B.
In particular we can take =r
X
A

, =r YA

with
1,
Z

 . Analogously spaces

,DUY ,

,DUY  

,

,BUY and

,BUY 



 are defined for
domains U in n R, for example, =v UU (see also §
1).
A generalized function '
f
B

we call a generalized
original, if there exist real numbers 11 <aa
 such that
for each 111 1
<,,,, <
nn
awwwwa

 the generalized
function
4)






exp ,
vU
v
ft qt

 is in

,
v
BUY 

for all


1
=,,
n
vv v,


1,1
j
v for every =1, ,jn for
each n
tR with 0
jj
tv  for each =1,,jn, where


11
1
=,,
vvnvn
n
qvwvw. By an image of such original we
call a function.
5)
  

,; ;:=,exp,;
n
Ffupf upt 

 of the vari-
able r pA

with the parameter r
A

, defined in the
domain



11
=: <<
fr
WpAaRepa

 by the fol-
lowing rule. For a given
f
pW

choose


111 1
<,,<<,,<
nn
awwRepww a


, then
6)




,exp, ;:=fupt 




 


exp,, exp,;,
vvU
vv
fqt uptqt







since
 



exp,;,,
vv
uptqt BUY 



,
where in each term


 

 
exp,,exp,;,
vvU
v
fqt uptqt 




the generalized function belongs to

,
v
BU Y



by Con-
dition (4), while the sum in (6) is by all admissible
vectors

1,1 n v .
2.20. Note and Examples
Evidently the transform

,; ;
n
Ffup

does not depend
on a choice of


11
,,,,
nn
ww ww

, since
  




 


exp(,, exp,;,
=exp, ,,exp,;, ,
vvU
v
vv vvU
v
fqt uptqt
fqtbt uptqtbt










for each n bR such that

11
<<< <
jjj j
awbRepw ba


for each =1,,jn, because


exp ,
v
bt R. At the
same time the real field R is the center of the Cayley-
Dickson algebra r
A
, where 2rN.
Let

be the Dirac delta function, defined by the
equation


DF







,:=0tt   
 for each B

. Then
1)


 


 


() ()
{1,1}
,; ;=[exp,,exp,;,)
nj j
nvvU
vv
Ftuptqt uptqt 
 



 

=
=1 exp,;
jj
tt
upt

  

,
since it is possible to take 11
<<0<<aa


 and
=0
k
w for each

1,1,2, 2,,,knn , where
n
R   is the parameter, || 11
11
:=
j
j
jj
ttt

 . In parti-
cular, for =0j we have
2)
   
,; ;=exp,;
n
Ftup up





 .
In the general case:
3)


 

 
11
12
|| 111
10120
1
1
2
01
,; ;=exp,0;
kj
j
kj n
j
jjk
nj n
neene
n
FtssupppSpSp
jS
kMp
 



 
 
 

in the r
A
spherical coordinates, or

S. V. LUDKOVSKY 80
3.1)







 
12
|| 1
101020
12
,; ;=exp,0;
j jj n
j
j
nj n
neene
n
Ftt tupppSppSppSup
 

in the r
A
Cartesian coordinates, where
1= n
jjj, 11 ,,,
n
kj j are nonnegative integers,
1
221
rr
n
 ,

:= !!!
llmlm
m
 
 
 denotes the bi-
nomial coefficient, 0! =1 , 1! = 1 , 2!=2;
!=1 2ll  for each 3l,

=;
jj
s
snt.
The transform


n
F
f of any generalized function
f
is the holomorphic function by
f
pW and by
r
A

, since the right side of Equation 19(5) is holo-
morphic by p in
f
W and by

in view of Theorem
4. Equation 19(5) implies, that Theorems 11-13 are
accomplished also for generalized functions.
For 11
=aa

the region of convergence reduces to the
vertical hyperplane in r
A
over R. For 11 <aa
 there
is no any common domain of convergence and


f
t
can not be transformed.
2.21. Theorem
If


f
t is an original function on n R,


;
n
Fp

is
its image,

|| 1
1
j
j
jn
n
f
ts s

 or

|| 1
1
j
j
jn
n
f
tt t
is an original, 1 =n jj j

, 1 0,,
n
jjZ,
1
221
rr
n


; then
1)

   


12
1
|| 111
1012
12
01
11
,; ;=,; ;
j kj n
j jjk
nj n
n
neene
n
kj
j
FftssupppSpS pSFftup
k








for

010
,; :=,;uptpsMpt

 given by 2(1,2, 2.1), or
1.1)

 






12
|| 1
101020
12
,; ;=,; ;
j jj n
j
j
nj n
n
neene
n
Fftttup ppSppSppSFftup

 
for

,;upt

given by 1(8,8.1) over the Cayley- Dick-
son algebra r
A
with 2<r. Domains, where For-
mulas (1,1.1) are true may be different from a domain of
the multiparameter noncommutative transform for
f
,
but they are satisfied in the domain


11
<<aRepa

,
where
 
 
|| 1
111 1
=min,: ,0
m m
mn
nll
aafaft mjmjl 


 ;
 
 
|| 1
111 1
=max,:,0
m m
mn
nll
aafaftmkmjl   ,
if 11
<aa
, where =
j
j
s

or =
j
j
t

for each j cor-
respondingly.
Proof. To each domain v U the domain v U

sym-
metrically corresponds. The number of different vectors

1,1 n v is even 2n. Therefore, for


00
=,;upt Mpt

 due to Theorem 12 the equality
2)




(,;) (,;)
d= d
upt upt
nn
jj
RR
f
tse sftset
 
 







(,;) (,;)
11
=dd d
ju ptju pt
nn
j
j
RR
tftet ftess










is satisfied in the r
A
spherical coordinates, since the
absolute value of the Jacobian

,
j
j
tts is unit.
Since for

11
<<aRepa
 the first additive is zero,
while the second integral converts with the help of
Formulas 12(2,2.1), Formula (1) follows for =1k:
3)




,; ;=
nj
Fftsup  






01, ,; ;,;;
nn
jje
j
pFftup pSFftup
 

To accomplish the derivation we use Theorem 14 so
that







  





011
0
011
0
,;
(,;) 1
0
,; ;,; ;
lim
=,;;,;;
lim
=d,
lim
nn
j
nn jj
uptp pipi
jj
upt
n
R
Fftup Ffteup
FftupFftupppipi
ft eet



























S. V. LUDKOVSKY 81
where

= 0,,0,1,0,,0n
j
eR with 1 on the j-th
place. If the original

|| 1
1
j
j
jn
n
f
ts s exists, then

|| 1
1
m m
mn
n
f
ts s is continuous for 01mj
with 0ll mj for each =1, ,ln, where 0:=
f
f.
The interchanging of 0
lim
  and n
R
 may change a
domain of convergence, but in the indicated in the
theorem domain

11
<<aRepa
, when it is non void,
Formula (3) is valid. Applying Formula (3) in the r
A
spherical coordinates by induction to


|| 1
1:,0
m m
mn
nll
f
tss mjmjl
with the corresponding order subordinated to

|| 1
1
j
j
jn
n
f
ts s, or in the r
A
Cartesian coor-
dinates using Formula 12(1.1) for the partial derivatives


|| 1
1):,0
m
m
mn
nll
f
tss mjmjl
with the corresponding order subordinated to

|| 1
1
/
j
j
jn
n
f
tt t we deduce Expressions (1) and (1.1)
with the help of Statement 6 from § XVII.2.3 [19]
about the differentiation of an improper integral by a
parameter and § 2.
2.22. Remarks
For the entire Euclidean space n R Theorem 21 for

j
f
ts gives only one or two additives on the right
side of 21(1) in accordance with 21(3).
Evidently Theorems 4, 11 and Proposition 10 are
accomplished for
 

1
;,, ,; ;
jjk
ktt
Ffup

 also.
Theorem 12 is satisfied for
 
1
;,,
j
jk
kt t
F
 and any


1, ,jjjk, so that

 
=;=
ll
j
ljk
s
skt tt for each 1lk

,
=0
m
p and =0
m

for each




11,,mjjk 
(the same convention is in 13, 14, 17, 21, see also below).
For
 
1
;,,
j
jk
kt t
F
 in Theorem 13 in Formula 13(1) it is
natural to put =0
m
t and =0
m
h for each






11,,mjjk , so that only


1k additives
with 0 h,

1,,
j
jk
hh on the right side generally may
remain. Theorems 14 and 17 and 21 modify for
 
1
;,,
jjk
kt t
F
 putting in 14(1) and 17(1,2) and 21(1)
=0
j
t and =0
j

respectively for each






1, ,jjjk.
To take into account boundary conditions for domains
different from v U, for example, for bounded domains
V in n R we consider a bounded noncommutative mul-
tiparameter transform
1)



  
,; ;=:,; ;
nn
VV
FftupFftup
 
.
For it evidently Theorems 4, 6-8, 11, 13, 14, 16, 17,
Proposition 10 and Corollary 4.1 are satisfied as well
taking specific originals
f
with supports in V.
At first take domains W which are quadrants, that is
canonical closed subsets affine diffeomorphic with
=1
=,
n
n
j
j
j
Qab




, where < jj
ab  ,


,:=:
j
jjj
abx Raxb 
 denotes the segment in
R. This means that there exists a vector n wR and a
linear invertible mapping C on n R so that


=CWw Q. We put

,1
1
:=,,,,: =
j
j
nj j
ttttta ,


,2
1
:=,,,,: =
j
j
nj j
tttttb . Consider


1
=,,n
n
ttt Q.
2.23. Theorem
Let


f
t be a function-original with a support by t
variables in n Q and zero outside n Q such that


j
f
tt

 also satisfies Conditions 1 (1-4). Suppose that


,;upt

is given by 2(1,2,2.1) or 1(8,8.1) over r
A
with 2<r

, 1 221
rr
n


. Then
1)
  




 


,2 ,1
1;,2,21;,1 ,1
,; ;=,; ;,; ;
jj
nntjjntjj
jn nn
QQ Q
FftttupFfttup Ffttup







 

 0
=1
,; ;
jn
ke n
kQ
k
ppSFfttup






in the r
A
spherical coordinates, or
 

 




,2 ,1
1;,2,21;,1 ,1
0
,; ;,; ;,; ;
jj
ntjjntjjn
nnjen
j QQ Q
Ff ttupFf ttupppSFfttup










in the r
A
Cartesian coordinates in a domain r WA;
if =
j
a or =
j
b, then the addendum with ,1
j
t
or ,2
j
t correspondingly is zero.
Proof. Here the domain n Q is bounded and
f
is
almost everywhere continuous and satisfies Conditions
1(1-4), hence








1
exp,;,,
nnr
f
tuptLRA  
for each r pA

, since
 
exp,;upt

 is continuous
and




n
s
upp f tQ.
Analogously to § 12 the integration by parts gives
2)




 

 


=
=
exp,;d=exp, ;exp, ;d,
tb
b b
jj
j j
j
jjj
a a
ta
j j
jj
f
t tupttftuptftupttt  

 


where

1
=,,
n
tt t. Then the Fubini's theorem implies:

S. V. LUDKOVSKY 82
3)








111
111
exp, ;d=exp,;dd
bbbbb
jjnj
j
njjj
Qaaaaa
jjnj
f
tt upttftt upttt 




  






 

 

 

,2,2,1 ,1
, =, =
1
01
=1
=exp,;d][exp,;d
exp,;d
j
jjj jj
nn
tQ tbtQ ta
jj jj
jbb
n
ke
kaa
n
k
f
tu pttf tu ptt
ppSft uptt



 
 
 
 

 





in the r
A
spherical coordinates or
3.1)
 

 
exp,;d
nj
Q
f
ttupt t   

 

 

  
,2,2,1 ,1
, =, =
1
01
=exp,;d exp,;d
exp,;d
j
jjjjj
nn
tQ tbtQ ta
jj jj
bb
n
je
jaa
n
f
tu pttf tu ptt
ppSftupt t




 



 





in the r
A
Cartesian coordinates, where as usually

111
=,,,0, ,,
j
j
jn
ttt tt

, 111 d=dd dd
jjj
tttt t

.
This gives Formulas (1,1.1), where
4)
 

,
1; ,,,
,,;;;
jk
n tjkjkjk
n
Q
Fft tuptp


 

111,,,
111
=exp,;d
bbbb
jjn
j
kjkjk
aaaa
jjn
f
tuptt 

 


is the non-commutative transform by ,
j
k
t,
1
221
rr
n
 , ,
d
j
k
t is the Lebesgue volume element
on 1n
R.
2.24. Theorem
If a function




n
Q
f
tt  is original together with its
derivative




1
nnn
Q
f
ttss 

 or




1
nnn
Q
f
tttt 

, where

;
n
u
Fp

is an image
function of




n
Q
f
tt 
over the Cayley-Dickson
algebra r
A
with 2rN

, 1 221
rr
n
 , for the
function


,;upt

given by 2(1,2,2.1) or 1(8,8.1),
=1
=0,
n
n
j
j
Qb




, >0
j
b for each j, then
1)
 
1
01 2
12 =0
;1
lim
nm
n
eeneu
n
pm
ppSpS pSFp 









 
1
1
()(0,0;)
01,
11 2
12
1<<; 1<<; ,
1
;=1 0
nmm
n
nm lu
jjejeje u
jjnmj
nm
jjnllnlj
ppSpSpSFp fe
















in the r
A
spherical coordinates, or
1.1)
 
1
01 020
12 =0
;1
lim
nm
n
ee neu
n
pm
ppS ppSppSFp 



 
 
 





 
1
1
()(0,0;)
000
12
12
1<<; 1<<; ,
1
;=1 0
lm
n
nm lu
jej ejeu
jjnmj
nm
jjn lnlj
nm
ppSppSppSFpfe









 
 
 





in the r
A
Cartesian coordinates, where
 
, 0
0=
lim n
tQ t
f
ft
 ,
p tends to the infinity inside the angle


<π2Arg p


for some 0< <π2

.
Proof. In accordance with Theorem 23 we have
Equalities 23(1,1.1). Therefore we infer that
2)
 

,
1; ,,,
,,;;;
jk
n tjkjkjk
n
Q
Fft tuptp


  

111
111
011 1
,
=ddddexp,;,
lim bb bb
jj n
jj n
aa aa
tjjn
jjk
ttttftupt



 
 

where ,1 ==0
jj
a

, ,2 =>0
jj
b

, =1,2k. Mention,
that


,
2, 2
1, 1
11, ,
1
=: =,,=
jl
j
l
lljjl
j
tttt 







for every 1jn. Analogously to § 12 we apply
Formula (2) by induction =1, ,jn, 1 221
rr
n
 ,
to






 
1
1,,
,,
nnj
njn
n
f
ts ssfts ss
fts s


 

 


S. V. LUDKOVSKY 83
instead of


j
f
ts s,

=;
jj
s
snt as in § 2, or
applying to the partial derivatives





1
1,, ,,
nnj
njnn
f
tt tftttftt

 
instead of


j
f
tt correspondingly. If >0
j
s for
some 1j, then 1>0s for n Q and


,; =0
lim
l
upt
pe
 
 for such ()l
t, where

1
=,,
n
tt t,



1,, n
lll, 1 =n ll l,

() ()()
1
=,,
ll l
n
tt t, () =
l
j
j
ta for =1
j
l and () =
l
j
j
tb
for =2
j
l, 121
r
j

. Therefore,


 
()
|| ()( ,; )
{1,2}; =1,,
(0,0; )
1
lim
=1 0,
l
llupt
pljn
j
nu
ft e
fe



 




since




,0;= 0,0;up u

, where



()
()
;
=lim l
ln
tQ tt
f
ft
 .
In accordance with Note 8 [4]


 

1
,|()|< π/2
,,;;;=0
lim nn nn
Q
pArgpFftsstuptp 

 
in the r
A
spherical coordinates and


 
 1
,|( )|<π/2
,,;;;=0
lim nn nn
Q
pArgp Ffttt tuptp 

 
in the r
A
Cartesian coordinates, which gives the state-
ment of this theorem.
2.25. Theorem
Suppose that
 
n
Q
ftt  is an original function,

;
n
Fp

is its image,
 
|| 1
1
j
j
jn
nn
Q
f
tttt   is
an original, 1 =n jj j, 1 0,,
n
jjZ,
1
221
rr
n
 , <
kk
ab  for each
=1, ,kn,
 
1
=,,
n
lll,

0,1, 2
k
l, =r WA for
bounded n Q. Let



1
=: <
r
WpAaRep for
=
k
b

for some k and finite k a for each k;




1
=: <
r
WpARepa

 for =
k
a for some k
and finite k b for each k;




11
=: <<
r
WpAaRepa

 when =
k
a

 and
=
l
b

 for some k and l;

() ()()
1
=,,
ll l
n
tt t.
We put () =
l
kk
tt and =0
k
q for =0
k
l, () =
l
kk
ta
for =1
k
l, () =
l
kk
tb for =2
k
l,
 
1
=,,
n
qqq,
1
=n qq q

,
 
 
|| 1
111 1
=max,: ,0
m m
mn
nkk
aafaftttmjmjk

 ,
 
 
|| 1
111 1
=min,:,0
m m
mn
nkk
aafaftttmjmjk

  if 11
<aa
.
If =
k
a and =
k
b for n Q with a given k,
then =0
k
l. If either >
k
a

 or <
k
b for a
marked k, then

0,1, 2
k
l. We also put
 
==
kk k
hhlsignl for each k, where


=1sign x

for <0x,

0=0sign ,


=1sign x for >0x,

=hhl, 1 =n hh h,
 

11
:=,,nn ljl signjlsignj.
Let the vector


l enumerate faces ()
n
l
Q in 1
n
k
Q


for

=1hlk, so that 1()
|()|=
=
nn
kl
hl k
QQ

,
()() =
nn
lm
QQ  for each




lm (see also more
detailed notations in § 28).
Let the shift operator be defined:

 


()1 1
;:= ;π2
mnn
TFpFp imim  ,
also the operator >
k
a


 
1
() 1
;:= ;
m mn
mee
n
SOSF pSSF p

,
where




1
=,, [0,)
nn
n
mmm R, () () =
k
mkm
SS
for each positive number 0<kR, 0=SI is the unit
operator for


=0m (see also Formulas 12(3.1-3.7)).
As usually let


1=1,0, ,0e,  ,

=0, ,0,1
n
e be
the standard orthonormal basis in n R so that


11
=nn mme me.
Theorem. Then
1)
 

 


|| 12
1
112
,,;;;= ,;;
j jjj j
nj n
n n
nn n ee e
n Q Q
FfttttuptpRRRFfttup







1|()|; =;0; 0; =; =0fo=0,foreach=1,,; (){0,1,2}
|( )||()| ||()()
12 1
1
12 ()
1,;;
n
kkkkkkkkkkkk
ljmqhjmqhsign ljqr ljknl
lj mq mm q
nhlj qljlj
nn
nn eee
nQlj
RRRFfttttup

 











S. V. LUDKOVSKY 84
for

,;upt

in the r
A
spherical coordinates or the
r
A
Cartesian coordinates over the Cayley-Dickson algebra
r
A
with 2<r, where
1.1) 01 2
212
:=
eee
RppSpS ,  ,
01 2
12
:=
eeene
nn
RppSpS pS 
in the r
A
spherical coordinates, while
1.2), 02
22
:=
ee
RppS,  , 0
:=
ene
nn
RppS in
the r
A
Cartesian coordinates, i.e.


=
ee
jj
RRp are
operators depending on the parameter p. If () =
l
j
t

for some 1jn

, then the corresponding addendum
on the right of (1) is zero.
Proof. In view of Theorem 23 we get the equality
2)


1
||1 (,;)
11 1
11 1d
mmm m m
mupt
kkk n
nkkkn
Q
f
ttttttet

















|| (,;)||(,;)
1 1
1 1
1 1
=d /dd
bkb
m m
m m
k
kmuptkmupt
n n
nn nn
nnkk
RQRQ a
k
ak
tftttetfttte tt
   
 
 
 
 




is satisfied for 0kk mj for each =1, ,kn with
<mj . On the other hand, for pW additives on the
right of (2) convert with the help of Formula 23(1). Each
term of the form

 
()
|()|
| |()()(,;)
1
1
()
dl
nhln
RQ
q q
ql lupt
n
nn
Ql
t
fttt te














can be further transformed with the help of (2) by the
considered variable k t only in the case =0
k
l.
Applying Formula (2) by induction to partial derivatives
|| 1
1
j
j
jn
n
f
tt, || 12
2
j
jj jn
n
f
tt

, ,
j
j
nn
n
f
t,,n
f
t as in § 21 and using Theorem
14 and Remarks 22 we deduce (1).
2.26. Theorem
Let




1, ,1
U
f
tt 
 be a function-original with values in
Ar with 2<r

, 1 221
rr
n


, u is given by
2(1,2,2.1) or 1(8,8.1),
1)
 
1
00
:=d,then
tt
n
gtfxx


2)



1, ,1,;;
n
U
Ff tup





12 1,,1
=,;;
n
ee eU
n
RRRFgttup



in the domain


1
>max ,0Re pa, where the operators
e
j
R are given by Formulas 25(1,1,1.2).
Proof. In view of Theorem 25 the equation
3)




1, ,112
,; ;=,; ;
nn
Ueee
n
FftupRRRF gtup
 







|( )||()| ()
12
12
1||; 01; =1; =; =1,,; =0,,=0
1
1,;;,
kkkk k
lm
mm nhll
n
eee
n
lm mh hsignl foreachknqq
n
RRR Fgtup


 



is satisfied, since



11, ,1
=
n
nU
g
tt tft    ,
where 1=1, ,=1,=1
nj
jjl for each =1,,jn. Equa-
tion (3) is accomplished in the same domain




1
>max ,0Re pa, since


0=0g and

g
t also ful-
fills conditions of Definition 1, while
  
11
<max,0agafb for each >0b, where
1
aR. On the other hand,

g
t is equal to zero on
1, ,1
U and outside 1, ,1 U in accordance with formula
(1), hence all terms on the right side of Equation (3) with
>0l disappear and

 
1, ,1
suppg tU. Thus we get
Equation (2).
2.27. Theorem
Suppose that

;
k
Fp

is an image


;, ,
1
1, ,1,;;
kt t
k
U
Fft tup


 of an original function

f
t for u given by 2(1,2,2.1) in the half space


1
:=: >
r
WpARepa with 2<r,
1=0p,  , 1=0 j
p; 1=π2

,,1=π2
j

 for
each 2j in the r
A
spherical coordinates or
1=0

,,1=0
j

 for each 2j in the r
A
Car-
tesian coordinates;
1) the integral

0;d
ijk
pi
jj
F
pz z 

 converges,
where 011 =kk r pp pipiA

 , j pR for each
=0, ,21
r
j

, 1 221
rr
k


,




1, ,111
:=,,: 0,, 0
k
kk
UttRtt


. Let also
2) the function


;
k
Fp

be continuous by the va-
riable r pA

on the open domain W, moreover, for
each 1 >wa there exist constants >0
w
C and >0
w

such that
3)




;'exp
k
ww
F
pC p   for each ()
R
n
pS

,




:=:
Rr
SzARezw,
 
0< <1Rn Rn for
each nN

,


=
limnRn
 , where 1
a is fixed,
00
=kk r iiA


 is marked, jR

 for each
=0,,jk. Then
4)

0;d=
ijk
pi
jj
F
pz z 


 


; ,...,
1
1, ,1,; ;,
kt t
k
eUj
j
SFfttup





S. V. LUDKOVSKY
85
where 1=0p,,1=0
j
p for each 2j;
1=π2

,,1=π2
j

 and

=;
jj
s
kt  in the r
A
spherical coordinates, while 1=0

,,1=0
j

 and
=
j
j
t

in the Ar Cartesian coordinates correspondingly
for each 1j.
Proof. Take a path of an integration belonging to the
half space

Re pw for some constant 1 >wa. Then
 

 

1, ,1
011
1, ,1
exp, ;d
expd <
U
k
U
ftupt t
Cpattt
 






converges, where =>0Cconst, 0 pw. For >0
j
t
for each =1, ,jk conditions of Lemma 2.23 [4] (that
is of the noncommutative analog over r
A
of Jordan’s
lemma) are satisfied. If j t, then j s, since all
1,, k
tt are non-negative. Up to a set 1, ,1 U of k

Lebesgue measure zero we can consider that
1>0,, >0 k
tt. If j s, then also 1 s. The
converging integral can be written as the following limit:
5)

0;d
ijk
pi
jj
F
pz z 




0
0<0
=;expd
lim
ijk
pi
jj
F
pz zz





for 1jk, since the integral

;d
Sk
S
F
wzz 






is absolutely converging and the limit

0exp= 1
lim z
  uniformly by z on each com-
pact subset in r
A
, where S is a purely imaginary
marked Cayley-Dickson number with =1S. Therefore,
in the integral
6)

0;d=
ijk
pi
jj
F
pz z 


 

0
1, ,1
exp, ;dd
ij
pi U
jj
f
tupzttz 








the order of the integration can be changed in accordance
with the Fubini’s theorem applied componentwise to an
integrand 00 =nn
g
gi gi with l
g
R for each
=0, ,ln:
7)

0;d
ijk
pi
jj
F
pz z 








0
0
1, ,1
,;
1, ,1
=d exp,;d
=dd.
ij
Upi
jj
iup zt
j
Upi
jj
tft upztz
ftez t














Generally, the condition 11
=0,,=0
j
pp

 and
11
=π2,,= π2
j


 in the r
A
spherical coordi-
nates or 11 =0,,=0
j


 in the r
A
Cartesian coor-
dinates for each 2j is essential for the convergence
of such integral. We certainly have
8)


*
cos d
bi
jj
jj j
pi
jj
iz z  



=
=
=
=
=sin
=cos π2
b
jj
jjjjp
jj
b
j
j
jj jj
p
j
j




 
 






and
9)


*
sin d
bi
jj
j
jjj
pi
jj
iz z  



=
=
=
=
=cos
=sinπ2
b
jj
jjjjp
jj
b
j
j
jj jj
p
j
j




 
 






for each > 0
j

and <<<
jj
pb

 and
=1,,jk. Applying Formulas (3-9) and 2(1,2,2.1) or
1(8,8.1) and 12(3.1-3.7) we deduce that:

  


1
0
1, ,1
;,,
1, ,1
;d
=exp,;d
=,;;,
k
ijk
pi
jj
ej
jU
kt t
eUj
j
Fp zz
Sft uptt
SFftt up


















where


1
=,,
k
tt t, =
j
jk
s
tt for each
1<jk

, =
kk
s
t, =
j
j
s

in the r
A
spherical coor-
dinates or =
j
j
t

in the r
A
Cartesian coordinates.
2.28. Application of the Noncommutative
Multiparameter Transform to Partial
Differential Equations
Consider a partial differential equation of the form:
1)






=
A
ft gt
, where
2)
  

|| 1
1
||
:= ,
j
j
jn
jn
j
Afta tfttt
 



j
at A

 are continuous functions, where 0
Z


,


1
=,,
n
jj j, 1 := n jj j

, 0k jZ,

is a
natural order of a differential operator
A
, 2r

,
1
221
rr
n


. Since


=;=
kkk n
s
snt tt

 for each =1, ,kn, the
operator
A
can be rewritten in
s
coordinates as
2.1)






A
fts
 



|| 1
1
||
:= .
j j
jn
jn
jbtfts ss  

That is, there exists 0
j
b for some j with
=j

and =0
j
b for >j

, while a function



1
1
,| |=
j
jn
j
n
jj btss s 
 is not zero identically on the
corresponding domain V. We consider that
(D1) U is a canonical closed subset in the Euclidean
space n R, that is


=UclIntU, where


I
nt U
denotes the interior of U and

cl U denotes the closure
of U.
Particularly, the entire space n
R may also be taken.

S. V. LUDKOVSKY
86
Under the linear mapping

 
11
,, ,,
nn
tt ss the
domain U transforms onto V.
We consider a manifold W satisfying the following
conditions (i-v).
i). The manifold W is continuous and piecewise C

,
where l
C denotes the family of l times continuously
differentiable functions. This means by the definition that
W as the manifold is of class 0
loc
CC

. That is W is
of class C

on open subsets 0,
j
W in W and

0,
\
j
j
WW
 has a codimension not less than one in
W.
ii). =0
=m
j
j
WW
, where 00,
=k
k
WW
, =
jk
WW


for each kj, =R mdimW, =
Rj
dim Wmj,
1
j
j
WW
 .
iii). Each Wi with =0, ,1jm is an oriented C

-
manifold,
j
W is open in =
m
k
kj
W
. An orientation of
1
j
W is consistent with that of
j
W for each
=0,1, ,2jm. For >0j the set
j
W is allowed to
be void or non-void.
iv). A sequence k W of C

orientable manifolds
embedded into n R, 1

, exists such that k W uni-
formly converges to W on each compact subset in n R
relative to the metric dist .
For two subsets B and E in a metric space X with
a metric

we put
3)

,:=distBE
   

max, ,,,
sup sup
bB eE
distbEdistBe

where




,:= ,
inf eE
distbEb e 
,
  
,:= ,
inf bB
distBebe 
, bB, eE.
Generally, =
R
dim Wmn. Let




1,,
kk
m
ex ex
be a basis in the tangent space k
x
TW at k
x
W con-
sistent with the orientation of k W, kN.
We suppose that the sequence of orientation frames
 

1,...,
kk
kmk
exex of k W at k
x
converges to
 

1,, m
exe x for each 0
x
W

, where
0
=
limkk
x
xW, while
 
1,, m
exex are linearly in-
dependent vectors in n R.
v). Let a sequence of Riemann volume elements k

on k W (see § XIII.2 [19]) induce a limit volume ele-
ment

on W, that is,


=lim k
k
BW BW 

 for each compact cano-
nical closed subset B in n R, consequently,

0
\=0WW  . We shall consider surface integrals of
the second kind, i.e. by the oriented surface W (see (iv)),
where each
j
W, =0,,1jm

 is oriented (see also §
XIII.2.5 [19]).
vi). Let a vector


wIntU exist so that -Uw is
convex in n R and let U be connected. Suppose that
a boundary U of U satisfies Conditions (i-v) and,
vii) let the orientations of k U and k U be con-
sistent for each kN (see Proposition 2 and Definition
3 [19]).
Particularly, the Riemann volume element λk on k U

is consistent with the Lebesgue measure on k U induced
from n
R for each k. This induces the measure

on
U

as in (v).
Also the boundary conditions are imposed:
4)




0
=,
U
f
tft






|| 1
1()
=
q
q
qn
nq
U
f
ts sft


  for 1q  ,
where


1
=,, n
n
s
ssR,
 
1
=,,
n
qq q,
1
=n qq q

, 0k qZ

 for each k, tU

 is
denoted by t

, 0
f
, ()q
f
are given functions. Generally
these conditions may be excessive, so one uses some of
them or their linear combinations (see (5.1) below). Fre-
quently, the boundary conditions
5)




0
=,
U
f
tft






=
ll
l
U
f
tft 



for 11l


 are also used, where

denotes a real
variable along a unit external normal to the boundary
U

at a point 0 tU


. Using partial differentiation in
local coordinates on U

and (5) one can calculate in
principle all other boundary conditions in (4) almost
everywhere on U

.
Suppose that a domain 1 U and its boundary 1 U

satisfy Conditions (D1, i-vii) and 11 =U gg

is an ori-
ginal on n R with its support in 1 U. Then any original g
on n R gives the original 2
2=:
U
g
g  on n R, where
21
=\
n
URU . Therefore, 12
g
g is the original on n R,
when 1
g
and 2
g
are two originals with their supports
contained in 1 U and 2 U correspondingly. Take now
new domain U satisfying Conditions (D1, i-vii) and
(D2-D3):
D2) 1 UU

and 1 UU

 ;
D3) if a straight line

containing a point 1 w (see
(vi)) intersects 1 U

at two points 1 y and 2 y, then
only one point either 1
y or 2 y belongs to U

, where
11
wU

, 1 Uw

and 11 Uw

are convex; if

in-
tersects 1 U

only at one point, then it intersects U

at the same point. That is,
D4) any straight line

through the point 1 w either
does not intersect U

or intersects the boundary U

only at one point.
Take now
g
with

s
upp gU, then


1
1
U
s
upp gU  . Therefore, any problem (1) on 1 U
can be considered as the restriction of the problem (1)
defined on U, satisfying (D1-D4, i-vii). Any solution
f
of (1) on U with the boundary conditions on U

gives the solution as the restriction 1
U
f
on 1 U with
the boundary conditions on U.
Henceforward, we suppose that the domain U satis-
fies Conditions (D1,D4, i-vii, which are rather mild and
natural. In particular, for n Q this means that either
=
k
a

 or =
k
b

 for each k. Another example is:
1
U is a ball in n
R with the center at zero,


11,1
=\
n
UU RU, 1=0w; or

S. V. LUDKOVSKY
87

1
=:
n
n
UU tRt

  with a marked number
0< <12

. But subsets ()l U in U can also be spe-
cified, if the boundary conditions demand it.
The complex field has the natural realization by 22

real matrices so that 0 1 =1 0 i



, 21 0 = 0 1 i



. The qua-
ternion skew field, as it is well-known, can be realized
with the help of 22 complex matrices with the gene-
rators 10 = 01 I


, 01 =1 0 J



, 0 =0
i
Ki




,
0
=0
i
Li





, or equivalently by 44 real matrices.
Considering matrices with entries in the Cayley-Dickson
algebra v
A
one gets the complexified or quaternionified
Cayley-Dickson algebras


vC
A
or

v
H
A with elements
=zaIbi or = zaIbJcKeL , where
,,,v abce A, such that each v aA commutes with the
generators i,
I
,
J
, K and L. When =2r,
f
and
g
have values in 2=
A
H and 24n and
coefficients of differential operators belong to 2
A
, then
the multiparameter noncommutative transform operates
with the associative case so that


=
nn
F
afaFf
for each aH

. The left linearity property




=
nn
F
afaFf for any ,,
J
KL
aH is also accom-
plished for either operators with coefficients in R or
=
i
RCIR iR

or ,, =
JKL
H
IR JR KRLR and
f
with values in v
A
with 121
v
n ; or vice versa
f
with values in i C or ,,
J
KL
H and coefficients a
in v
A
but with 14n

. Thus all such variants of ope-
rator coefficients
j
a and values of functions
f
can be
treated by the noncommutative transform. Henceforward,
we suppose that these variants take place.
We suppose that


g
t is an original function, that is
satisfying Conditions 1(1-4). Consider at first the case of
constant coefficients
j
a on a quadrant domain n Q. Let
n
Q be oriented so that =
k
a and =
k
b

 for
each kn


; either =
k
a or =
k
b for each
>kn


, where 0n


 is a marked integer number.
If conditions of Theorem 25 are satisfied, then
6)



 



12
12
||
,; ;=,; ;
j jj n
n n
je een
nQ
j
FAftupaRpR pR pFfttup







 

 


1|()|; =; 0; 0; =(); =0=0,=1,...,; (){0,1,2}
|( )|12 |()| ||()()1
1
12 ()
1,; ;
=
kkkkkkkkk kkkn
ljmqhjmqhsignljqforl jforeachknl
m mm n
lj q
q
nhljqljljn
ee enn
nQlj
n
RpR pRpFftttt up
Fgt


 





 




 





,; ;
n
Qtup

for

,;upt

in the r
A
spherical or r
A
Cartesian
coordinates, where the operators

j
e
Rp are given by
Formulas 25(1.1) or 25(1.2). Here

l enumerates faces
()
n
l
Q in 1
n
k
Q  for

=1hlk, so that
1()
|()|=
=
nn
kl
hl k
QQ

, ()() =
nn
lm
QQ  for each
 
lm in accordance with § 25 and the notation of
this section.
Therefore, Equation (6) shows that the boundary con-
ditions are necessary:


|| ()1
1
()
q
q
ql n
nn
Ql
fttt   for j

,

1lj ,
0
j
a, =0
k
q for =0
kk
lj , =
kkkk
mqh j,

=
kkk
hsignlj , =1, ,kn, ()
()
ln
l
tQ. But
=1
n
R
dim Qn for n Q, consequently,


|| ()1
1
()
q q
ql n
nn
Ql
fttt   can be calculated if know


|| ()1
(1)( )
()
lm
mn
Ql
ft tt
 



 for =q  , where


1
=,,
m

 ,


=mhl, a number

k  corres-
ponds to ()>0
k
l

, since =0
k
q for =0
k
l and
>0
k
q only for >0
kk
lj and > kn  . That is,
(1)( )
,,m tt

 are coordinates in n R along unit vectors
orthogonal to ()
n
l
Q.
Take a sequence k U of sub-domains
1kk
UUU

 for each kN so that each
()
,
=1
=mk
kn
kl
l
UQ
 is the finite union of quadrants ,
n
kl
Q,


mkN

. We choose them so that each two different
quadrants may intersect only by their borders, each k U
satisfies the same conditions as U and
7)


,=0
lim k
kdistU U  .
Therefore, Equation (6) can be written for more gen-
eral domain U also.
For U instead of n Q we get a face ()l U instead
of ()
n
l
Q and local coordinates (1)() ,,m


  orthogo-
nal to ()l U

instead of (1)( ) ,,m tt

 (see Conditions
(i-iii) above).
Thus the sufficient boundary conditions are:

S. V. LUDKOVSKY
88
5.1)
 


|| ()()
1
(1)(),( )
()
(|=
lj lj
m
mU lj
lj
f
tt
 





for =q  , where

=mhlj, j

,

1lj ,
0
j
a, =0
k
q for =0
kk
lj , =
kkkk
mqh j ,

=
kkk
hsign lj, 01
kk
qj for >kn  ;

()
,()
l
lt 
 are known functions on ()l U, ()
()
l
l
tU.
In the half-space 0
n
t only
5.2)

=0
ntn
ft t


are necessary for =<q

 and q as above.
Depending on coefficients of the operator
A
and the
domain U some boundary conditions may be dropped,
when the corresponding terms vanish in Formula (6). For
example, if 2
12
=
A
tt, 1,1 =UU, =2n, then
0
U
f 

 is not necessary, only the boundary condition
U
f

is sufficient.
If =n UR, then no any boundary condition appears.
Mention that
5.3)



 
0(,;)
;; ;=upa Ffaup fae


,
which happens in (6), when () =l at and

=hln.
Conditions in (5.1) are given on disjoint for different (i)
submanifolds ()l U

in U

and partial derivatives are
along orthogonal to them coordinates in n R, so they are
correctly posed.
In r
A
spherical coordinates due to Corollary 4.1
Equation (6) with different values of the parameter

gives a system of linear equations relative to unknown
functions




() ,; ;
n
m
SF ftup

, from which




,; ;
n
Fftup

can be expressed through a family
 
 

| ()|| |()()1
()()1
()
,;;; ,;;:
q q
nnhlqll n
n
mm nn
Ql
SFgtupSFftttt upmZ  


 


 


 

and polynomials of p,