Heisenberg group and Lewy operator.

1. Introduction

The existence theorems, elliptic el·lip·tic   or el·lip·ti·cal
1. Of, relating to, or having the shape of an ellipse.

2. Containing or characterized by ellipsis.

3.
a.
boundary problems and boundary value problems in the theory of differential operators (with constant or variable coefficients) on [R.sup.n], have a wide applications in the whole sciences. The results for differential operators with constant coefficients In mathematics, constant coefficients is a term applied to differential operators, and also some difference operators, to signify that they contain no functions of the independent variables, other than constant functions.  were first obtained by, Malgrange [11], Treves [13], Atiyah [1], Hormander [8] and many others. Elliptic operators and boundary value problems were studied by [2, 10, 12], hypoelliptic and hyperbolic hy·per·bol·ic   also hy·per·bol·i·cal
1. Of, relating to, or employing hyperbole.

2. Mathematics
a. Of, relating to, or having the form of a hyperbola.

b.
equations with constant coefficients were developed by Hormander [8].

Unfortunately, L. Hormander in 1960 [8, P. 156] by his necessary condition had discovered that the situation is completely different when the coefficients are variables P(x,D)u = f on [R.sup.n], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] has a locally solution. By applying this condition on the operator

P = i[[partial derivative].sub.t] + [[partial derivative].sub.y] - 2iy[[partial derivative].sub.z] - 2x[[partial derivative].sub.z] (1.1)

the hypotheses of this condition are not fulfilled for every open set [OMEGA] of [R.sup.4]. So the operatorP is not locally solvable.

In dealing with the non existence of solutions of partial differential operators it was customary during the last fifty years and it still is to day in larger applications, to appeal to the necessary Hormander condition which guarantees the non existence of solutions on [R.sup.4]. The goal of this paper is to construct a solvable algebra of partial differential equations on [R.sup.2n+1] for which the equation P is among of its elements. Thanks to the magic (and obvious) relation between the 2n + 1-dimensional Heisenberg group In mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the form

and its vector group A Vector group is the International Electrotechnical Commission (IEC) method of categorizing the primary and secondary winding configurations of three-phase transformers. Within a polyphase system power transformer it indicates the windings configurations and the difference in  [R.sup.2n+1]. Then one has to do so that the major business of this algebra is to solve the phenomena of the equation (1.1).

2. A Solvable Algebra

A linear partial differential operator differential operator

In mathematics, any combination of derivatives applied to a function. It takes the form of a polynomial of derivatives, such as D2xxD2xyD
in 2n +1 independent variables z,[x.sub.1],[x.sub.2],...,[x.sub.n],[y.sub.1], [y.sub.2],...,[y.sub.n], with constant coefficients defined on [R.sup.2n+1] is polynomial polynomial, mathematical expression which is a finite sum, each term being a constant times a product of one or more variables raised to powers. With only one variable the general form of a polynomial is a0xn+a  in partial differentiations and has the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.1)

here [beta] and [gamma] are multi-index, that is, an n-tuple of integers [[beta].sub.j] [greater than or equal to] 0 (reesp. [[gamma].sub.j] [greater than or equal to] 0) and a [member of] N; [absolute value of [alpha] + [beta] + [gamma]] denotes their length [alpha] + [[beta].sub.1] + [[beta].sub.2] + ... + [[beta].sub.n] + [[gamma].sub.1] + [[gamma].sub.2] + ... + [[gamma].sub.n] = m. Also

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The integer m is usually the order of the operator; this assumes that for some multi-index [alpha], [beta], [gamma] with length [absolute value of [alpha] + [beta] + [gamma]] = m the coefficient [C.sub.[alpha],[beta],[gamma]] [member of] C is not identically equal to zero. Denote by [D.sub.c] the algebra of all linear partial differential operator in 2n + 1 independent variables z,[x.sub.1],[x.sub.2],...,[x.sub.n],[y.sub.1],[y.sub.2],...,[y.sub.n], with constant coefficients defined on [R.sup.2n+1].

Definition 2.1. For every f [member of] [C.sup.[infinity]]([R.sup.2n+1]), one can define a function [??](f) [member of] [C.sup.[infinity]]([R.sup.2n+1]), by the following manner

[??](f)(z,y,x) = f(z - 2<x,y>,y - x) (2.2)

we see immediately that the mapping [??]: [C.sup.[infinity]]([R.sup.2n+1]) [right arrow] [C.sup.[infinity]]([R.sup.2n+1]) is topological isomorphism isomorphism (ī'səmôr`fĭzəm), of minerals, similarity of crystal structure between two or more distinct substances. Sodium nitrate and calcium sulfate are isomorphous, as are the sulfates of barium, strontium, and lead.  from [C.sup.[infinity]]([R.sup.2n+1]) onto [C.sup.[infinity]]([R.sup.2n+1]) and [[??].sup.2] = I, where I is the identity operator of [C.sup.[infinity]]([R.sup.2n+1]).

Theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance.  2.2. Let Q([partial derivative]/[[partial derivative].sub.z], [partial derivative]/[partial derivative]y, [partial derivative]/[[partial derivative].sub.x]) be a linear partial differential operator with constant coefficients on [R.sup.2n+1], then there is a differential operator P([partial derivative]/[[partial derivative].sub.z], [partial derivative]/[partial derivative]y, [partial derivative]/[[partial derivative].sub.x]) with variable coefficients, such that

[??]Pf = Q[??]f (2.3)

where X = (z,y,x) and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.4)

Proof. Let f [member of] [C.sup.[infinity]]([R.sup.2n+1]), and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the partial differentiation the variable [x.sub.j], then we have for any 1 [less than or equal to] j [less than or equal to] n

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.5)

where x + [t.sub.j] = ([x.sub.1],[x.sub.2],...,[x.sub.j] + [t.sub.j],...,[x.sub.n]). A similar for the partial differentiations [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](1 [less than or equal to] j [less than or equal to] n) and [partial [derivative].sub.z], we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.6)

for every f [member of] [C.sup.[infinity]]([R.sup.2n+1]). Which is the theorem.

Corollary 2.3. Let D be the algebra of partial differential operators with variable coefficients on [R.sup.2n+1], then the mapping

T: [D.sub.c] [right arrow] D (2.7)

defined by T(Q) = [??]Q[??] is one-to-one algebra homomorphism A homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A,
• F(kx) = kF(x)
• F(x + y
. The prove results immediately from the theorem 2.2.

2.1. Application to the Local Solvability solv·a·ble
Possible to solve: solvable problems; a solvable riddle.

solv
of the Equation (1.1)

Definition 2.4. For every f [member of] [C.sup.[infinity].sub.0]([R.sup.4]), one can define a function [??]f [member of] [C.sup.[infinity]]([R.sup.4]) as follows:

[??]f(z,y,x,t) = f(z - 2xy,y,0,t + x) (2.8)

for any (z,y,x,t) [member of] L. Note that the function [??]f is invariant (programming) invariant - A rule, such as the ordering of an ordered list or heap, that applies throughout the life of a data structure or procedure. Each change to the data structure must maintain the correctness of the invariant.  in the following sense:

[??]f(k(z,y),x - k, t + k) = [??]f(z,y,x,t) (2.9)

for any z [member of] R, y [member of] R, x [member of] R, t [member of] R and k [member of] R.

Let D'([R.sup.4]) be the space of distributions on [R.sup.4] and let S(y,x) be a distribution on [R.sup.2], such that QS(y,x) = [delta](y,x) where Q = i[[partial derivative].sub.x] + [[partial derivative].sub.y] and [delta](y,x) is the Dirac measure In mathematics, a Dirac measure is a measure δx on a set X (with any σ-algebra of subsets of X) that gives the singleton set the measure 1, for a chosen element x ∈ X:
on [R.sup.2] at (0,0). Now let T be the distribution on [R.sup.4] defined by

<T(z,y,x,t),f(z,y,x,t)> = <[delta](z)S(y,x)[delta](t), f(z,y,x,t)> (2.10)

for all f [member of] [C.sup.[infinity]]([R.sup.4]) and (z,y,x,t) [member of] [R.sup.4], where [delta](z) (resp.[delta](t)) is the Dirac measure on R at 0

Theorem 2.5. For every f [member of] [C.sup.[infinity]]([R.sup.4)], let [??]T be the distribution on [R.sup.4] defined by

<[??]T(z,y,x,t), f(z,y,x,t)> = <T(z,y,x,t), [??]f(z,y,x,t)> (2.11)

Then [??]T is a fundamental solution of the equation P = i[[partial derivative].sub.t] + [[partial derivative].sub.y] - 2iy[[partial derivative].sub.z] - 2x[[partial derivative].sub.z]

P = i[[partial derivative].sub.t] + [[partial derivative].sub.y] - 2iy[[partial derivative].sub.z] - 2x[[partial derivative].sub.z] (2.12)

Proof. For any f [member of] [C.sup.[infinity]]([R.sup.4]), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2.13)

Definition 2.6. [6] One can define a topological isomorphism H of [C.sup.[infinity]]([R.sup.3] x [R.sup.*.sub.+]),as follows

H(f)(z,y,x,t) = f(z - [lambda]xy, ty, -[t.sup.-1] x, [t.sup.-1]) (2.14)

Corollary 2.7. Let Q([partial derivative]/[[partial derivative].sub.z], [partial derivative]/[partial derivative]y, [partial derivative]/[[partial derivative].sub.x],[partial derivative]/[partial derivative]t) be a linear partial differential operator with constant coefficients on [R.sup.3] x [R.sup.*.sub.+], then there is a differential operator P{X, [partial derivative]/[[partial derivative].sub.z], [partial derivative]/[partial derivative]y, [partial derivative]/[[partial derivative].sub.x],[partial derivative]/[partial derivative]t), and X = (z,y,x,t).

HPf = QHf (2.15)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The proof of this corollary is similar to the proof of theorem 2.1.

3. Lewy Operator and Heisenberg Group

Lewy in 1957 [9] had proved that if the equation

LT = (-[[partial derivative].sub.x] - i[[partial derivative].sub.y] + 2ix[partial derivative]z - 2y[partial derivative]z)T = f (3.1)

for a real function f [member of] [C.sup.1](R) have any solution function T, with T in [C.sup.1]([R.sup.3]). Then f is analytic.

In dealing with the non existence of solutions of partial differential operators it was customary during the last fifty years and it still is to day in larger applications, to appeal to the example of the Lewy operator which guarantees the non existence of solutions for any f [member of] [C.sup.[infinity]]([R.sup.3]), where [C.sup.[infinity]]([R.sup.3]) is the space of [C.sup.[infinity]]-functions on [R.sup.3]. Understanding the nature of these kind of partial differential operators and their invariance in·var·i·ant
1. Not varying; constant.

2. Mathematics Unaffected by a designated operation, as a transformation of coordinates.

n.
An invariant quantity, function, configuration, or system.
on the Heisenberg group requires the admission of solutions. It was therefor a matter of considerable surprise to the author, to discover that this inference is returned erroneous. More precisely, the Lewy operator L is solvable.

Theorem 3.1. The Lewy operator L is globally solvable.

Proof. Let [??] be the transformation of [C.sup.[infinity]]([R.sup.3]) defined by

[??](f)(z,y,x) = f(z - 2xy, y, -x) (3.2)

and let Q = [[partial derivative].sub.x] - i[[partial derivative].sub.y] be the Cauchy-Riemann operator, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.3)

where [sigma]f(z,y,x) = f(z,y,-x). Since the operator R = [[partial derivative].sub.x] - 2y[[partial derivative].sub.z] - i[[partial derivative].sub.y] + 2ix[[partial derivative].sub.z] is solvable because [sigma]([??]([[partial derivative].sub.x] - i[[partial derivative].sub.y])[??]f)(z,y,x) = R([sigma]f)(z,y,x), i.e for any function g [member of] [C.sup.[infinity]]([R.sup.3]), there is a function [psi] [member of] [C.sup.[infinity]]([R.sup.3]) such that R[psi](z,y,x) = g(z,y,x), and for any [psi] [member of] [C.sup.[infinity]]([R.sup.3]), there is f [member of] [C.sup.[infniity]]([R.sup.3]), such that [sigma]f(z,y,x) = [psi](z,y,x). Thus we get

(-[[partial derivative].sub.x] - i[[partial derivative].sub.y] + 2x[partial derivative]z - 2iy[[partial derivative].sub.z])f(z,y,-x) = R([sigma]f)(z,y,x) = g(z,y,x)

Hence the Theorem.

Theorem 3.2. The following conditions are verified:

(i) The extended Lewy operator is globally solvable.

(ii) The Lewy operator L has a fundamental solution.

Proof. (i) results immediately from equations (16). To prove (ii), we use the fact that the operator P = -[[partial derivative].sub.x] - 2y[[partial derivative].sub.z] - i[[partial derivative].sub.y] - 2ix[[partial derivative].sub.z], is locally solvable. Let T [member of] D'(G) and let [??] be the distribution defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.4)

for any [phi] [member of] [C.sup.[infinity].sub.0]([R.sup.3]). Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.5)

So if T is a fundamental solution of P then [??] is a fundamental solution of L.

3.1. Two Examples

In the followings will give two examples of different type of the Heisenberg group on which the Lewy operator L is invariant on each one. For each type we find a relationship between L and the Cauchy-Reimann operator which gives the solvability of L.

Example 3.3. G. B. Folland, In his book [4, P.56] had considered the following Lewy operator

L = -[[partial derivative].sub.x] + i[[partial derivative].sub.y] - [1/2]y[[partial derivative].sub.z] - [i/2]x[[partial derivative].sub.z] (3.6)

as a left invariant on the 3-dimensional Heisenberg group [R.sup.3] with law

(z,y x)(z',y',x') = (z + z' + [1/2](xy' - x'y), y + y', x + x') (3.7)

To solve this operator, we consider the mapping [LAMBDA]: [C.sup.[infinity]]([R.sup.3]) [right arrow] [C.sup.[infinity]]([R.sup.3]), which is defined by

[LAMBDA](f)(z,y,x) = f(z - [1/2]xy, y, -x) (3.8)

and the Cauchy-Reimann operator

Q = [[partial derivative].sub.x] + i[[partial derivative].sub.y] (3.9)

The mapping [LAMBDA] is topological isomorphism of [C.sup.[infinity]]([R.sup.3]), So we obtain the following theorem.

Theorem 3.4. The Lewy operator L verifies the following properties

L[C.sup.[infinity]]([R.sup.3]) = [C.sup.[infinity]]([R.sup.3]) (3.10)

Proof. In fact if f [member of] [C.sup.[infinity]]([R.sup.3]), then we have

[LAMBDA]([[partial derivative].sub.x]+i[[partial derivative].sub.y])[LAMBDA]f(z,y,-x) = (-[[partial derivative].sub.x] -[1/2]y[[partial derivative].sub.z] + i[[partial derivative].sub.y]-[i/2]x[[partial derivative].sub.z]) f(z,y,-x) (3.11)

So the solvability of L.

Example 3.5. F. Rouviere had proved in his paper [3] the following Lewy operator

P = -[[partial derivative].sub.x] - i[[partial derivative].sub.y] + 2y[[partial derivative].sub.z] - 2ix[[partial derivative].sub.z] (3.12)

is left invariant on the 3-dimensional Heisenberg group

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3.13)

where x [member of] R, y [member of] R, and z [member of] R. Let R = [[partial derivative].sub.x] - i[[partial derivative].sub.y] and define an operator T on [C.sup.[infinity]]([R.sup.3]), carrying a function f(x) into the function defined by

Tf(z,y,x) = f (-[[xy]/2] + [z/4],y, -x) (3.14)

It is clear that the mapping f [right arrow] Tf is a topological isomorphism of [C.sup.[infinity]]([R.sup.3]), and by a similar calculation as in Example 3.3 we obtain the following equation and the solvability of the operator P

Pf(z,-y,x) = JRJf(z,-y,x) (3.15)

Acknowledgements

Author would like to thank Abdaleziz Majed Al-Enad, for his support.

References

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Usually, an invariant differential operator
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abbr.
International Standard Serial Number
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General
• Unsolved problems in linguistics
• Unsolved problems in economics
• Unsolved problems in mathematics
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Mathematical statement that contains one or more derivatives. It states a relationship involving the rates of change of continuously changing quantities modeled by functions.
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Kahar El-Hussein

Department of Mathematics, Faculty of Science, Al-Jouf University, KSA KSA Kingdom of Saudi Arabia
KSA Korean Student Association (student organization providing cultural awareness and community empowerment)
KSA Knowledge, Skills & Abilities
KSA Knowledge, Skills and Attitudes
KSA Korean Standards Association

E-mail: kahar_h990@yahoo.com, kumath@hotmail.com
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