# Heisenberg group and Lewy operator.

1. Introduction

The existence theorems, elliptic 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 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 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] 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 and its vector group [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 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 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 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 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. The prove results immediately from the theorem 2.2.

2.1. Application to the Local Solvability 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 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 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 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

[1] Atiyah M.F., 1970, Resolution of Singularities and Division of Distributions, Comm. on Pure and App. Math., 23:145-150.

[2] Agmon S., 1965, Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathenmatical Studies 2, Princeton, NJ.

[3] Cerezo et A., Rouviere F., 1971, Resolubilite Local d'um operateur differentiel invariant du prenuier ordre, Ann. Scient. E c. Norm. sup [4.sup.e] serie, t.4, 21-30.

[4] Folland G.B., 1995, Introduction to Partial Differential Equations, Princeton University Press.

[5] El-Hussein K., 2006, On the Existence Theorem for Invariant Differential Operators on the Heisenberg Group, Global Journal of Pure and Applied Mathematics, ISSN 0973-1768, 2(2):103-109.

[6] El-Hussein K., 2009, A Fundamental Solution of an Invariant Differential Operator on the Heisenberg Group, International Mathematical Forum, 4(12):601-612.

[7] El-Hussein K., 2009, Research Announcements. Unsolved Problems, International Mathematical Forum, 4(12):597-600.

[8] Hormander L., 1963, Linear Partial Differential Operators, Springer-Verlag, Berlin.

[9] Lewy H., 1957, An Example of a Smooth Linear Partial Differential Equation Without Solution, Ann. Math., 66(2):155-158.

[10] Lions J.L. and Magenes E., 1972, Non-homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin.

[11] Malgrange B., 1955, Existence and Approximation des Solutions des Equations aux Derivees Partielles et des Equations de Convolutions, Ann. Inst. Fourier Grenoble, Vol. 6, pp. 271.

[12] Nerenberg L., 1959, On Elliptic Partial Differential Equations, Ann. Scuola Norm. Sup. Pisa, 13(3):115-162.

[13] Treves F., 1966, Linear Partial Differential Equations with Constant Coefficients, Gordon and Breach.

Kahar El-Hussein

Department of Mathematics, Faculty of Science, Al-Jouf University, KSA

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