Plancherel Theorem on the Symplectic Group SP(4,R)
aa r X i v : . [ m a t h . C A ] A ug Plancherel Theorem on the Symplectic Group
S P (4 , R ) Kahar El-HusseinDepartment of Mathematics, Faculty of Science and Arts at Al Qurayat,Al Jouf University, KSA & Forat University, Deir el Zore, SyriaE-mail: [email protected], [email protected] 24, 2018
Abstract
Let SL (4 , R ) be the 15 − dimensional connected semisimple Liegroup and let SL (4 , R ) = KAN be the Iwasawa decomposition. Let R ⋊ SL (4 , R ) be the group of the semidirect product of SL (4 , R ) withthe real vector group R . The goal of this paper is to define the Fouriertransform on SL (4 , R ) in order to obtain the Plancherel theorem on SL (4 , R ) and so on R ⋊ SL (4 , R ) . Since the symplectic group SP (4 , R )is a subgroup of SL (4 , R ), then it will be easy to get the Planchereltheorem on SP (4 , R ) and so on its inhomogeneous group. To this end,we obtain some interesting results on its nilpotent symplectic group Key words : Semisimple Lie group SL (4 , R ) , Symplectic Lie group SP (4 , R ) , Nilpotent Symplectic Group , Fourier transform and Plancherel Theorem on SL (4 , R ) , Plancherel Theorem on SP (4 , R ) and on its Inhomogeneous group. AMS 2000 Subject Classification: A D As well known the connected semisimple Lie group SL ( n, R ), consists ofthe following matrices SL ( n, R ) = { A = GL ( n, R ); det A = 1 } (1)1he group Sp (2 n, R ) is a subgroup of SL (2 n, R ) , which is SP (2 n, R ) = { g ∈ SL (2 n, R ); gg t = } (2)where is the symplectic matrix defined by = (cid:18) I n − I n (cid:19) (3)and 0 and I are the n × n zero and identity matrices. It is clear det = 1 , = I, and t = − = − The 10 − demesional symplectic group SP (4 , R ). If g ∈ SP (4 , R ), then g = x x x x x x x x x x − x − x x x − x − x , x ij ∈ R , 1 ≤ i, j ≤ SP (4 , R ) and its inhomogeneous group.Therefore, I will define the Fourier transform on SL (4 , R ), and I will prove itsPlancherel theorem. Besides, I will demonstrate the existence theorem andhypoellipticity of the partial differential equations on its nilpotent symplectic In the following and far away from the representations theory of Liegroups we use the Iwasawa decomposition of SL (4 , R ) , to define the Fouriertransform and to get the Plancherel formula on the connected real semisimpleLie group SL (4 , R ) . Therefore let SL (4 , R ) be the complex Lie group, whichis SL (4 , R ) = { A = a a a a a a a a a a a a a a a a : a ij ∈ R , ≤ i, j ≤ and det A = 1 } (5)2et G = SL (4 , R ) = KN A be the Iwasawa decomposition of G , where K = SO (4) N = { x x x x x x i x i ∈ R , ≤ i ≤ } A = { a a a
00 0 0 a : a i ∈ R ⋆ + , ≤ i ≤ , a a a a = 1 } (6)Hence every g ∈ SL (4 , R ) can be written as g = kan ∈ SL (4 , R ) , where k ∈ K, a ∈ A, n ∈ N. We denote by L ( SL (4 , R )) the Banach algebra thatconsists of all complex valued functions on the group SL (4 , R ), which are inte-grable with respect to the Haar measure dg of SL (4 , R ) and multiplication isdefined by convolution product on SL (4 , R ) , and we denote by L ( SL (4 , R ))the Hilbert space of SL (4 , R ). So we have for any f ∈ L ( SL (4 , R )) and φ ∈ L ( SL (4 , R )) φ ∗ f ( h ) = Z G f ( g − h ) φ ( g ) dg (7)The Haar measure dg on a connected real semi-simple Lie group G = SL (4 , R ),can be calculated from the Haar measures dn, da and dk on N ; A and K ;respectively, by the formula Z SL (4 , R ) f ( g ) dg = Z A Z N Z K f ( ank ) dadndk (8) Keeping in mind that a − ρ is the modulus of the automorphism n → ana − of N we get also the following representation of dg Z SL (4 , R ) f ( g ) dg = Z A Z N Z K f ( ank ) dadndk = Z N Z A Z K f ( nak ) a − ρ dndadk (9)where ρ = 2 − X α i m ( α ) α m ( α ) denotes the multiplicity of the root α and ρ = the dimension of thenilpotent group N. Furthermore, using the relation R G f ( g ) dg = R G f ( g − ) dg, we receive Z SL (4 , R ) f ( g ) dg = Z K Z A Z N f ( kan ) a ρ dndadk (10) N Let N be the real group consisting of all matrices of the form x x x x x x (11)where ( x , x , x , x , x , x ) ∈ R . The group can be identified with thegroup ( R ⋊ ρ R ) ⋊ ρ R be the semidirect product of the real vector groups R , R and R , where ρ is the group homomorphism ρ : R → Aut ( R ) , whichis defined by ρ ( x , x )( y , y , y ) = ( y + x y , y + x y , y ) (12)and ρ is the group homomorphism ρ : R → Aut ( R ⋊ ρ R ) , which is givenby ρ ( x )( y , y , y , y , y ) = ( y + x y , y , y , y + x y , y ) (13)where Aut ( R ) ( resp.Aut ( R ⋊ ρ R )) is the group of all automorphisms of R ( resp. ( R ⋉ ρ R )) , see [6] . L = R × R × R × R × R be the group with law: X.Y = ( x , x , x , x , x , t , t , x , t )( y , y , y , y , y , s , s , y , s )= (( x , x , x , x , x , t , t )( ρ ( t )( y , y , y , y , y , s , s ) , y + x , s + t )= (( x , x , x , x ) ρ ( t , t )( y + t y , y , y , y , s + t s , s ) , x + y , y + x , s + t )= (( x , x , x ) + ( y + t y + t y , y + t y , y ) , x + y , s + t s ,s + t , x + y , y + x , s + t )= ( x + y + t y + t y , x + y + t y , x + y , x + y , t + s + t s ,y + x , s + t , y + x , s + t ) (14)for all ( X, Y ) ∈ L . In this case the group N can be identified with the closedsubgroup R × { } ⋊ ρ R × { } ⋊ ρ R of L and B with the closed subgroup R × R × { }× R × { } of L, where B = R × R × R the group, which isthe direct product of the real vector groups R , R and R Let C ∞ ( N ) , D ( N ) , D ′ ( N ) , E ′ ( N ) be the space of C ∞ - functions, C ∞ withcompact support, distributions and distributions with compact support on N respectively . We denote by L ( N ) the Banach algebra that consists of allcomplex valued functions on the group N , which are integrable with respectto the Haar measure of N and multiplication is defined by convolution on N ,and we denote by L ( N ) the Hilbert space of N . Definition 3.1.
For every f ∈ C ∞ ( N ), one can define function e f ∈ C ∞ ( L ) as follows: e f ( x, x , x , t , t , x , t ) = f ( ρ ( x )( ρ ( x , x )( x ) , t + x , t + x ) , t ) (15) for all ( x, x , x , t , t , x , t ) ∈ L, here x = ( x , x , x ) ∈ R . Remark 3.1.
The function e f is invariant in the following sense: e f (( ρ ( h )(( ρ ( r, k )( x ) , x − r, x − k, t + r, t + k ) , x − h, t + h )= e f ( x, x , x , t , t , x , t ) (16) for any ( x, x , x , t , t , x , t ) ∈ L , h ∈ R and ( r, k ) ∈ R , where x =( x , x , x ) ∈ R . So every function ψ ( x, x , x , x ) on N extends uniquely asan invariant function e ψ ( x, x , x , t , t , x , t ) on L. Theorem 3.1.
For every function F ∈ C ∞ ( L ) invariant in sense (16) and for every ϕ ∈ D ( N ), we have u ∗ F ( x, x , x , t , t , x , t ) = u ∗ c F ( x, x , x , t , t , x , t ) (17)5 or every ( X, x , x , t , t , x , t ) ∈ L , where ∗ signifies the convolution prod-uct on N with respect the variables ( x, t , t , t ) and ∗ c signifies the commu-tative convolution product on B with respect the variables ( x, x , x , x ) . Proof : In fact we have ϕ ∗ F ( x, x , x , t , t , x , t )= Z N F (cid:2) ( y, y , y , s ) − ( X, x , x , t , t , x , t ) (cid:3) u ( y, y , y , s ) dydy dy ds = Z N F (cid:2) ( ρ ( s − )( y, y , y ) − , − s )( x, x , x , t , t , x , t ) (cid:3) u ( y, y , y , s ) dydy dy ds = Z N F [( ρ ( s − )(( ρ ( y , y ) − (( − y ) + ( x ))) , x , x , t − y , t − y ) , x , t − s )] u ( y, y , y , s ) dydy dy ds (18)Since F is invariant in sense (16) , then for every ( x, x , x , t , t , x , t ) ∈ L we get ϕ ∗ F ( x, x , x , t , t , x , t )= Z N F [( ρ ( s − )( ρ ( y , y ) − ( − y + x ) , x , x , t − y , t − y ) ,x , t − s )] u ( y, y , y , s ) dydy dy ds = Z N F [ x − y, x − y , x − y , t , t , x − s, t ] u ( y, y , y , s ) dydy dy ds = ϕ ∗ c F ( x, x , x , t , t , x , t ) (19)As in [6], we will define the Fourier transform on G . Therefore let S ( N )be the Schwartz space of N which can be considered as the Schwartz spaceof S ( B ) , and let S ′ ( N ) be the space of all tempered distributions on N. Definition 3.2. If f ∈ S ( N ) , one can define its Fourier transform F f by the Fourier transform on its vector group : F f ( ξ ) = Z N f ( X ) e − i h ξ , X i dX (20)6 or any ξ = ( ξ , ξ , ξ , ξ , ξ , ξ ) ∈ R , and X = ( x , x , x , x , x , x ) ∈ R , where h ξ , X i = ξ x + ξ x + ξ x + ξ x + ξ x + ξ x and dX = dx dx dx dx dx dx is the Haar measure on N . The mapping f → F f isisomorphism of the topological vector space S ( N ) onto S ( R ) . Theorem 3.2.
The Fourier transform F satisfies : ∨ ϕ ∗ f (0) = Z R F f ( ξ ) F u ( ξ ) dξ (21) for every f ∈ S ( N ) and ϕ ∈ S ( N ) , where ∨ ϕ ( X ) = u ( X − ) , ξ = ( ξ , ξ , ξ , ξ , ξ , ξ ) ,dξ = dξ dξ dξ dξ dξ dξ , is the Lebesgue measure on R , and ∗ denotes theconvolution product on N .Proof: By the classical Fourier transform, we have: ∨ ϕ ∗ f (0) = Z R F ( ∨ ϕ ∗ f )( ξ ) dξ = Z R Z N ∨ ϕ ∗ f ( X ) e − i h ξ,X i dXdξ = Z R Z N Z N f ( Y X ) u ( Y ) e − i h ξ,X i dY dXdξ. (22)By change of variable Y X = X ′ with Y = ( x , x , x , x , x , x ) and X ′ = ( y , y , y , y , y , y ), we get X = Y − X ′ = ( x , x , x , x , x , x ) − ( y , y , y , y , y , y )= ( y − x + x x − x y − x x x + x x − x y + x x y , y − x + x y − x x ,x + y , y − x − x y + x x , y − x , y − x )and − i h ξ, X i = − i (cid:10) ξ, Y − X ′ (cid:11) = − i [( y − x + x x − x y − x x x + x x − x y + x x y ) ξ + ( y − x + x y ) ξ − x x ξ + ( y − x ) ξ + ( y − x − x y + x x ) ξ + ( y − x ) ξ + ( y − x ) ξ ]= − i [( x ξ − y ξ ) + ( − x x ξ + x y ξ − y ξ + x ξ + x ξ − y ξ − ξ ) x − y ξ +( y ξ − x ξ ) + ( y ξ − x ξ − ξ ) x + y ξ + y ξ + ( x ξ − y ξ − ξ ) x + ( y − x ) ξ
7o we obtain e − i ( y − x + x x − x y − x x x + x x − x y + x x y ) ξ e − i (( y − x + x y − x x ) ξ +( y − x ) ξ ) e − i (( y − x − x y + x x ) ξ +( y − x ) ξ +( y − x ) ξ ) = e − i (( x ξ − y ξ )+( − x x ξ + x y ξ − y ξ + x ξ + x ξ − y ξ − ξ ) x − y ξ ) e − i (( y ξ − x ξ )+( y ξ − x ξ − ξ ) x + y ξ ) − i ( y ξ +( x ξ − y ξ − ξ ) x +( y − x ) ξ ) By the invariance of the Lebesgue,s measures dξ , dξ and dξ , we get ∨ ϕ ∗ f (0) = Z N Z N Z R f ( X ′ ) e − i (( x ξ − y ξ )+( − x x ξ + x y ξ − y ξ + x ξ + x ξ − y ξ − ξ ) x + y ξ ) e − i (( y ξ − x ξ )+( y ξ − x ξ − ξ ) x + y ξ ) e − i ( y ξ +( x ξ − y ξ − ξ ) x +( y − x ) ξ ) ϕ ( Y ) dY dX ′ dξ = Z N Z N Z R f ( X ′ ) e − i ( y ξ − x ξ + y ξ − x ξ + y ξ − x ξ + y ξ − ξ x − ξ x + y ξ − ξ x + y ξ ) ϕ ( Y ) dY dX ′ dξ = Z R F f ( ξ ) F ϕ ( ξ ) dξ for any Y = ( x , x , x , x , x , x ) ∈ R and X ′ = ( y , y , y , y , y , y ) ∈ R , where 0 = (0 , , , , ,
0) is the identity of N . The theorem is proved. Corollary 3.1.
In theorem if we replace ϕ by f, we obtain thePlancherel,s formula on N ∨ f ∗ f (0) = Z N | f ( X ) | dX = Z R |F f ( ξ ) | dξ (23) SL (4 , R ) Let k be the Lie algebra of K = SO (4). Let ( X , X , X , X ) a basis of k , such that the both operators∆ = X i =1 X i , D q = X ≤ l ≤ q − X i =1 X i ! l (24)8re left and right invariant (bi-invariant) on K, this basis exist see [4 , p. l ∈ N . Let D l = (1 − ∆) l , then the family of semi-norms { σ l , l ∈ N } such that σ l ( f ) = Z K (cid:12)(cid:12) D l f ( y ) (cid:12)(cid:12) dy ) , f ∈ C ∞ ( K ) (25)define on C ∞ ( K ) the same topology of the Frechet topology defined by thesemi-normas k X α f k defined as k X α f k = Z K ( | X α f ( y ) | dy ) , f ∈ C ∞ ( K ) (26)where α = ( α , ....., α m ) ∈ N m , see [4 , p. b K be the set of all irreducible unitary representations of K. If γ ∈ b K ,we denote by E γ the space of the representation γ and d γ its dimension thenwe get Definition 4.1.
The Fourier transform of a function f ∈ C ∞ ( K ) isdefined as T f ( γ ) = Z K f ( x ) γ ( x − ) dx (27) where T is the Fourier transform on K Theorem (A. Cerezo [4] ) 4.1.
Let f ∈ C ∞ ( K ) , then we have theinversion of the Fourier transform f ( x ) = X γ ∈ b K dγtr [ T f ( γ ) γ ( x )] (28) f ( I K ) = X γ ∈ b K dγtr [ T f ( γ )] (29) and the Plancherel formula k f ( x ) k = Z K | f ( x ) | dx = X γ ∈ b K d γ k T f ( γ ) k H.S (30) for any f ∈ L ( K ) , where I K is the identity element of K and k T f ( γ ) k H.S is the Hilbert- Schmidt norm of the operator
T f ( γ )9 efinition 4.2 . For any function f ∈ D ( G ) , we can define a function Υ( f ) on G × K = G × SO (4) by Υ( f )( g, k ) = Υ( f )( kna, k ) = f ( gk ) = f ( knak ) (31) for g = kna ∈ G, and k ∈ K . The restriction of Υ( f ) ∗ ψ ( g, k ) on K ( G ) is Υ( f ) ∗ ψ ( g, k ) ↓ K ( G ) = f ( nak ) = f ( g ) ∈ D ( G ) , and Υ( f )( g, k ) ↓ SO (4) = f ( kna ) ∈ D ( G ) Remark 4.1 . Υ( f ) is invariant in the following senseΥ( f )( gh, h − k ) = Υ( f )( g, k ) (32) Definition 4.3 . If f and ψ are two functions belong to D ( G ) , then wecan define the convolution of Υ( f ) and ψ on G × SO (4) as Υ( f ) ∗ ψ ( g, k )= Z G Υ( f )( gg − , k ) ψ ( g ) dg = Z SO (4) Z N Z A Υ( f )( knaa − n − k − k ) ψ ( k n a ) dk dn da and so we getΥ( f ) ∗ ψ ( g, k ) ↓ K ( G ) = Υ( f ) ∗ ψ ( I K na, k )= Z SO (4) Z N Z A f ( naa − n − k − k ) ψ ( k n a ) dk dn da = Υ( f ) ∗ ψ ( na, k )where g = k n a Definition 4.3 . For f ∈ D ( G ), let Υ( f ) be its associated function , wedefine the Fourier transform of Υ( f )( g, k ) by F Υ( f ))( I SO (4) , ξ, λ, γ )= Z N Z A [ Z SO (4) ( T Υ( f )( I SO (4)) na, k ) γ ( k − ) dk ] a − iλ e − i h ξ, n i dadn = Z SO (3) Z N Z A [Υ( f )( I SO (3) na, k )] a − iλ e − i h ξ, n i γ ( k − ) dadndk (33)10 here F is the Fourier transform on AN and T is the Fourier transform on SO (4) , I SO (3) is the identity element of SO (4), and n = ( x , x , x , x , x , x ) , n =( y , y , y , y , y , y ) , ξ = ( ξ , ξ , ξ , ξ , ξ , ξ ) , a = b b b and a = a a a Plancherel’s Theorem 4.2 . For any function f ∈ L ( G ) ∩ L ( G ) , weget Z G | f ( g ) | dg = Z A Z N Z SO (4) | f ( kna ) | dadndk = X γ ∈ \ SO (4) d γ Z R Z R k T F f ( α, ξ, γ ) k dαdξ (34) f ( I A I N I S ) = Z N Z A X γ ∈ b K d γ T F f ( α, ξ, γ )] dαdξ = X γ ∈ b K d γ Z R Z R T F f ( α, ξ, γ ) dαdξ (35) where , I A , I N , and I K are the identity elements of A , N and K respectively, F is the Fourier transform on AN and T is the Fourier transform on K, Proof:
First let ∨ f be the function defined by ∨ f ( kna ) = f (( kna ) − ) = f ( a − n − k − ) (36)Then we have Z SL (4 , R ) | f ( g ) | dg = Υ( f ) ∗ ∨ f ( I SO (4) I N I A , I SO (4) )= Z G Υ( f )( I SO (4) I N I A g − , I SO (4) ) ∨ f ( g ) dg = Z A Z N Z SO (4) Υ( f )( a − n − k − , I SO (4) ) ∨ f ( k n a ) da dn dk = Z A Z N Z SO (4) f ( a − n − k − ) f (( k n a ) − ) da dn dk = Z A Z N Z SO (4) | f ( a n k ) | da dn dk (37)11econdlyΥ( f ) ∗ ∨ f ( I SO (4) I N I A , I SO (4) )= Z R X γ ∈ \ SO (4) dγ Z SO (4) tr (Υ( f ) ∗ ∨ f ( I SO (4) na, k ) γ ( k − )) a − iα e − i h ξ, n i dadndk dλdξ = X γ ∈ \ SO (4) dγ Z SO (4) Z R tr [Υ( f ) ∗ ∨ f ( I SO (3 na, k ) dka − iα e − i h ξ, n i γ ( k − )] dadndk dλdξ = Z R X γ ∈ \ SO (4) Z SO (4) tr [Υ( f )( I SO (4) nab − n − k − , k ) ∨ f ( k n b ) γ ( k − ) dk ] a − iα e − i h ξ, n i dndadn dbdλdξ where e − i ( y − x + x x − x y − x x x + x x − x y + x x y ) ξ e − i (( y − x + x y − x x ) ξ +( y − x ) ξ ) e − i (( y − x − x y + x x ) ξ +( y − x ) ξ +( y − x ) ξ ) = e − i ( y ξ − x ξ + y ξ − x ξ + y ξ − x ξ + y ξ − ξ x − ξ x + y ξ − ξ x + y ξ ) n = ( x , x , x , x , x , x ) , n = ( y , y , y , y , y , y ) , ξ = ( ξ , ξ , ξ , ξ , ξ , ξ ) , a = b b b and a = a a a Using the fact that Z A Z N Z SO (4) f ( kna ) dadndk = Z N Z A Z SO (4) f ( kan ) a dndadk (38)12nd Z R Z A Z N Z SO (4) f ( kna ) e − i h ξ, n i dadndkdξ = Z R Z A Z N Z SO (4) f ( kan ) e − i h ξ, an a − i a dadndkdξ = Z R Z A Z N Z SO (4) f ( kan ) e − i h aξa − , n i a dadndkdξ = Z R Z A Z N Z SO (4) f ( kan ) e − i h ξ, n i dadndkdξ (39)Then we haveΥ( f ) ∗ ∨ f ( I SO (4) I N I A , I SO (4) )= Z R Z SO (4) Z A Z N X γ ∈ \ SO (3) d γ Z SO (4) f ( nab − n − k − , k ) ∨ f ( k n b ) γ ( k − ) dk dk a − iλ e − i h ξ, n i dndadn da dλdξ = Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ab − nn − k − , k ) ∨ f ( k n b ) γ ( k − ) dk dk a − iλ e − i h ξ, n i dndadn da dλdξ = Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ank − , k ) ∨ f ( k n b ) γ ( k − ) dk dk ab − iλ e − i h ξ, nn i dndadn da dλdξ = Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ank − k ) ∨ f ( k n b ) γ ( k − ) dk dk ab − iλ e − i h ξ, nn i dndadn da dλdξ = Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ank − ) ∨ f ( k n b ) γ ( k − ) γ ( k − ) dk dk a − iλ b − iλ e − i h ξ, n i e − i h ξ, n i dndadn da dλdξ f ) ∗ ∨ f ( I SO (4) I N I A , I SO (4) )= Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ank − ) ∨ f ( k n b ) γ ( k − ) γ ( k − ) dk dk a − iλ b − iλ e − i h ξ, n + n i dndadn da dλdξ = Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ank − ) f ( b − n − k − ) γ ( k − ) γ ( k − ) dk dk a − iλ e − i h ξ, n i b − iλ e − i h ξ, n i dndadn da dλdξ = Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ank − ) f ( bn k ) γ ( k − ) γ ( k − ) dk dk a − iλ e − i h ξ, n i b − iλ e i h ξ, n i dndadn da dλdξ = Z R X γ ∈ \ SO (4) d γ Z SO (4) Z SO (4) f ( ank − ) f ( bn k ) γ ( k − ) γ ( k − ) dk dk a − iλ e − i h ξ, n i b − iλ e − i h ξ, n i dndadn dbdλdξ = Z R X γ ∈ \ SO (4) d γ T F f ( λ, ξ, γ ) T F f ( λ, ξ, γ ) dλdξ = Z R X γ ∈ \ SO (4) d γ | T F ( f )( λ, ξ, γ ) | dλdξ Hence theorem of Plancherel on SL (4 , R ) is ProvedLet SP (4 , R ) = KN A be the Iwasawa decomposition of the symplectic SP (4 , R ) . My state result is
Corollary 4.1.
For any function f ∈ L ( SP (4 , R )) ∩ L ( SP (4 , R )) , weget Z SP (4 , R ) | f ( v, g ) | dvdg = Z N Z A X γ ∈ b K d γ kF R T F F ( ξ, λ, γ ) k dηdλdξ (40)Which is the Plancherel theorem on the symplectic SP (4 , R ) R ⋊ SL (4 , R ) Let P = R ⋊ ρ SL (4 , R ) be the 14 − dimensional affine group . Let ( v, g ) and( v ′ , g ′ ) be two elements belong P, then the multiplication of ( v, g ) and ( v ′ , g ′ )14s given by ( v, g )( v ′ , g ′ ) = ( v + ρ ( g )( v ′ ) , gg ′ ) = ( v + gv ′ , gg ′ ) (41)for any ( v, v ′ ) ∈ R × R and ( g, g ′ ) ∈ SL (4 , R ) × SL (4 , R ) , where gv ′ = ρ ( g )( v ′ ) . To define the Fourier transform on P , we introduce the followingnew group Definition 5.1 . Let Q = R × SL (4 , R ) × SL (4 , R ) be the group withlaw : X · Y = ( v, h, g )( v ′ , h ′ , g ′ )= ( v + gv ′ , hh ′ , gg ′ ) (42) for all X = ( v, h, g ) ∈ Q and Y = ( v ′ , h ′ , g ′ ) ∈ Q. Denote by A = R × SL (4 , R ) the group of the direct product of R with the group SL (4 , R ) . Then the group A can be regarded as the subgroup R × SL (4 , R ) ×{ I SL (4 , R ) } of Q and P can be regarded as the subgroup R ×{ I SL (4 , R ) } × SL (4 , R ) of Q. Definition 5.2 . For any function f ∈ D ( P ) , we can define a function e f on Q by e f ( v, g, h ) = f ( gv, gh ) (43) Remark 5.1.
The function e f is invariant in the following sense e f ( q − v, g, q − h ) = e f ( v, gq − , h ) (44) Theorem 5.1.
For any function ψ ∈ D ( P ) and e f ∈ C ∞ ( Q ) invariantin sense (32) , we get ψ ∗ e f ( v, h, g ) = e f ∗ c ψ ( v, h, g ) (45) where ∗ signifies the convolution product on P with respect the variable ( v, g ) , and ∗ c signifies the convolution product on A with respect the variable ( v, h ) 15 roof : In fact for each ψ ∈ D ( P ) and e f ∈ C ∞ ( Q ) , we have ψ ∗ e f ( v, h, g )= Z R Z SL (4 , R ) e f (( v ′ , g ′ ) − ( v, h, g )) ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z SL (4 , R ) e f [( g ′− ( − v ′ ) , g ′− )( v, h, g )] ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z SL (4 , R ) e f [( g ′− ( − v ′ ) , g ′− )( v, h, g )] ψ ( v ′ , g ′ ) dv ′ dg ′ (46)= Z R Z SL (4 , R ) e f [( g ′− ( v − v ′ ) , h, g ′− g )] ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z SL (4 , R ) e f [ v − v ′ , hg ′− , g ] ψ ( v ′ , g ′ ) dv ′ dg ′ = e f ∗ c ψ ( v, h, g ) (47) Corollary 5.1.
From theorem , the equation turns as ψ ∗ e f ( v, h, I G )= ψ ∗ c e f ( v, h, I SL (3 , R ) ) = Z R Z SL (3 , R ) e f [ v − v ′ , hg ′− , g ] ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z SL (4 , R ) f [ hg ′− ( v − v ′ ) , hg ′− ] ψ ( v ′ , g ′ ) dv ′ dg ′ = h ( f ) ∗ c ψ ( v, h )(48)where h ( f )( v, g ) = f ( gv, g ) (49) Definition 5 . . Let Υ F be the function on P × SL (4 , R ) defined by Υ F ( v, ( g, k )) = F ( v, gk ) (50) Definition 5.4.
Let ψ ∈ D ( P ) and F ∈ D ( P ) , then we can define a onvolution product on the Affine group P as ψ ∗ c Υ F ( v, ( g, k ))= Z R Z SL (4 , R ) Υ F ( v − v ′ , ( gg ′− , k )) ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z K Z N Z A F ( v − v ′ , kna ( k ′ n ′ a ′ ) − k )) ψ ( v ′ , k ′ n ′ a ′ ) dv ′ dk ′ dn ′ da ′ where g = kna and g ′ = k ′ n ′ a ′ Corollary 5.2.
For any function F belongs to D ( P ) , we obtain ψ ∗ c Υ h ( F )( v, ( g, k ))= Z R Z SL (4 , R ) Υ h ( F )( v − v ′ , ( gg ′− , k ) ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z SL (4 , R ) Υ h ( F )( v − v ′ , ( gg ′− , k ) ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z SL (4 , R ) h ( F )( v − v ′ , gg ′− k ) ψ ( v ′ , g ′ ) dv ′ dg ′ = Z R Z SL (4 , R ) F ( gg ′− k ( v − v ′ ) , gg ′− k ) ψ ( v ′ , g ′ ) dv ′ dg ′ Corollary 5.3.
For any function F belongs to D ( P ) , we obtain F ∗ Υ h ( ∨ F )(0 , ( I SL (4 , R ) , I K )) = Z R Z SL (4 , R ) | f ( v, g ) | dgdv = k f k (51) Proof: If F ∈ D ( P ) , then we get 17 ∗ Υ h ( ∨ F )(0 , ( I SL (4 , R ) , I K ))= Z R Z SL (4 , R ) Υ ~ ( ∨ F )[(0 − v ) , ( I G g − , I S )] F ( v, g ) dgdv = Z R Z SL (4 , R ) ~ ( ∨ F )[(0 − v ) , I G g − I S ] F ( v, g ) dgdv = Z SO (3) Z R ~ ( ∨ F )[( − v ) , g − ] F ( v, g ) dgdv = Z R Z SL (4 , R ) ∨ F [ g − ( − v ) , g − ] F ( v, g ) dgdv = Z R Z SL (4 , R ) F [ g − ( − v ) , g − ] − F ( v, g ) dgdv = Z R Z SL (4 , R ) F [ v, g ] F ( v, g ) dgdv = Z R Z SL (4 , R ) | f ( v, g ) | dgdv Definition 5.5.
Let f ∈ D ( P ) , we define its Fourier transform by F R T F f ( η, γ, ξ, λ ) = Z R Z A Z N Z K f ( v, kna ) e − i h η, v i γ ( k − ) a − iλ e − i h ξ, n i dkdadndv where F R is the Fourier transform on R , kna = g, η = ( η , η , η , η ) ∈ R ,v = ( v , v , v , v ) ∈ R , and dv = dv dv dv dv is the Lebesgue measure on R and h ( η , η , η , η ) , ( v , v , v , v ) i = X i =1 η i v i (52) Plancherel’s Theorem 5.2 . For any function f ∈ L ( P ) ∩ L ( P ) , weget Z P | f ( v, g ) | dvdg = Z R X γ ∈ \ SO (4) d γ kF R T F F ( η, γ, ξ, λ ) k dηdλdξ (53)18 roof: Let Υ ~ ( ∨ F ) be the function defined asΥ ~ ( ∨ F ) ( v ; ( g, k )) = ~ ( ∨ F ) ( v ; gk )= ∨ F ( gk v ; gk ) = F ( gk v ; gk ) − ) (54)then, we have F ∗ Υ ~ ( ∨ F ) (0 , ( I SL (4 , R ) I N I A , I SO (4) ))= Z R Z R Z R F R F ( F ∗ Υ ~ ( ∨ F )( η, ( I SO (4) ξ, λ, I SO (4) )) dλdξdη = Z R X γ ∈ \ SO (4) d γ Z SO (4) F R T F ( F ∗ Υ ~ ( ∨ F ))(( η, ( I SO (4) na, k )) γ ( k − ) dk ) e − i h η, v i a − iλ e − i h ξ, n i dadndvdλdξdη = Z SL (4 , R ) Z R X γ ∈ \ SO (3) d γ Z SO (4) Υ ~ ( ∨ F )(( v − w ) , ( I SO (4) nag − , k )) γ ( k − ) dk F ( w, g ) dg e − i h η, v i a − iλ e − i h ξ, n i dadndvdwdλdξdη = Z R Z SO (4) X γ ∈ \ SO (4) d γ Z SO (4) ~ ( ∨ F )(( v − w ) , ( nab − n − k − k )) γ ( k − ) dk F ( w, k n b ) dk e − i h η, v i a − iλ e − i h ξ, n i dbdn dadndwdvdλdξdη = Z R Z SO (4) X γ ∈ \ SO (4) d γ Z SO (4) ~ ( ∨ F )( v, ( ank )) γ ( k − ) dk F ( w, k n b ) γ ( k − ) dk e − i h η, v + w i a − iλ b − iλ e − i h ξ, n i e − i h ξ, n i da dn dadndwdvdλdξdη Z R Z SO (4) X γ ∈ \ SO (4) d γ Z SO (4) ( ∨ F )( ank v, ank ) γ ( k − ) dk F ( w, k n a ) γ ( k − ) dk e − i h η, v i e − i h η, w i e − i h λ,a i e − i h λ,a i e − i h ξ, n i e − i h ξ, n i da dn dadndwdvdλdξdη = Z R Z SO (4) X γ ∈ \ SO (4) d γ Z SO (4) F ( − v, k − n − a − ) F ( w, k n a ) γ ( k − ) γ ( k − ) γ ( k − ) e − i h η, v i e − i h η, w i a − iλ a − iλ e − i h ξ, n i e − i h ξ, n i dk dk da dn dadndwdvdλdξdη = Z R Z SO (4) X γ ∈ \ SO (4) d γ Z SO (4) F ( v, k na ) γ ( k ) F ( w, k n a ) γ ( k − ) dk dk e i h η, v i e − i h η, w i e i h λ,a i e − i h λ,a i e i h ξ, n i e − i h ξ, n i da dn dadndwdvdλdξdη Z R X γ ∈ \ SO (4) d γ kF R T F F ( η, γ, ξ, λ ) k H.S dηdλdξ
Hence the theorem is proved on the R ⋊ SL (4 , R ) . Corollary 5.3.
For any function f ∈ L ( R ⋊ ρ SP (4 , R )) ∩ L ( R ⋊ ρ SP (4 , R )) , we get Z R ⋊ ρ SP (4 , R ) | f ( v, g ) | dvdg = Z R Z N Z A X γ ∈ b K d γ kF R T F F ( η, γ, ξ, λ ) k dηdλdξ (55)Which is the Plancherel theorem on the inhomogeneous group R ⋊ ρ SP (4 , R ) of the symplectic SP (4 , R ) , where KN A is the Iwasawa decompo-sition of the symplectic group SP (4 , R ) . Denote by SP N the nilpotent symplectic subgroup of the group SP (4 , R )consists of all matrices of the form SP N = x y z z − xt t − x , ( x, y, z, t ) ∈ R (56)20e denote by N the nilpotent symplectic subgroup of SP N , formed bythe following matrix N = x y z z
00 0 1 00 0 − x , ( x, y, z ) ∈ R (57)The group N is isomorphic onto the group G = R ⋊ ρ R semidirect oftwo groups R and R , where ρ : R → Aut ( R ) is the group homomorphismdefined by ρ ( x )( z, y ) = ( z + xy, y ) . The multiplication of two elements X = ( z, y, x ) and Y = ( c, b, a ) is given by( z, y, x )( c, b, a )= ( z + c + xb − ay, y + b, x + a )= ( z + c + (cid:12)(cid:12)(cid:12)(cid:12) x ya b (cid:12)(cid:12)(cid:12)(cid:12) , y + b, x + a ) (58) . Our aim is to prove the solvability and hypoellipticity of the following Lewyoperators L = ( − ∂ x − i∂ y − y∂ z + 2 ix∂ z ) (59) L ⋆ = ( − ∂ x + i∂ y − y∂ z − ix∂ z ) (60) Definition 6.1.
One can define a transformation ~ : D ′ ( R ) → D ′ ( R ) ~ Ψ( z, y , x ) = Ψ( z − xy , y , − x ) (61)It results from this definition that ~ = ~ Theorem 6.1.
Let Q = ∂ x − i∂ y be the Cauchy - Riemann operator, thenwe have for any f ∈ C ∞ ( R )( Lf )( z, y, − x ) = ~ Q ~ f ( z, y, − x ) (62)For the proof of this theorem see [6] . Corollary 6.1.
The Lewy operator L is solvable roof: In fact the Cauchy-Riemann operator Q = ∂ x − i∂ y is solv-able, because QC ∞ ( R ) = C ∞ ( R ) , and ~ C ∞ ( R ) = C ∞ ( R ) . So, I have LC ∞ ( H ) = C ∞ ( H ) Definition 6.1.
Let G be a Lie group an operator Γ : D ′ ( G ) → D ′ ( G ) is called hypoelliptic if Γ ϕ ∈ C ∞ ( G ) = ⇒ ϕ ∈ C ∞ ( G ) (63) for every distribution ϕ ∈ D ′ ( G ) . Theorem 6.2.
The Lewy operator is hypoellipticProof:
First the operator ~ is hypoelliptic, and the Cauchy- Riemannoperator ∂ x − i∂ y is hypoelliptic. So if ϕ ∈ D ′ ( R ) and if Lϕ ( z, y, − x ) = ~ Q ~ ϕ ( z, y, − x ) ∈ C ∞ ( R ) , then I get Lϕ ∈ C ∞ ( R ) = ⇒ ~ Q ~ ϕ ∈ C ∞ ( R )= ⇒ Q ~ ϕ ∈ C ∞ ( R ) = ⇒ ~ ϕ ∈ C ∞ ( R )= ⇒ ϕ ∈ C ∞ ( R ) (64) Theorem 6.3.
Let Q ⋆ be the operator L ⋆ = ( − ∂ x + i∂ y − y∂ z − ix∂ z ) (65) Q ⋆ = ∂ x + i∂ y (66) then for every ϕ ∈ C ∞ ( R ) , I have ~ ( ∂ x − i∂ y )( ∂ x + i∂ y ) ~ ϕ ( z, y, − x ) = ~ ∆ ~ ϕ ( z, y, − x )= [( − ∂ x − y∂ z ) + ( − i∂ y + 2 ix∂ z )(( − ∂ x − y∂ z ) + ( i∂ y − ix∂ z ) ϕ ]( z, y, − x )= LL ⋆ ϕ ( z, y, − x ) (67)where ∆ and L ⋆ are the operators∆ = ∂ ∂ x + ∂ ∂ y (68) L ⋆ = ( i∂ y − ix∂ z ) + ( − ∂ x − y∂ z ) (69) L ⋆ is called the conjugate of the Lewy operator, which can be consideredanother form of the Lewy operator. As in theorem we can easily see22hat L ⋆ C ∞ ( R ) = C ∞ ( R ) . The operator LL ⋆ can be regarded as the squareof the Lewy operator on the 3 − dimensional Heisenberg group. Corollary 6.1.
The operators LL ⋆ and L ⋆ are hypoellipticProof: From the above we deduce the following L ⋆ ϕ ∈ C ∞ ( R ) = ⇒ ~ Q ⋆ ~ ϕ ∈ C ∞ ( R )= ⇒ Q ⋆ ~ ϕ ∈ C ∞ ( R ) = ⇒ ~ ϕ ∈ C ∞ ( R )= ⇒ ϕ ∈ C ∞ ( R ) (70)In other hand we have LL ⋆ ϕ ∈ C ∞ ( R ) = ⇒ ~ QQ ⋆ ~ ϕ ∈ C ∞ ( R )= ⇒ QQ ⋆ ~ ϕ ∈ C ∞ ( R ) = ⇒ Q ⋆ ~ ϕ ∈ C ∞ ( R )= ⇒ ~ ϕ ∈ C ∞ ( R ) = ⇒ ϕ ∈ C ∞ ( R ) (71) Theorem 6.4.
The following left invariant differential operators on
Gy∂ z + ∂ x + i∂ y + ix∂ z (72) ∂ ∂ x − ∂ ∂ y − x ∂∂ z ∂∂y + 2 y ∂∂ z ∂∂x + ( y − x ) ∂ ∂ z + ∂ ∂ z (73) are solvable and hypoellipticProof: The solvability results from theorem
For the hypoellipticity,we consider the mapping Γ : D ′ ( G ) → D ′ ( G ) defined byΓ φ ( z, y, x ) = φ ( z − xy, y, x ) (74)The operator Γ is hypoelliptic and its inverse isΓ − φ ( z, y, x ) = φ ( z + xy, y, x ) (75)thus we getΓ( ∂ x + i∂ y )Γ − φ ( z, y, x ) = ( y∂ z + ∂ x + i∂ y + ix∂ z ) φ ( z, y, x (76)and 23( ∂ ∂ x + ∂ ∂ y + ∂ ∂ z )Γ − φ ( z, y, x ) (77)= ( ∂ ∂ x − ∂ ∂ y − x ∂∂ z ∂∂y + 2 y ∂∂ z ∂∂x + ( y − x ) ∂ ∂ z + ∂ ∂ z ) φ ( z, y, x )(78)Since the operators Γ , ∂ x + i∂ y , ∂ ∂ x + ∂ ∂ y + ∂ ∂ z and Γ − are hypoelliptic,then the hypoellipticity of the operators ( y∂ z + ∂ x + i∂ y + ix∂ z ) and ∂ ∂ x − ∂ ∂ y − x ∂∂ z ∂∂y + 2 y ∂∂ z ∂∂x + ( y − x ) ∂ ∂ z + ∂ ∂ z is fulfilled Hormander condition for the hypoellipticity
By the sufficient condition of the hypoellipticity given by the Hormandertheorem [3 , page 11], we oblige already quoted the sublaplacian ∂ ∂ x + ∂ ∂ y + 4 x ∂∂ z ∂∂y − y ∂∂ z ∂∂x + 4( y + x ) ∂ ∂ z (79)which is hypoelliptic by the Hormander theorem, while the operator ∂ ∂ x + ∂ ∂ y + 4 x ∂∂ z ∂∂y − y ∂∂ z ∂∂x + 4( y + x ) ∂ ∂ z − i ∂∂ z (80)is not hypoelliptic because the Hormander condition is not fulfilled. Bycontrast all our results, which are obtained by above theorems, contradictthe Hormonder conditions for the solvability and the hypoellipticity.The basis of the Lie algebra of the group N is given by the followingvector fields Z = ∂∂ z , Y = ( x ∂∂ z + ∂∂y ) , X = ( − y ∂∂ z + ∂∂x ). Since [ X, Y ] = 2 Z, and X, Y, [ X, Y ] span the Lie algebra of N . Then the Hormander theoremin [5], gives the hypoellipticity of the operator X + Y = ( x ∂∂ z + ∂∂y ) + ( − y ∂∂ z + ∂∂x ) (81)While my results prove the solvability and hypoellipticity operators X + Y + Z = ( x ∂∂ z + ∂∂y ) + ( − y ∂∂ z + ∂∂x ) + ∂ ∂ z (82)24s well known the Laplace operator∆ = X i =1 ∂ ∂ x i (83)on the real vector group R is solvable and hypoelliptic. This operator as aleft invariant differential on the group N is nothing but the following operator∆ h = X i =1 ∂ ∂ z + ( x ∂∂ z + ∂∂ y ) + ( − y ∂∂ z + ∂∂ x ) (84)and as a right invariant on N is the operator∆ h = X i =1 ∂ ∂ z + ( − x ∂∂ z + ∂∂ y ) + ( y ∂∂ z + ∂∂ x ) (85)where ∆ h ( resp. ∆ h ) is the left (resp. right) invariant differential operatorassociated to ∆ . The operators ∆ h and ∆ h can be regarded as the Laplacianoperators on the 3 − dimensional Symplectic Nilpotent group N .My aim result is Theorem 6.4.
The Laplace operators ∆ h and ∆ h on the Heisenberggroup are hypoellipticProof: We consider the following mappings from D ′ ( N ) → D ′ ( N ) definedby ΛΨ( z, y , x ) = Ψ( z + xy , y , x ) (86) τ Ψ( z, y , x ) = Ψ( z + xy , − y , x ) (87) π Ψ( z, y , x ) = Ψ( z + xy , y , − x ) (88)These operators has the property of hypoellipticity, because if ΛΨ( z, y, x ) =Ψ( z + xy, − y, x ) ∈ C ∞ ( N ) , then Ψ( z, y, x ) ∈ C ∞ ( N ), so on τ and π. In otherside we have τ ∆ΛΨ( z, y, x ) = ∆ h Ψ( z, − y, x ) (89) π ∆ΛΨ( z, y, x ) = ∆ h Ψ( z, y, − x ) (90)Since ∆, Λ , τ and π are hypoelliptic, then the hypoellipticity of ∆ h and∆ h are accomplished 25 On the Existence Theorem on N Out of the proofs of my book [6], I solve here by different method the equation
P C ∞ ( G ) = C ∞ ( G )For this, I introduce two groups: The first is the group G × R , which isthe direct product of the group G with the real vector group R . The secondis the group E = R × R × R with law: g · g ′ = ( X, x, y )( X ′ , x ′ , y ′ ) = ( X + X ′ + yX ′ , , x + x ′ , y + y ′ ) (91)for all g = ( X, x, y ) ∈ R , g ′ = ( X ′ , x ′ , y ′ ) ∈ R , X ∈ R and X ′ ∈ R . Inthis case the group G can be identified with the closed sub − group R × { } × R of E and the group A = R × R , direct product of the group R by thegroup R with the closed sub − group R × R × { } of E Definition 7.1.
For every φ ∈ C ∞ ( G ) , one can define a functions τ φ belong to C ∞ ( G × R ) , and ιφ belong to E as follows: τ φ ( X, x, y ) = φ ( x − X, x + y ) (92) ιφ ( X, x, y ) = φ ( xX, x + y ) (93) for any ( X, x, y ) ∈ G × R m . The functions τ φ and ιφ are invariant in thefollowing sense τ φ ( kX, x + k, y − k ) = φ ( z, x, y ) (94) ιφ ( kX, x − k, y + k ) = φ ( z, x, y ) (95)Now, I state my theorem Theorem 7.1.
Let P be a right invariant differential on G, and let u bethe distribution associated to P. Then the equation
P φ ( X, x ) = u ∗ φ ( X, x ) = Z G φ (( w, v ) − ( X, x ) u ( X, x ) dwdv = ϕ ( X, x ) (96) has a solution φ ∈ C ∞ ( G ) , for any function ϕ ∈ C ∞ ( G ) , where ∗ signifiesthe convolution product on G. roof: Consider the operator P as a differential operator Q on the abeliangroup A = R × { } × R . By the theory of partial differential equations withconstant coefficients on R × R , then for any function g ∈ C ∞ ( R × R ) , thereexist a function ψ on R × R , such that Qψ ( X, x ) = u ∗ c ψ ( X, x ) = Z R ψ ( X − a, x − b ) u ( X, x ) dadb = g ( X, x ) (97)Using the extension of the function ψ on the group G × R , then for each f ∈ C ∞ ( R × R ) , I get= ( u ∗ c τ ψ )( X, , y ) ↓ A = f ( X, y ) (98)Let τ f be the extension of the function f on the group G × R , that means Qτ ψ ( X, , y ) ↓ A = ( u ∗ c τ ψ )( X, , y ) ↓ A = (99) τ f ( X, , y ) ↓ A = f ( X, y )where ( u ∗ c ψ )( X, y ) = Z R ψ ( X − a, y − b ) u ( X, y ) dadb (100)= τ f ( X, , y ) ↓ A = f ( X, y ) (101)Let ⊤ x be the right translation of the group G , which is defined as= ⊤ x Ψ( X, t ) = Ψ((
X, t )((0 , x )) = Ψ(
X, t + x ) (102)Then I have τ ψ is the solution of the equation( u ∗ τ ψ )( X, x, ↓ G = f ( x − X, x ) (103)In fact, we have= ⊤ x ( u ∗ c τ ψ )( X, , u ∗ τ ψ )( X, x, ↓ G = ( u ∗ τ ψ )( X, x, ⊤ x τ f ( X, , ↓ G = τ f ( X, x,
0) = f ( x − X, x ) (104)So I get, if ψ is the solution of the equation on the abelian group A = R × R ( Qψ )( X, y ) = f ( X, y ) (105)27n the abelian group A = R × R , then the function τ ψ is the solution of theequation ( P τ ψ )( X, x ) = f ( x − X, x ) (106)on the group G. Let e ψ ( X, x ) be the function, which is defined as e ψ ( X, x ) = ψ ( xX, x ) (107)In the same way, I have proved by in [6] , if e ψ ( X, x ) is the solution of theequation Q e ψ ( X, x ) = e ϕ ( X, x ) (108)on the group A, then the function ψ is the solution of the equation P ψ ( X, x ) = ϕ ( X, x ) (109)on the group G. Corollary 7.1.
The Lewy equation is solvable in the sense, for any g ∈ C ∞ ( R ) there is a function f ∈ C ∞ ( R ) , such that L = ( − ∂∂x − i ∂∂y + 2 i ( x + i y ) ∂∂z ) f = g (110)The Lewy equation is invariant on the 3 − dimensional nilpotent symplec-tic group N = R ⋊ ρ R . So it is solvable. The Example of Hormander for the non solvability
Homander had considered in his book [16 , p. , another form of theLewy operator, which is P ( x, D ) = ( − i∂ x + ∂ y − x∂ z − iy∂ z ) (111)He constructed his example the operator of real variable coefficients,which is Q ( x, D ) = P ( x, D ) P ( x, D ) P ( x, D ) P ( x, D ) (112)and proved Q ( x, D ) is unsolvable see [16 , p P ( x, D ) is the operatordefined by P ( x, D ) = ( i∂ x + ∂ y − x∂ z + 2 iy∂ z ) (113)My result is: Theorem 7.2.
The operator Q ( x, D ) is solvable roof: Let R be the following Cauchy-Riemann operator R = − i∂ x + ∂ y (114)and let φ be any function infinitely differentiable on R , then we get= ~ ( − i∂ x ) ~ φ ( z, y, − x ) = − i∂ x ~ ϕ ( z + 2 yx, y, x )= ( − i ddt ) ~ φ ( z + 2 yx, y, x + t )= ( − i ddt ) φ ( z − yt ) , y, − x − t )= ( − i∂ x − yi∂ z ) φ ( z, y, − x ) (115)and ~ ( ∂ y ) ~ φ ( z, y, − x ) = ∂ y ~ φ ( z + 2 xy, y, x )= ( dds ) ~ φ ( z + 2 yx, y + s, x )= ( dds ) φ ( z − sx, y + s, − x )= ( ∂ y − x∂ z ) φ ( z, y, − x )So, we get( P ( x, D ) φ )( z, y, − x ) = ( − i∂ x + ∂ y − x∂ z − iy∂ z ) φ = ~ R ~ φ ( z, y, − x ) (116)In the same manner, I prove( P ( x, D ) φ )( z, y, − x ) = ( i∂ x + ∂ y − x∂ z + 2 iy∂ z ) φ = ~ R ⋆ ~ φ ( z, y, − x ) (117)where R ⋆ R ⋆ = i∂ x + ∂ y (118)Finally, I find(( P ( x, D )( P ( x, D ) φ )( z, y, − x ) = ~ R ⋆ R ~ φ ( z, y, − x ) (119)(( P ( x, D )( P ( x, D )) φ )( z, y, − x ) = ~ RR ⋆ ~ φ ( z, y, − x ) (120)29((( P ( x, D ) P ( x, D ) P ( x, D )))))( P ( x, D ) φ )( z, y, − x )= ~ RR ⋆ R ⋆ R ~ φ ( z, y, − x ) = Q ( x, D ) φ ( z, y, − x ) (121)Hence the solvability of the operator Q ( x, D ) . Also the operator X + iY − iZ = ix ∂∂ z + i ∂∂y − y ∂∂ z + ∂∂x − i ∂∂ z (122)is solvable. So the invalidity of the Hormander condition for the non solv-ability Any invariant differential operator has the form P = X α,β a α,β X α Y β (123)on the Lie group G = R × ρ R , where X α = ( X α , X α ) , Y β , α i ∈ N ∈ N (1 ≤ i ≤
2) and X = ( X , X ) , are the invariant vectors field on G, whichare the basis of the Lie algebra g of G and a α,β ∈ C . Any invariant partialdifferential equation on the 3 − dimensional group G = R × ρ R is solvable.So the invalidity of the Hormander condition for the non existence.Over fifty years ago where there are a lot of books and lot of publishedpapers by many mathematicians as [2 , , , , , , , , are all based ona non careful mathematical ideas. Especially those research published after2006 the date of opening my new way in Fourier analysis on non abelianLie groups. Unfortunately, some of those research books and articles werepublished in the famous scientific centers such as Springer [3 , , The operator [3 , p. y − z ) ∂ u∂x + (1 + x )( ∂ u∂y − ∂ u∂z ) − xy ∂ u∂x∂y −− ∂ ( xyu ) ∂x∂y + xz ∂ u∂x∂z + ∂ ( xyu ) ∂x∂z (124)30an be solved Is the operator X + Y − iZ = ( x ∂∂ z + ∂∂y ) + ( − y ∂∂ z + ∂∂x ) − i ∂∂ z (125)solvable and hypoelliptic, and the operator L − αi∂ z = ∂ x + 2 y∂ z + i∂ y − ix∂ z − αi∂ z , α ∈ R (126)is hypoelliptic Open Question.
Consider the Kannai operators [3 , p. D = ∂∂ x + x ∂ ∂ y , D = ∂∂ x − x ∂ ∂ y (127)I believe, the first can be solved on the 3 − dimensional Heisenberg group H and the second can be hypoelliptic. References [1] E. Barletta , S. dragomir, On Lewy’s Unsolvability Phenomenon, inComplex variables and Elliptic Equations · January 2011, PublisherFrancis &Taylor[2] U. N. Bassey and M. E. Egwe, “Non Solvability of Heisenberg Laplacianby Factorization,” Journal of Mathematical Sciences, Vol. 21, No. 1,2010, pp. 11-15.[3] M. Bramanti, An Invitation to Hypoelliptic Operators and Hormander’sVector Fields, Series: Springer Briefs in Mathematics, 2014.[4] A. Cerezo and F. Rouviere, (1969)”Solution elemetaire d’un operatordifferentielle lineare invariant agauch sur un group de Lie reel compact”Annales Scientiques de E.N.S. 4 serie, tome 2, n o , p 561-581.[5] L. Corwin, L.P. Rothschild, Necessary Conditions for Local Solvabilityof Homogeneous Left Invariant Operators on Nilpotent Lie Groups, ActaMath., 147 (1981), pp. 265–288.316] K. El- Hussein., ( Notes on Differential Geometry and Lie Groups , De-partment of Computer and Information Science, University of Pennsyl-vania Philadelphia, PA 19104, USA.[12] Harish-Chandra; (1952),
Plancherel formula for × real unimodulargroup , Proc. nat. Acad. Sci. U.S.A., vol. 38, pp. 337-342.[13] Harish-Chandra, The Plancherel formula for complex semisimple Liegroups , Trans. Amer. Math. Soc., Vol. 76, No. 3, 1954, 458-528.[14] S. Helgason., (2005),
The Abel, Fourier and Radon Transforms on Sym-metric Spaces . Indagationes Mathematicae. 16, 531-551.[15] S. Helgason, (1984),