On Kac's principle of not feeling the boundary for the Kohn Laplacian on the Heisenberg group
aa r X i v : . [ m a t h . A P ] M a r ON KAC’S PRINCIPLE OF NOT FEELING THE BOUNDARY FORTHE KOHN LAPLACIAN ON THE HEISENBERG GROUP
MICHAEL RUZHANSKY AND DURVUDKHAN SURAGAN
Abstract.
In this note we construct an integral boundary condition for the KohnLaplacian in a given domain on the Heisenberg group extending to the setting ofthe Heisenberg group M. Kac’s “principle of not feeling the boundary”. This alsoamounts to finding the trace on smooth surfaces of the Newton potential associatedto the Kohn Laplacian. We also obtain similar results for higher powers of theKohn Laplacian. Introduction
In a bounded domain of the Euclidean space Ω ⊂ R d , d ≥ , it is very well knownthat the solution to the Laplacian equation(1.1) ∆ u ( x ) = f ( x ) , x ∈ Ω , is given by the Green formula (or the Newton potential formula)(1.2) u ( x ) = Z Ω ε d ( x − y ) f ( y ) dy, x ∈ Ω , for suitable functions f supported in Ω. Here ε d is the fundamental solution to ∆ in R d given by(1.3) ε d ( x − y ) = (cid:26) − d ) s d | x − y | d − , d ≥ , π log | x − y | , d = 2 , where s d = π d Γ( d ) is the surface area of the unit sphere in R d .An interesting question having several important applications is what boundaryconditions can be put on u on the (smooth) boundary ∂ Ω so that equation (1.1)complemented by this boundary condition would have the solution in Ω still given bythe same formula (1.2), with the same kernel ε d given by (1.3). It turns out that theanswer to this question is the integral boundary condition(1.4) − u ( x ) + Z ∂ Ω ∂ε d ( x − y ) ∂n y u ( y ) dS y − Z ∂ Ω ε d ( x − y ) ∂u ( y ) ∂n y dS y = 0 , x ∈ ∂ Ω , where ∂∂n y denotes the outer normal derivative at a point y on ∂ Ω. A converse questionto the one above would be to determine the trace of the Newton potential (1.2) on
Mathematics Subject Classification.
Key words and phrases. sub-Laplacian, Kohn Laplacian, integral boundary conditions, Heisen-berg group, Newton potential.The authors were supported in parts by the EPSRC grant EP/K039407/1 and by the LeverhulmeGrant RPG-2014-02, as well as by the MESRK grant 5127/GF4. the boundary surface ∂ Ω, and one can use the potential theory to show that it hasto be given by (1.4).The boundary condition (1.4) appeared in M. Kac’s work [10] where he called itand the subsequent spectral analysis “the principle of not feeling the boundary”.This was further expanded in Kac’s book [11] with several further applications tothe spectral theory and the asymptotics of the Weyl’s eigenvalue counting function.In [12] by using the boundary condition (1.4) the eigenvalues and eigenfunctions ofthe Newton potential (1.2) were explicitly calculated in the 2-disk and in the 3-ball.In general, the boundary value problem (1.1)-(1.4) has various interesting propertiesand applications (see, for example, Kac [10, 11] and Saito [19]). The boundary valueproblem (1.1)-(1.4) can also be generalised for higher degrees of the Laplacian, see[13, 14].In this note we are interested in and we give answers to the following questions: • What happens if an elliptic operator (the Laplacian) in (1.1) is replaced bya hypoelliptic operator? We will realise this as a model of replacing theEuclidean space by the Heisenberg group and the Laplacian on R d by a sub-Laplacian (or the Kohn-Laplacian) on H n − . We will show that the boundarycondition (1.4) is replaced by the integral boundary condition (2.5) in thissetting (see also (1.11)). • Since the theory of boundary value problems for elliptic operators is wellunderstood, we know that the single condition (1.4) on the boundary ∂ Ω of abounded domain Ω guarantees the unique solvability of the equation (1.1) inΩ. Is this uniqueness preserved in the hypoelliptic model as well for a suitablychosen replacement of the boundary condition (1.4)? The case of the secondorder operators is favourable from this point of view due to the validity of themaximum principle, see Bony [1]. The Dirichlet problem has been consideredby Jerison [9]. The answer in the case of the boundary value problem in oursetting is given in Theorem 2.1. • What happens if we consider the above questions for higher order equations?In general, it is known that for higher order Rockland operators on stratifiedgroups, fundamental solutions may be not unique, see Folland [5] and Geller[6], and for a unifying discussion see also the book [2]. However, for powersof the Kohn Laplacian we still have the uniqueness provided that we imposehigher order boundary conditions in a suitable way, see Theorem 3.1.We now describe the setting of this paper. The Heisenberg group H n − is the space C n − × R with the group operation given by(1.5) ( ζ , t ) ◦ ( η, τ ) = ( ζ + η, t + τ + 2 Im ζ η ) , for ( ζ , t ) , ( η, τ ) ∈ C n − × R . Writing ζ = x + iy with x j , y j , j = 1 , ..., n − , the realcoordinates on H n − , the left-invariant vector fields˜ X j = ∂∂x j + 2 y j ∂∂t , j = 1 , ..., n − , ˜ Y j = ∂∂y j − x j ∂∂t , j = 1 , ..., n − , OUNDARY CONDITIONS FOR THE KOHN LAPLACIAN 3 T = ∂∂t , form a basis for the Lie algebra h n − of H n − .On the other hand, H n − can be viewed as the boundary of the Siegel upper halfspace in C n , H n − = { ( ζ , z n ) ∈ C n : Im z n = | ζ | , ζ = ( z , ..., z n − ) } . Parameterizing H n − by z = ( ζ , t ) where t = Re z n , a basis for the complex tangentspace of H n − at the point z is given by the left-invariant vector fields X j = ∂∂z j + iz ∂∂t , j = 1 , . . . , n − . We denote their conjugates by X j ≡ X j = ∂∂z j − iz ∂∂t . The operator(1.6) (cid:3) a,b = n − X j =1 ( aX j X j + bX j X j ) , a + b = n − , is a left-invariant, rotation invariant differential operator that is homogeneous ofdegree two (cf. [3]). This operator is a slight generalisation of the standard sub-Laplacian or Kohn-Laplacian (cid:3) b on the Heisenberg group H n − which, when actingon the coefficients of a (0 , q )-form can be written as (cid:3) b = − n − n − X j =1 (( n − − q ) X j X j + qX j X j ) . Folland and Stein [4] found that a fundamental solution of the operator (cid:3) a,b is aconstant multiple of(1.7) ε ( z ) = ε ( ζ , t ) = 1( t + i | ζ | ) a ( t − i | ζ | ) b , and defined the Newton potential (volume potential) for a function f with compactsupport contained in a set Ω ⊂ H n − by(1.8) u ( z ) = Z Ω f ( ξ ) ε ( ξ − z ) dν ( ξ ) , with dν being the volume element (the Haar measure on H n − ), coinciding with theLebesgue measure on C n − × R . More precisely, they proved that (cid:3) a,b u = c a,b f, where the constant c a,b is zero if a and b = − , − , . . . , n, n + 1 , . . . , and c a,b = 0 if a or b = − , − , . . . , n, n + 1 , . . . In fact, then we can take c a,b = 2( a + b )Vol( B )(2 i ) n ( n − a ( a − ... ( a − n + 1) (1 − exp( − iaπ ))for a Z , see the proof of Theorem 1.6 in Romero [16]. Similar conclusions by adifferent methods were obtained by Greiner and Stein [8]. For a more general analysisof fundamental solutions for sub-Laplacians we can refer to Folland [5] as well as toa discussion and references in Stein [20]. The Kohn Laplacian and its generalisations MICHAEL RUZHANSKY AND DURVUDKHAN SURAGAN may be considered as natural models for dealing with sums of squares also on moregeneral manifolds, as it is now well known, see e.g. Rothschild and Stein [17].In the above notation, the distribution c a,b ε is the fundamental solution of (cid:3) a,b ,while ε satisfies the equation(1.9) (cid:3) a,b ε = c a,b δ. However, although we could have rescaled ε for it to become the fundamental solution,we prefer to keep the notation yielding (1.9) in order to follow the notation of [4] and[16] to be able to refer to their results directly.Throughout this paper we assume that c a,b = 0, i.e. both a and b = − , − , . . . , n, n + 1 , . . . . In addition, without loss of generality we may also assume that a, b ≥ R d , we consider the hypoelliptic boundary value problem for the sub-Laplacian (cid:3) a,b on H n − , namely the equation(1.10) (cid:3) a,b u = c a,b f in a bounded set Ω ⊂ H n − with smooth boundary ∂ Ω. The first aim of this paperis to find a boundary condition of the Newton potential u on ∂ Ω such that with thisboundary condition the equation (1.10) has a unique solution, which is the Newtonpotential (1.8).Basing our arguments on the analysis of Folland and Stein [4] and Romero [16] weshow that the boundary condition (1.4) for the Laplacian in R d is now replaced bythe integral boundary condition (2.5) in this setting, namely by the condition(1.11) ( c a,b − H.R ( z )) u ( z ) − Z ∂ Ω ε ( ξ, z ) h∇ b,a u ( ξ ) , dν ( ξ ) i + p.v. W u ( z ) = 0 , z ∈ ∂ Ω , on the boundary ∂ Ω, where
H.R ( z ) is the so-called half residue, and where the secondand the third term can be interpreted as coming from the suitably defined respectivelysingle and double layer potentials S and W for the problem. See Section 2 for thedefinitions and the precise statement.In Section 2 by using properties of fundamental solutions we construct a well-posed boundary value problem for the differential equation (1.10) with the requiredproperties. In Section 3 we generalise this result for higher powers of the KohnLaplacian. Throughout this paper we may use notations from [16], [15] and [18].2. The Kohn Laplacian
Let Ω ⊂ H n − be an open bounded domain with a smooth boundary ∂ Ω ∈ C ∞ .Consider the following analogy of the Newton potential on the Heisenberg group(2.1) u ( z ) = Z Ω f ( ξ ) ε ( ξ, z ) dν ( ξ ) in Ω , where ε ( ξ, z ) = ε ( ξ − z ) is the rescaled fundamental solution (1.7) of the sub-Laplacian,satisfying (1.9). As we mentioned u is a solution of (1.10) in Ω. The aim of this sec-tion is to find a boundary condition for u such that with this boundary condition the OUNDARY CONDITIONS FOR THE KOHN LAPLACIAN 5 equation (1.10) has a unique solution in C (Ω), say, and this solution is the Newtonpotential (2.1).We recall a few notions and properties first. For z = ( ζ , t ) ∈ H n − , we define itsnorm by | z | := ( | ζ | + | t | ) / . As any (quasi-)norm on H n − , this satisfies a triangleinequality with a constant, and allows for a polar decomposition. For 0 < α < α (Ω) byΓ α (Ω) = f : Ω → C : sup z ,z ∈ Ω z = z | f ( z ) − f ( z ) || z − z | α < ∞ . For k ∈ N and 0 < α <
1, one defines Γ k + α (Ω) as the space of all f : Ω → C suchthat all complex derivatives of f of order k belong to Γ α (Ω).A starting point for us will be that if f ∈ Γ α (Ω) for α > u defined by (2.1) istwice differentiable in the complex directions and satisfies the equation (cid:3) a,b u = c a,b f .We refer to Folland and Stein [4], Greiner and Stein [8], and to Romero [16] for threedifferent approaches to this property. Moreover, Folland and Stein have shown thatif f ∈ Γ α (Ω , loc ) and (cid:3) a,b u = c a,b f , then f ∈ Γ α +2 (Ω , loc ). These results extend thoseknown for the Laplacian, in suitably redefined anisotropic H¨older spaces.We record relevant single and double layer potentials for the problem (1.10). In[9], Jerison used the single layer potential defined by S g ( z ) = Z ∂ Ω g ( ξ ) ε ( ξ, z ) dS ( ξ ) , which, however, is not integrable over characteristic points. On the contrary, thefunctional Sg ( z ) = Z ∂ Ω g ( ξ ) ε ( ξ, z ) h X j , dν ( ξ ) i , where h X, dν i is the canonical pairing between vector fields and differential forms, isintegrable over the whole boundary ∂ Ω. Moreover, it was shown in [16, Theorem 2.3]that if the density of g ( ξ ) h X j , dν i in the operator S is bounded then Sg ∈ Γ α ( H n − )for all α <
1. Parallel to S , it is natural to use the operator(2.2) W u ( z ) = Z ∂ Ω u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i as a double layer potential. Our main result for the sub-Laplacian is the followingjustification of formula (1.11) in the introduction: Theorem 2.1.
Let ε ( ξ, z ) = ε ( ξ − z ) be the rescaled fundamental solution to (cid:3) a,b , sothat (2.3) (cid:3) a,b ε = c a,b δ on H n − . For any f ∈ Γ α (Ω) , the Newton potential (2.1) is the unique solution in C (Ω) ∩ C (Ω) of the equation (2.4) (cid:3) a,b u = c a,b f MICHAEL RUZHANSKY AND DURVUDKHAN SURAGAN with the boundary condition (2.5) ( c a,b − H.R ( z )) u ( z ) + lim δ → Z ∂ Ω \{| ξ − z | <δ } u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i− Z ∂ Ω ε ( ξ, z ) h∇ b,a u ( ξ ) , dν ( ξ ) i = 0 , for z ∈ ∂ Ω , where H.R ( z ) is the so-called half residue given by the formula (2.6) H.R ( z ) = lim δ → Z ∂ Ω \{| ξ − z | <δ } h∇ a,b ε ( ξ, z ) , dν ( ξ ) i , with ∇ a,b g = n − X j =1 ( aX j gX j + bX j gX j ) . The half residue
H.R. ( z ) in (2.6) appears in the jump relations for the problem(2.4) in the following way. The double layer potential W u in (2.2) has two limits W + u ( z ) = lim z → z z ∈ Ω Z ∂ Ω u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i and W − u ( z ) = lim z → z z Ω Z ∂ Ω u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i , and the principal value W u ( z ) = p.v. W u ( z ) = lim δ → Z ∂ Ω \{| ξ − z | <δ } u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i . We note that this principal value enters as the second term in the integral boundarycondition (2.5). It was proved in [16, Theorem 2.4] that for sufficiently regular u (e.g. u ∈ Γ α (Ω)) and z ∈ ∂ Ω these limits exist and satisfy the jump relations W + u ( z ) − W − u ( z ) = c a,b u ( z ) ,W u ( z ) − W − u ( z ) = H.R. ( z ) u ( z ) ,W + u ( z ) − W u ( z ) = ( c a,b − H.R. ( z )) u ( z ) , (2.7)the last property (2.7) following from the first two by subtraction. Proof of Theorem 2.1.
Since the solid potential(2.8) u ( z ) = Z Ω f ( ξ ) ε ( ξ, z ) dν ( ξ )is a solution of (2.4), from the aforementioned results of Folland and Stein it followsthat u is locally in Γ α +2 (Ω , loc ) and that it is twice complex differentiable in Ω. Inparticular, it follows that u ∈ C (Ω) ∩ C (Ω).The following representation formula can be derived from the generalised secondGreen’s formula (see Theorem 4.5 in [16] and cf. [15]), for u ∈ C (Ω) ∩ C (Ω) we OUNDARY CONDITIONS FOR THE KOHN LAPLACIAN 7 have(2.9) c a,b u ( z ) = c a,b Z Ω f ( ξ ) ε ( ξ, z ) dν ( ξ ) + Z ∂ Ω u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i− Z ∂ Ω ε ( ξ, z ) h∇ b,a u ( ξ ) , dν ( ξ ) i , for any z ∈ Ω . Since u ( z ) given by (2.8) is a solution of (2.4), using it in (2.9) we get(2.10) Z ∂ Ω u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i − Z ∂ Ω ε ( ξ, z ) h∇ b,a u ( ξ ) , dν ( ξ ) i = 0 , for any z ∈ Ω . It is easy to see that the fundamental solution, i.e. the function ε ( z ) in (1.7) ishomogeneous of degree − n + 2, that is ε ( λz ) = λ − a − b ε ( z ) = λ − n +2 ε ( z ) for any λ > , since a + b = n −
1. It follows that ε and its first order complex derivatives are locallyintegrable. Since ε ( ξ, z ) = ε ( ξ − z ), we obtain that as z approaches the boundary, wecan pass to the limit in the second term in (2.10).By using this and the relation (2.7) as z ∈ Ω approaches the boundary ∂ Ω frominside, we find that(2.11) ( c a,b − H.R ( z )) u ( z ) + lim δ → Z ∂ Ω \{| ξ − z | <δ } u ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i− Z ∂ Ω ε ( ξ, z ) h∇ b,a u ( ξ ) , dν ( ξ ) i = 0 , for any z ∈ ∂ Ω . This shows that (2.1) is a solution of the boundary value problem (2.4) with theboundary condition (2.5).Now let us prove its uniqueness. If the boundary value problem has two solutions u and u then the function w = u − u ∈ C (Ω) ∩ C (Ω) satisfies the homogeneousequation(2.12) (cid:3) a,b w = 0 in Ω , and the boundary condition (2.5), i.e.(2.13) ( c a,b − H.R ( z )) w ( z ) + lim δ → Z ∂ Ω \{| ξ − z | <δ } w ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i− Z ∂ Ω ε ( ξ, z ) h∇ b,a w ( ξ ) , dν ( ξ ) i = 0 , for any z ∈ ∂ Ω . Since f ≡ c a,b w ( z ) = Z ∂ Ω w ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i − Z ∂ Ω ε ( ξ, z ) h∇ b,a w ( ξ ) , dν ( ξ ) i MICHAEL RUZHANSKY AND DURVUDKHAN SURAGAN for any z ∈ Ω. As above, by using the properties of the double and single layerpotentials as z → ∂ Ω, we obtain(2.15) c a,b w ( z ) = ( c a,b − H.R ( z )) w ( z )+lim δ → Z ∂ Ω \{| ξ − z | <δ } w ( ξ ) h∇ a,b ε ( ξ, z ) , dν ( ξ ) i − Z ∂ Ω ε ( ξ, z ) h∇ b,a w, dν ( ξ ) i for any z ∈ ∂ Ω . Comparing this with (2.13) we arrive at(2.16) w ( z ) = 0 , z ∈ ∂ Ω . The homogeneous equation (2.12) with the Dirichlet boundary condition (2.16)has only trivial solution w ≡ C (Ω) ∩ C (Ω). This completes the proof of Theorem 2.1. (cid:3) Powers of the Kohn Laplacian
As before, let Ω ⊂ H n − be an open bounded domain with a smooth boundary ∂ Ω ∈ C ∞ . For m ∈ N , we denote (cid:3) ma,b := (cid:3) a,b (cid:3) m − a,b . Then for m = 1 , , . . . , weconsider the equation(3.1) (cid:3) ma,b u ( z ) = c a,b f ( z ) , z ∈ Ω . Let ε ( ξ, z ) = ε ( ξ − z ) be the rescaled fundamental solution of the Kohn Laplacianas in (2.3). Let us now define(3.2) u ( z ) = Z Ω f ( ξ ) ε m ( ξ, z ) dν ( ξ )in Ω ⊂ H n − , where ε m ( ξ, z ) is a rescaled fundamental solution of (3.1) such that (cid:3) m − a,b ε m = ε. We take, with a proper distributional interpretation, for m = 2 , , . . . ,(3.3) ε m ( ξ, z ) = Z Ω ε m − ( ξ, ζ ) ε ( ζ , z ) dν ( ζ ) , ξ, z ∈ Ω , with ε ( ξ, z ) = ε ( ξ, z ) . A simple calculation shows that the generalised Newton potential (3.2) is a solutionof (3.1) in Ω. The aim of this section is to find a boundary condition on ∂ Ω such thatwith this boundary condition the equation (3.1) has a unique solution in C m (Ω),which coincides with (3.2).Although fundamental solutions for higher order hypoelliptic operators on theHeisenberg group may not have unique fundamental solutions, see Geller [6], in thecase of the iterated sub-Laplacian (cid:3) ma,b we still have the uniqueness for our problemin the sense of the following theorem, and the uniqueness argument in its proof. OUNDARY CONDITIONS FOR THE KOHN LAPLACIAN 9
Theorem 3.1.
For any f ∈ Γ α (Ω) , the generalised Newton potential (3.2) is a uniquesolution of the equation (3.1) in C m (Ω) ∩ C m − (Ω) with m boundary conditions (3.4) ( c a,b − H.R ( z )) (cid:3) ia,b u ( z )+ m − i − X j =0 lim δ → Z ∂ Ω \{| ξ − z | <δ } (cid:3) j + ia,b u ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i− m − i − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) j + ia,b u ( ξ ) dν ( ξ ) i = 0 , z ∈ ∂ Ω , for all i = 0 , , . . . , m − , where ∇ a,b g = n − X j =1 ( aX j gX j + bX j gX j ) and H.R ( z ) is the half residue given by the formula (2.6).Proof. By applying Green’s second formula for each z ∈ Ω, as in (2.9), we obtain(3.5) c a,b u ( z ) = c a,b Z Ω f ( ξ ) ε m ( ξ, z ) dν ( ξ ) = Z Ω (cid:3) ma,b u ( ξ ) ε m ( ξ, z ) dν ( ξ ) = Z Ω (cid:3) m − a,b u ( ξ ) (cid:3) a,b ε m ( ξ, z ) dν ( ξ ) − Z ∂ Ω (cid:3) m − a,b u ( ξ ) h∇ a,b ε m ( ξ, z ) , dν ( ξ ) i + Z ∂ Ω ε m ( ξ, z ) h∇ b,a (cid:3) m − a,b u ( ξ ) , dν ( ξ ) i = Z Ω (cid:3) m − a,b u ( ξ ) (cid:3) a,b ε m ( ξ, z ) dν ( ξ ) − Z ∂ Ω (cid:3) m − a,b u ( ξ ) h∇ a,b (cid:3) a,b ε m ( ξ, z ) , dν ( ξ ) i + Z ∂ Ω (cid:3) a,b ε m ( ξ, z ) h∇ b,a (cid:3) m − a,b u ( ξ ) , dν ( ξ ) i− Z ∂ Ω (cid:3) m − a,b u ( ξ ) h∇ a,b ε m ( ξ, z ) , dν ( ξ ) i + Z ∂ Ω ε m ( ξ, z ) h∇ b,a (cid:3) m − a,b u ( ξ ) , dν ( ξ ) i = ... = c a,b u ( z ) − m − X j =0 Z ∂ Ω (cid:3) ja,b u ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i + m − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) ja,b u ( ξ ) , dν ( ξ ) i , z ∈ Ω . This implies the identity(3.6) m − X j =0 Z ∂ Ω (cid:3) ja,b u ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i− m − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) ja,b u ( ξ ) , dν ( ξ ) i = 0 , z ∈ Ω . By using the properties of the double and single layer potentials as z approachesthe boundary ∂ Ω from the interior, from (3.6) we obtain( c a,b − H.R ( z )) u ( z ) + m − X j =0 lim δ → Z ∂ Ω \{| ξ − z | <δ } (cid:3) ja,b u ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i− m − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) ja,b u ( ξ ) , dν ( ξ ) i = 0 , z ∈ ∂ Ω . Thus, this relation is one of the boundary conditions of (3.2). Let us derive theremaining boundary conditions. To this end, we write(3.7) (cid:3) m − ia,b (cid:3) ia,b u = c a,b f, i = 0 , , . . . , m − , m = 1 , , . . . , and carry out similar considerations just as above. This yields c a,b (cid:3) ia,b u ( z ) = c a,b Z Ω f ( ξ ) (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) = Z Ω (cid:3) m − ia,b (cid:3) ia,b u ( ξ ) (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) = Z Ω (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) (cid:3) a,b (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) − Z ∂ Ω (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) h∇ a,b (cid:3) ia,b ε m ( ξ, z ) , dν ( ξ ) i + Z ∂ Ω (cid:3) ia,b ε m ( ξ, z ) h∇ b,a (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) , dν ( ξ ) i = Z Ω (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) (cid:3) a,b (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) − Z ∂ Ω (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) h∇ a,b (cid:3) a,b (cid:3) ia,b ε m ( ξ, z ) , dν ( ξ ) i + Z ∂ Ω (cid:3) a,b (cid:3) ia,b ε m ( ξ, z ) h∇ b,a (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) , dν ( ξ ) i− Z ∂ Ω (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) h∇ a,b (cid:3) ia,b ε m ( ξ, z ) , dν ( ξ ) i + Z ∂ Ω (cid:3) ia,b ε m ( ξ, z ) h∇ b,a (cid:3) m − i − a,b (cid:3) ia,b u ( ξ ) , dν ( ξ ) i = ... = Z Ω (cid:3) ia,b u ( ξ ) (cid:3) m − ia,b (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) − OUNDARY CONDITIONS FOR THE KOHN LAPLACIAN 11 m − i − X j =0 Z ∂ Ω (cid:3) ja,b (cid:3) ia,b u ( ξ ) h∇ a,b (cid:3) m − i − − ja,b (cid:3) ia,b ε m ( ξ, z ) , dν ( ξ ) i + m − i − X j =0 Z ∂ Ω (cid:3) m − i − − ja,b (cid:3) ia,b ε m ( ξ, z ) h∇ b,a (cid:3) ja,b (cid:3) ia,b u ( ξ ) , dν ( ξ ) i = c a,b (cid:3) ia,b u ( z ) − m − i − X j =0 Z ∂ Ω (cid:3) j + ia,b u ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i + m − i − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) j + ia,b u ( ξ ) , dν ( ξ ) i , z ∈ Ω , where, as usual, ε m ( ξ, z ) = ε m ( ξ − z ), and (cid:3) ia,b ε m is a rescaled fundamental solutionof the equation (3.7), i.e., (cid:3) m − ia,b (cid:3) ia,b ε m = c a,b δ, i = 0 , , . . . , m − . From the previous relations, we obtain the identities m − i − X j =0 Z ∂ Ω (cid:3) j + ia,b u ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i− m − i − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) j + ia,b u ( ξ ) , dν ( ξ ) i = 0for any z ∈ Ω and i = 0 , , . . . , m − . By using the properties of the double andsingle layer potentials as z approaches the boundary ∂ Ω from the interior of Ω, wefind that( c a,b − H.R ( z )) (cid:3) ia,b u ( z )+ m − i − X j =0 lim δ → Z ∂ Ω \{| ξ − z | <δ } (cid:3) j + ia,b u ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i− m − i − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) j + ia,b u ( ξ ) , dν ( ξ ) i = 0 , z ∈ ∂ Ω , are all boundary conditions of (3.2) for each i = 0 , , . . . , m − w ∈ C m (Ω) ∩ C m − (Ω) satisfies theequation (cid:3) ma,b w = f and the boundary conditions (3.4), then it coincides with thesolution (3.2). Indeed, otherwise the function v = u − w ∈ C m (Ω) ∩ C m − (Ω) , where u is the generalised Newton potential (3.2), satisfies the homogeneous equation(3.8) (cid:3) ma,b v = 0 and the boundary conditions (3.4), i.e. I i ( v )( z ) := ( c a,b − H.R ( z )) (cid:3) ia,b v ( z )+ m − i − X j =0 lim δ → Z ∂ Ω \{| ξ − z | <δ } (cid:3) j + ia,b v ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i− m − i − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) j + ia,b v ( ξ ) , dν ( ξ ) i = 0 , i = 0 , , . . . , m − , for z ∈ ∂ Ω . By applying the Green formula to the function v ∈ C m (Ω) ∩ C m − (Ω)and by following the lines of the above argument, we obtain0 = Z Ω (cid:3) ma,b v ( z ) (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) = Z Ω (cid:3) m − ia,b (cid:3) ia,b v ( z ) (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) = Z Ω (cid:3) m − a,b v ( z ) (cid:3) a,b (cid:3) ia,b ε m ( ξ, z ) dν ( ξ ) − Z ∂ Ω (cid:3) m − a,b v ( z ) h∇ a,b (cid:3) ia,b ε m ( ξ, z ) , dν ( ξ ) i + Z ∂ Ω (cid:3) ia,b ε m ( ξ, z ) h∇ a,b (cid:3) m − a,b v ( z ) , dν ( ξ ) i = ... = c a,b (cid:3) ia,b v ( z ) − m − i − X j =0 Z ∂ Ω (cid:3) j + ia,b v ( ξ ) h∇ a,b (cid:3) m − − ja,b ε m ( ξ, z ) , dν ( ξ ) i + m − i − X j =0 Z ∂ Ω (cid:3) m − − ja,b ε m ( ξ, z ) h∇ b,a (cid:3) j + ia,b v ( ξ ) , dν ( ξ ) i , i = 0 , , . . . , m − . By passing to the limit as z → ∂ Ω, we obtain the relations(3.9) (cid:3) ia,b v ( z ) | z ∈ ∂ Ω = I i ( v )( z ) | z ∈ ∂ Ω = 0 , i = 0 , , . . . , m − . Assuming for the moment the uniqueness of the solution of the boundary valueproblem(3.10) (cid:3) ma,b v = 0 , (cid:3) ia,b v | ∂ Ω = 0 , i = 0 , , . . . , m − , we get that v = u − w ≡
0, for all z ∈ Ω, i.e. w coincides with u in Ω. Thus (3.2) isthe unique solution of the boundary value problem (3.1), (3.4) in Ω.It remains to argue that the boundary value problem (3.10) has a unique solutionin C m (Ω) ∩ C m − (Ω). Denoting ˜ v := (cid:3) m − a,b v , this follows by induction from theuniqueness in C (Ω) ∩ C (Ω) of the problem (cid:3) a,b ˜ v = 0 , ˜ v | ∂ Ω = 0 . The proof of Theorem 3.1 is complete. (cid:3)
OUNDARY CONDITIONS FOR THE KOHN LAPLACIAN 13
Remark 3.2.
It follows from Theorem 3.1 that the kernel (3.3), which is a rescaledfundamental solution of the equation (3.1), is the Green function of the boundaryvalue problem (3.1), (3.4) in Ω. Therefore, the boundary value problem (3.1), (3.4)can serve as an example of an explicitly solvable boundary value problem in anydomain Ω (with smooth boundary) on the Heisenberg group.
References [1] J.-M. Bony. Principe du maximum, in´egalite de Harnack et unicit´e du probl`eme de Cauchy pourles op´erateurs elliptiques d´eg´en´er´es. Ann. Inst. Fourier (Grenoble), 19 (1969), pp. 277-304.[2] V. Fischer and M. Ruzhansky, Quantization on nilpotent Lie groups, to appear in Progress inMathematics, Birkh¨auser, 2015.[3] G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy Riemann Complex,Princeton University Press, Princeton, NJ, 1972.[4] G. B. Folland and E. M. Stein, Estimates for the ∂ b complex and analysis on the Heisenberggroup, Comm. Pure Appl. Math. 27 (1974), pp. 429-522.[5] G. B. Folland. Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat., 13(1975), pp. 161-207.[6] D. Geller. Liouville’s theorem for homogeneous groups. Comm. Partial Differential Equations,8 (1983), pp. 1665-1677.[7] D. Geller, Analytic pseudodifferential operators for the Heisenberg group and local solvability.Mathematical Notes, 37. Princeton University Press, Princeton, NJ, 1990.[8] P. C. Greiner and E. M. Stein, Estimates for the ∂ b -Neumann problem. Princeton UniversityPress, Princeton, NJ, 1977.[9] D. S. Jerison, The Dirichlet problem for the Kohn Laplacian on the Heisenberg group. I andII. J. Funct. Anal. 43 (1981), pp. 97-142, 224-257.[10] M. Kac, On some connections between probability theory and differential and integral equations,Proceedings of the Second Berkeley Symposium on Mathematical Statistics and Probability,1950, University of California Press, Berkeley and Los Angeles, 1951, pp. 189–215.[11] M. Kac, Integration in function spaces and some of its applications, Lezioni Fermiane. Ac-cademia Nazionale dei Lincei, Pisa, 1980. 82 pp.[12] T. Sh. Kal’menov and D. Suragan, To spectral problems for the volume potential, DokladyMathematics, 80 (2009), pp. 646–649.[13] T. Sh. Kal’menov and D. Suragan, Boundary conditions for the volume potential for the poly-harmonic equation, Differential Equations, 48 (2012), pp. 604–608.[14] T. Sh. Kalmenov and D. Suragan, A boundary condition and Spectral Problems for the NewtonPotentials, Operator Theory: Advances and Applications, Vol. 216 (2011), pp. 187-210.[15] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press,Cambridge, UK, 2000.[16] C. Romero, Potential theory for the Kohn Laplacian on the Heisenberg group, PhD thesis,University of Minnesota, 1991.[17] L. P. Rothschild and E. M. Stein. Hypoelliptic differential operators and nilpotent groups, ActaMath., 137 (1976), pp. 247–320.[18] M. Ruzhansky and V. Turunen, Pseudo-differential operators and symmetries: Backgroundanalysis and advanced topics, Birkhauser, Basel, 2010.[19] N. Saito, Data analysis and representation on a general domain using eigenfunctions of Lapla-cian, Appl. Comput. Harmon. Anal. (2008), pp. 68–97.[20] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals,Princeton University Press, Princeton, NJ, 1993. Michael Ruzhansky:Department of MathematicsImperial College London
180 Queen’s Gate, London SW7 2AZUnited Kingdom
E-mail address [email protected]
Durvudkhan Suragan:Institute of Mathematics and Mathematical Modelling125 Pushkin str.050010 AlmatyKazakhstan