ON EXTENSION PROBLEM, TRACE HARDY AND HARDY’S INEQUALITIES FOR SOME FRACTIONAL LAPLACIANS

. We obtain generalised trace Hardy inequalities for fractional powers of general operators given by sums of squares of vector ﬁelds. Such in- equalities are derived by means of particular solutions of an extended equation associated to the above-mentioned operators. As a consequence, Hardy in- equalities are also deduced. Particular cases include Laplacians on stratiﬁed groups, Euclidean motion groups and special Hermite operators. Fairly explicit expressions for the constants are provided. Moreover, we show several characterisations of the solutions of the extension problems associated to operators with discrete spectrum, namely Laplacians on compact Lie groups, Hermite and special Hermite operators.

1. Introduction and main results. The aim of this paper is two-fold. On the one hand, we prove trace Hardy and Hardy's inequalities for fractional powers of operators that are given as sums of squares of vector fields on a Lie group. In some particular cases, the results will be more precise, showing explicit constants. On the other hand, we obtain several characterisations of solutions of the extension problem when the operator involved has discrete spectrum.
Hardy's inequality for fractional powers of the Laplacian ∆ on R n has been investigated by many authors, see [21,40,16,4,5], and there is a vast literature on this topic. There are several ways of proving such a Hardy's inequality, which reads as valid for 0 < s < 2 and suitable functions. Though the constant is known to be sharp, it is never achieved. On the other hand, there is another version of Hardy's inequality where the homogeneous weight function |x| −s is replaced by a non-homogeneous one: |f (x)| 2 (δ 2 +|x| 2 ) s dx, δ > 0, (1.2) where again the constant is sharp and equality is achieved for the functions (δ 2 + |x| 2 ) −(n−s)/2 and their translates. Though (1.2) is known to the experts in the field, we are not able to locate a reference where it is actually proved (a proof will be given in Remark 2.6).
Ramifications of Hardy's inequality (1.1) have been investigated in the literature and in [38] Tzirakis has improved (1.1) with sharp error terms. The main idea used in [38] is the use of trace Hardy inequality. For suitable functions u(x, ρ) on R n+1 + := R n × R + , where R + := [0, ∞), the trace Hardy inequality states that |u(x, 0)| 2 (δ 2 + |x| 2 ) s dx, δ > 0. (1.4) Such inequalities are known as trace Hardy inequalities with non-homogeneous weights. As we see later, trace Hardy inequalities lead to Hardy inequalities and such inequalities with non-homogenous weights lead to uncertainty principles for fractional powers (see e.g. Garofalo-Lanconelli [18] and Roncal-Thangavelu [27]). In this paper we will see that a more general version of the inequalities (1.3) and (1.4) can be proved by means of the use of solutions of the initial value problem ∆ + ∂ 2 ρ + 1 − s ρ ∂ ρ v(x, ρ) = 0, x ∈ R n , ρ > 0; v(x, 0) = ϕ(x), x ∈ R n . (1.5) In the above initial value problem we take ϕ ∈ L 2 (R n ) and the limit is taken in the L 2 sense. Let us explain more. The problem (1.5) is known as the extension problem for the Laplacian in the literature and has been studied e.g. by Caffarelli and Silvestre [7] (see also, for instance, [25,9]). One of the most interesting facts shown in [7] about solutions of the above problem is that when both ϕ and (−∆) s/2 ϕ are in L 2 (R n ), for 0 < s < 2. Here (−∆) s/2 is the fractional power of the Laplacian and the limit is understood in the L 2 sense. Indeed, it can be seen that, by taking into account the solutions of (1.5), we can obtain the generalised trace Hardy inequality and deduce a version of Hardy's inequality for fractional powers of L. Here we assume that X j , j = 1, 2, ..., m are left invariant vector fields on a Lie group G which satisfies Hörmander's condition, that is, the vector fields are smooth and their commutators up to certain order span the tangent space at each point. Then it is known (see [23]) that the second order differential operator L is essentially selfadjoint and hypoelliptic i.e., whenever Lf = g and g ∈ C ∞ , then f is in C ∞ . Some examples for these operators are the Euclidean Laplacian, the sublaplacian on the Heisenberg group (or more generally the sublaplacian on stratified Lie groups), Laplacian on motion groups, or Laplacian on compact Lie groups.
In order to prove a trace Hardy inequality for L we need to find solutions of the extension problem The extension problem for general second order partial differential operators has been extensively studied by Stinga-Torrea [31], Galé et al [17] and Banica et al [3].
In the above C 1 0 ([0, ∞), C 1 0 (G)) stands for the space of all compactly supported C 1 functions on [0, ∞) taking values in C 1 0 (G). The following theorem is our first main result: a trace Hardy inequality concerning operators of the form (1.7) given by sums of Hörmander vector fields on general Lie groups G. Theorem 1.1 (General trace Hardy inequality). Let 0 < s < 2 and let ϕ be a real valued function in the domain of (−L) s/2 . Assume also that ϕ −1 (−L) s/2 ϕ is locally integrable. Then for any real valued function u ∈ W s 0 (G × R + ), we have the inequality It is enough to prove the inequality in Theorem 1.1 for functions from . Then standard density arguments guarantee the validity for u ∈ W s 0 (G × R + ). Equality is attained when u is a solution of the extension problem, under an extra assumption on the heat kernel associated to the operator L, see Proposition 2.5. We will consider several particular cases (Laplacians on stratified groups, Euclidean motion groups and special Hermite operators) in which this extra assumption is easily seen to be satisfied.
In order to prove Theorem 1.1 we will strongly use the solutions of the extension problem (1.8) and the characterisation of the fractional powers of the operators L by means of this extension problem due to Stinga and Torrea, see [31]. This technique was recently developed in the Euclidean setting (by using the characterisation for the Euclidean fractional Laplacian in [7]) to get improved trace Hardy inequalities on bounded domains [38], or domains satisfying suitable geometric assumptions [12,13], see also [26].
From Theorem 1.1, we can prove the following Hardy type inequality for (−L) s/2 .
for any real valued function ϕ in the domain of (−L) s/2 such that the right hand side is finite.
We will study several particular settings throughout the paper to get more precise inequalities. In order to obtain these inequalities, we will start from the general forms obtained in Theorem 1.1 and Corollary 1.2. Then, the ideas to follow will be different from those used in [12,13,38,26]. Instead, we will make use of the solutions of the corresponding extensions problems.
First, we will consider the case of the sublaplacian L on stratified groups (see for instance [15,6] for definitions and facts on stratified Lie groups). In this case, good Gaussian bounds for the associated heat kernel q t are known (see e.g. Theorem 3.4 in [10]) and we can obtain more explicit results. Indeed, by means of Theorem 1.1, we can prove an analogue of (1.3), i.e., a trace Hardy inequality with a homogeneous weight function for L. Let Q stand for the homogeneous dimension of the stratified group G. Using properties of the kernels R α of the Riesz potentials (−L) −α/2 (see Subsection 2.4) we will prove the following result. Theorem 1.3 (Trace Hardy inequality for stratified groups). There exists a positive weight function w s (x) which is homogenous of degree s such that is valid for real valued functions u ∈ W s 0 (G × R) and 0 < s < 2. As a corollary of Theorem 1.3, it can be proved (after an application of Corollary 1.2) that, when u satisfies the extension problem with initial condition f , the left hand side of (1.9) reduces to ((−L) s/2 f, f ), leading to the following Hardy inequality.
Then, there exists a positive weight function w s (x) which is homogenous of degree s such that Though the function w s has the right homogeneity we do not have any way of computing it explicitly. However, it is possible to estimate w s in terms of a fixed homogeneous norm from above and below. It is also possible to write w s as a ratio of two different homogeneous norms.
We point out that in [8] the authors already obtained Hardy-type inequalities on stratified Lie groups. The results are slightly different and they estimate the constants arising.
We will also consider trace Hardy inequalities associated to the Euclidean motion group M (n), which is the semidirect product of R n with the special orthogonal group SO(n) with its natural action on R n . In the case of M (2) our results are more explicit. Identifying R 2 with C and SO(2) with the circle group S 1 we write the coordinates on M (2) as (z, e iθ ) and let ∆ G = ∆ + ∂ 2 θ , where ∆ is the Laplacian on R 2 , and define ∇ = (∂ x , ∂ y , ∂ θ ), with z = x + iy.
The trace Hardy inequality for the motion group M (2) can now be stated in the following form. It is also possible to prove a similar inequality for M (n).
Another operator that we will consider is the special Hermite operator L 1 . We will exploit the relation between L 1 and the sublaplacian L red on the reduced Heisenberg group H n red , namely, the quotient group H n /Γ, where H n is the Heisenberg group and the central subgroup Γ is given by Γ = {(0, 2kπ) : k ∈ Z} (see [34,Chapter 4]). The reduced Heisenberg group can be identified with C n × T, where T is the torus. It happens that when considering functions independent of τ ∈ T, the sublaplacian in H n red coincides with the Euclidean Laplacian on R 2n . Then, we can obtain a trace Hardy inequality in H n red , and also Hardy's inequality for the special Hermite operator. The latter reads as follows.
Theorem 1.6. Let 0 < s < 2. Let f be a real valued function on C n such that f and L s/2 1 f are both in L 2 (C n ). Then we have In [1] the authors proved a sharp Hardy-Sobolev inequality for the twisted Laplacian on C n using the fundamental solutions, see Corollary 1.2 therein. This technique was already used in [2] to obtain Hardy-Sobolev inequalities for the Laplace-Beltrami operator on certain manifolds and for the sublaplacian on the Heisenberg group. Observe that the weight function involved in [2, Corollary 1.2] is different from the one obtained in Theorem 1.6. Remark 1.7. We do not state the optimality of the constants obtained in Theorems 1.3, 1.6 and Corollary 1.4, since we do not know about that. Anyway, intuitively, the constants in Theorem 1.3 and Corollary 1.4 seem to be the right ones, because they match with the optimal constants in the Euclidean setting. Moreover, in the statement we are taking general homogeneous weights, and new constants. However, the constant in Theorem 1.5 is sharp. Actually, for general inequalities of the form where u s,δ are defined in (2.2), it can be easily checked that the function u −s,δ optimize the above inequality.
We emphasize that, in the results above, a precise knowledge on the solution of the extension problem is crucial. We will take the chance to further study several characterisations of solutions of the extension problem (1.8) in the case of compact Lie groups, Hermite operator and special Hermite operator. This will be the second important part of our paper. Characterisations of L p solutions of the extension problem were already carried out in [31]. We try to go beyond and, in the case of compact Lie groups and the Hermite operator, we find all the solutions when the initial data is a tempered distribution and, in the case of special Hermite operator, when the initial data belongs to a suitable Sobolev space. Moreover, we will characterise all the solutions of the extension problem which are tempered distributions. Our characterisations concern operators with discrete spectrum. Characterisations of solutions of the extension problem when the operator involved has continuous spectrum are more involved and they will appear elsewhere. We do not state the results on characterisations here, for the sake of easy reading. Instead, we defer all the related set-up, results and the proofs to Section 3.
The structure of the paper is as follows. Trace Hardy and Hardy's inequalities, and the proofs of the theorems stated in the introduction, are given in Section 2. In Section 3 we deal with the characterisations of all solutions of the extension problem in the case in which the associated operators have discrete spectrum.
2. An extension problem, trace Hardy and Hardy's inequalities for fractional Laplacians. In this section we prove the results stated in the introduction related to trace Hardy and Hardy's inequalities.
2.1. A basic lemma. The proof of the trace Hardy inequality in Theorem 1.1 depends on the following result, which is well known to experts in Partial Differential Equations. We include it here for the convenience of the readers. We introduce the gradient ∇ = X 1 , · · · , X m , ∂ ∂ρ on G × R + . Recall that we are dealing with left invariant vector fields satisfying Hörmander's condition, and they are skew-symmetric: where dx stands for the left Haar measure on G. Denote by L s the operator With these notations we have the following lemma. It is initially stated for C 1 0 functions, but it remains valid for functions u coming from a suitable Sobolev space.
Lemma 2.1. Let 0 < s < 2. Let u(x, ρ) be a real valued function from C 1 0 ([0, ∞), C 1 0 (G)) and let v(x, ρ) be another real valued function for which lim Proof. Consider the following integral: Integrating by parts and using (2.1), we get On the other hand, a similar calculation with the ρ-derivative gives Adding and then taking all integrations into account we get our result, in view of (2.1).
Remark 2.2. The assumption in Lemma 2.1 that u is from C 1 0 ([0, ∞), C 1 0 (G)), i.e. it is a compactly supported C 1 function on [0, ∞) taking values in C 1 0 (G) allows to guarantee that integrations by parts are justified leaving no boundary terms.
In Lemma 2.1, if v satifies the extension problem (1.8), i.e., the equation L s v = 0 on G × R + with a given initial condition v(x, 0) = ϕ(x), then we get the inequality In view of the above, we need to solve the extension problem for L with a given initial conditon ϕ. We also need to compute lim ρ→0 ρ 1−s ∂ ρ v(x, ρ) in terms of L and ϕ.

2.2.
An extension problem for sum of squares of vector fields. We will look at the extension problem (1.8) more carefully. A good reference for this subsection is the article by Stinga-Torrea [31]. Let X j , j = 1, 2, ..., m, be left invariant vector fields on a Lie group G satisfying the Hörmander's condition. Let q t be the heat kernel associated to the Laplacian For s > 0 the integral defines an L 1 function and G u s,ρ (x)dx = 1. But for −2 < s < 0 the function need not be integrable. However, if we assume that q t 2 ≤ C t −γ , γ > 1, the integral defining u s,ρ converges even for −2 < s < 0 and defines an L 2 function. Indeed, In Theorem 2.4 below a formula for the solution to the extension problem is given, see [31, Theorem 1.1] (see also [17] and [11]). We sketch a proof of the theorem for the sake of completeness, and because some parts of the proof will be referred later.
First of all, we have the following lemma.
Lemma 2.3. For −2 < s < 2, the functions u s,ρ and u −s,ρ are related via the equation Proof. This can be proved, as in [31], by considering (−L) s/2 u −s,ρ , g and using the spectral definition of (−L) s/2 . In fact, the proof depends on the numerical identity valid for λ > 0 (which is true by a simple change of variables).
Proof. We first show that u s,ρ solves the extension problem: To see this, note that u s,ρ = ρ s v s,ρ , with v s,ρ =

Hence by a simple calculation we get
On the other hand, since Lq t (x) = ∂ t q t (x) we see that and simplifying we get Combining (2.6) and (2.7) we finish the proof of the first claim.
On the other hand, by a simple change of variables, we have As f * q t converges to f in L p as t tends to 0 it follows that converges to f in L p , and therefore it solves the extension problem with initial condition f (in other words, u s,ρ is an approximate identity). With the relation (2.4) at hand, it is easy to calculate the limit of ρ 1−s ∂ ρ (f * u s,ρ ) as ρ goes to zero. Indeed, by the relation (2.5) we have Since it is easy to see that

Proofs of Theorem 1.1 and Corollary 1.2.
Proof of Theorem 1.1. As remarked earlier, it is enough to prove the result when . We take v = ϕ * u s,ρ and observe that, by Theorem 2.4, v solves the equation Then, by taking this v in Lemma 2.1 and taking into account Theorem 2.4, we obtain the inequality as desired.
Moreover, we claim the equality in Theorem 1.1 under certain assumption on the heat kernel associated to L. Proposition 2.5. Under the same hypotheses as in Theorem 1.1, let q t be the heat kernel associated to the Laplacian L = m j=1 X 2 j and assume that q t 2 ≤ C t −γ , γ > 1. Let ϕ ∈ L 2 (G) be such that (−L) s/2 ϕ ∈ L 2 (G) and let u = ϕ * u s,ρ be the solution of the extension problem (1.8) with initial condition ϕ.
Proof. Note that if f and g are both from L 2 (G), then their convolution is uniformly continuous and vanishes at infinity. This can be proved by approximating f and g by a sequence of compactly supported smooth functions. Since ϕ and u s,ρ are from L 2 (G), due to (2.3), it follows that the solution u of the extension problem given by u = ϕ * u s,ρ vanishes at infinity as a function of x for any fixed ρ. Moreover, X j u s,ρ ∈ L 2 (G), and the same is true for ∂ ρ [ρ −s u s,ρ ]. Integrating by parts and using the fact that u vanishes at infinity, we have Now, we sum up and use the fact that u solves the extension problem with initial condition ϕ. The result follows.
, and taking into account that u = f * u s,ρ solves the differential equation, we have that Then, by Theorem 2.4, the right hand side of the above identity reduces to On the other hand, by Theorem 1.1, we have that Combining all these facts, we conclude the result.
2.4. Sublaplacian on stratified groups and proof of Theorem 1.3. In this subsection we consider L = n j=1 X 2 j to be the hypoelliptic sublaplacian on a stratified Lie group G. A Lie group G is called stratified if it is nilpotent and simply connected and its Lie algebra g admits a vector space direct sum decomposition g = g 1 ⊕ · · · ⊕ g m such that [g 1 , g k ] = g k+1 for 1 ≤ k < m and [g 1 , g m ] = {0}, where g 1 = [g, g]. We refer to Folland-Stein [15] for more about stratified groups and sublaplacians. Theorem 1.1 says that Let q t be the associated heat kernel of L. Using the functions u s,ρ defined in the previous subsection in (2.2) and Lemma 2.3 we could prove the following trace Hardy inequality: Remark 2.6. In the case of the Laplacian ∆ on R n the heat kernel is explicitly given by q t (x) = (4πt) −n/2 e − 1 4t |x| 2 and hence the expression in (2.2) can be explicitly computed to be for −2 < s < 2. Then observe that inequality (1.2) follows from Corollary 1.2 (indeed, if f ∈ L 2 and (−∆) s/2 ∈ L 2 , the solution of the extension problem given by u(x, ρ) = f * u s,ρ (x) belongs to W s,2 0 (R n+1 ), see [28,Proposition 3.13]), after choosing ϕ = u −s,δ as above, and taking into account Lemma 2.3 (with L to be the Euclidean Laplacian ∆).
In the case of stratified Lie groups, we do not have an explicit expression for the heat kernel, so we are not able to arrive at a sharp constant. The situation is the same even in the case of the Heisenberg group H n where the heat kernel is almost explicit. Now, consider the following functions which are kernels of the Riesz potentials: Then it is known ( [14] or [8]) that they are the kernels associated to (−L) −α/2 . They can be explicitly calculated (e.g. in the case of ∆ on R n ) once we have a good expression for the heat kernel. Moreover, they are homogeneous of degree α − Q and satisfy R α * R β = R α+β .
Proof of Theorem 1.3. The proof is very similar to the one given in the context of H-type groups, in [28, Proof of Theorem 1.5]. We sketch here the main ideas.
In view of the relation in Lemma 2.3 we obtain that Recalling the definition of u −s,δ in (2.2) we see that δ s u −s,δ converges pointwise to 2 s |Γ(s/2)| |Γ(−s/2)| R s as δ tends to zero (it actually converges in the sense of distributions. Both convergences are easy to see from the definition). Since (ψ R α ) * u s,δ converges to ψ R α (because u s,δ is an approximate identity, see the proof of Theorem 2.4) we obtain the following inequality, from (2.8), by letting δ → 0 : As 0 ≤ ψ ≤ 1 and ψ R α * R s ≤ R α * R s = R α+s , the latter reads as (2.10) By letting tend to 0 we obtain the inequality Let us define The choice α = Q−s 2 leads to the inequality Remark 2.7. For each γ we define the function |x| (γ) by the equation .
Then it is clear that |x| (γ) is a homogeneous norm on G. With this definition, we see that and the trace Hardy inequality reads as Since all the homogeneous norms on G are equivalent we have C 1 |x| (−s) ≤ |x| ≤ C 2 |x| (s) for some constants where |x| is a standard homogeneous norm. With this we can replace the weight w s (x) by C −1 s |x| s and get the inequality The values of the constant depend on good estimates of the heat kernel in terms of Gaussian from above and below.
Proof of Corollary 1.4. By Corollary 1.2 and the computations in the proof of Theorem 1.3, one immediately obtains Remark 2.8. Observe that we could get also the following Hardy's inequality for stratified Lie groups In Consequently, by choosing α = (n − s)/2 in (2.10), the trace Hardy inequality for R n takes the form This inequality is known to be sharp, though equality is never attained (see [38]). Similarly, Hardy inequality (1.1) on R n follows from Corollary 1.2 and the computation starting from (2.9), choosing again α = (n − s)/2: which coincides with (1.1).

2.5.
Euclidean motion groups and proof of Theorem 1.5. In this subsection we consider the Euclidean motion group M (n) which is the semidirect product of R n with the orthogonal group SO(n) acting on R n . Then the Laplacian on G = M (n) is given by ∆ G = ∆ R n + ∆ K where ∆ K is the Laplacian on K = SO(n). By a suitable choice of the inner product on the Lie algebra of K we can assume that ∆ K = m j=1 X 2 j where X j are vector fields on K forming an orthonormal basis for its Lie algebra. Thus ∆ G is a sum of squares of vector fields and the results proved in Subsection 2.2 apply to the case of M (n). In particular, the solution of the extension problem for −2 < s < 2. In the above formula, q t is the heat kernel associated to ∆ G , which is just the product of the Euclidean heat kernel p t (x) with the heat kernel h t (k) associated to ∆ K . Thus we have The kernels u s,ρ (x, k) can be expressed in terms of known functions as follows. We need to set up some notation here; we refer the reader to Subsection 3.2 for more details. Given a compact Lie group K we letK stand for the unitary dual consisting of equivalence classes of irreducible unitary representations of K. If π ∈K then χ π will stand for the character associated to π. Then it is known that χ π are eigenfunctions of the Laplacian ∆ K and the eigenvalues are denoted by −λ 2 π . The heat kernel h t is given by the expansion where d π is the dimension of π. In view of this, the function u s,ρ on the motion group is given by where the functions u s,ρ (x; π) are u s,ρ (x; π) = ρ s (4π) n/2 |Γ(s/2)| ∞ 0 e − 1 4t (ρ 2 +|x| 2 ) e −tλ 2 π t −(n+s)/2−1 dt.
We can easily see that u s,ρ (x; π) = ρ −n ϕ (n+s)/2 (ρ −1 x, ρλ π ) where By recalling the integral representation of the Macdonald functions K (n+s)/2 (see (3.8) for the definition) we obtain Thus we have the following formula for the kernel u s,ρ (x, k): Then, by choosing ϕ(x, k) = u −s,δ (x, k) in Theorem 1.1, we obtain the following.
Theorem 2.10. Let 0 < s < 2 and fix δ > 0. Then for any real valued function where the functions u ±s,δ are the ones in (2.11).
In the case of M (2), we can get a more precise expression for the kernel u s,ρ (x, k) using Poisson summation formula for the heat kernel h t (k). We identify K = SO(2) with the circle group S 1 and R 2 with C so that M (2) is the semidirect product of C with S 1 . The elements of G = M (2) will be written as (z, e iθ ) rather than (x, k). Proof. In the case of M (2) the heat kernel is given by In view of the Poisson summation formula (see [30, Chapter VII]) we also have the representation Thus, the integral representation for u s,ρ (z, e iθ ) gives By evaluating the integral the proposition is proved.
Remark 2.12. There is also Poisson summation formula available for SO(n) (see [39,Theorem 3]). Thus, we could obtain analogous results as in Proposition 2.11 and Theorem 1.5 in higher dimensions. In order to avoid additional notation, we leave the details to the reader.
2.6. The special Hermite operator and proof of Theorem 1.6. In this section we prove trace Hardy and Hardy's inequalities for fractional powers of the special Hermite operator (also known as twisted Laplacian) on C n . This operator is related to the sublaplacian on the Heisenberg group, and so we will define it starting from the Heisenberg group setting. Let us set up the notation. Let H n = C n × R denote the (2n + 1)-dimensional Heisenberg group with the group operation (z, t)(w, s) = z + w, t + s + 1 2 Im(z ·w) . Its Lie algebra h n is generated by the (2n + 1) left invariant vector fields x j ∂ ∂t , j = 1, 2, . . . , n.
The operator L = − n j=1 (X 2 j + Y 2 j ) is called the sublaplacian on H n . Written explicitly We can also write L as are the Cauchy-Riemann operators.
For each nonzero λ ∈ R, let the special Hermite operators L λ (or twisted Laplacian) be defined by the equation (2.12) Then it is clear from the definition that L λ = −∆ C n + λ 2 4 |z| 2 + iλN and we can also write L λ as where Z j (e iλt f (z)) = e iλt Z j (λ)f (z) andZ j (e iλt f (z)) = e iλtZ j (λ)f (z). Each of these L λ is an elliptic differential operator on C n whose spectral decomposition is known explicitly and is given by special Hermite functions. We refer to [33] for the theory of special Hermite expansions. Though this operator is not a sum of squares of real vector fields, it is related to the sublaplacian L on H n (which is a sum of squares) via the relation (2.12) and we make use of this connection.
There is an orthonormal basis for L 2 (C n ) consisting of the "special Hermite functions" Φ α,β , α, β ∈ N n , which are eigenfuncions of L 1 , more precisely, L 1 Φ α,β = (2|β| + n)Φ α,β . Note that the spectrum of L 1 consists of (2k + n), k ∈ N and each eigenspace is infinite dimensional. The projection onto the corresponding eigenspace can be written in a compact form. If stand for Laguerre functions of type (n − 1) then we have where the twisted convolution ϕ × ψ of two functions on C n is defined by More generally, the λ-twisted convolution is defined by which is related to the convolution on the Heisenberg group via the identity (f * g) λ (z) = f λ * λ g λ (z) where f * g is the convolution of f and g on the Heisenberg group H n and Actually, it is convenient to work with the reduced Heisenberg group H n red rather than the Heisenberg group. This group is defined as the quotient group H n /Γ where H n is the Heisenberg group and the central subgroup Γ is given by Γ = {(0, 2kπ) : k ∈ Z} (see [34,Chapter 4]). The reduced Heisenberg group can be identified with C n × T where T is the torus with group law written as (z, e it )(w, e iτ ) = (z + w, e i(t+τ + 1 2 Im z·w) ). We let L red stand for the sublaplacian as H n red which is simply the sublaplacian on H n acting on functions f (z, w) which are 2π-periodic in the last variable.
We need also to relate the heat semigroups associated to the sublaplacian and to the special Hermite operator. The sublaplacian is a self-adjoint, non-negative, hypoelliptic operator, and it generates a contraction semigroup which we denote by e −tL . It is known (and it can be deduced from the facts shown above) that (e −tL f ) λ (z) = e −tL λ f λ (z) = f λ * λ q λ t (z), where The operator L red generates a diffusion semigroup e −tL red whose heat kernel q t is given by where q t is the heat kernel on H n . We then have, for k ∈ Z, k = 0 2π 0 q t (z, τ )e ikτ dτ = q k t (z) and when k = 0 we also have which is just the Euclidean heat kernel on C n . Consider the extension problem for the Laplacian ∆ C n . We can also think of this v as a solution of the extension problem on the reduced Heisenberg group. This is so simply because L red v(z, ρ) = ∆ C n v(z, ρ) for functions independent of τ . Theorem 1.1 reads in this context as follows.
Theorem 2.13. For real valued functions ϕ on C n and u on H n red with mild decay conditions we have the inequality From Theorem 2.13 we can also prove the following trace Hardy inequality for functions on H n red : Theorem 2.14. Let 0 < s < 2. Then, for real valued functions u on H n Proof. The proof is analogous to that of Theorem 1.3 for stratified Lie groups or more precisely, as in the Euclidean case as explained in Remark 2.9. Indeed, it is enough to observe that C n = R 2n .
We know that when u satisfies the extension problem (2.13) with the initial condition f the left hand side of the above integral reduces to L s/2 red f, f (i.e., we have Corollary 1.2 in this context). By taking f in the particular form g(z) cos(τ ), where g is a real valued function we can obtain the Hardy's inequality for fractional powers of the special Hermite operator L 1 stated in Theorem 1.6. Now we show the proof.

From this we get
e −rL red f (z, τ ) = Re(e −iτ e −rL1 g(z)) and consequently, L On the other hand, by following the analogous reasoning as for the Euclidean case in Remark 2.9, we obtain that the expression on the right hand side of (2.14) is precisely This completes the proof.
3. Operators with discrete spectrum: characterisations of solutions to the extension problem. Our aim in this section is to characterise all the solutions of the extension problem when the associated operator has discrete spectrum. We remark that the initial considerations in the present section have been already presented in the literature (see [31,3]). The starting point will be the most general framework, and then we will study several particular cases. Let us consider a Riemannian manifold (M, g), with or without boundary. On M we have a natural second-order partial differential operator, namely the Laplace-Beltrami operator ∆ g , defined by ∆ g f = − div(∇f ). We assume that the spectrum of ∆ g is discrete. This is the case e.g. when M is compact. Let {λ 2 k } be the spectrum. Let ϕ k be the associated eigenfunctions ∆ g ϕ k = −λ 2 k ϕ k normalised so that {ϕ k } forms an orthonormal basis of L 2 (M ).
We consider the associated extension problem Let us initially impose mild conditions on the initial data, say, f be a distribution. Then we have the equation which shows that the Fourier coefficients u k (ρ) := u(·, ρ), ϕ k M = M u(x, ρ)ϕ k (x)dx satisfy the equation It is well known (see [24,Chapter 5,Section 5.7]) that the functions (ρλ k ) s/2 I s/2 (ρλ k ) and (ρλ k ) s/2 K s/2 (ρλ k ), where I ν is the modified Bessel function of first kind and K ν is the Macdonald's function, cf. [24, Chapter 5, Section 5.7] (see (3.3) and (3.4) for the definitions), are two linearly independent solutions of the equation (3.2), i.e., Then, for the time being formally, we have We also know that u(x, ρ) converges to the distribution f (x) as ρ goes to zero. Therefore, u k (ρ) converges to f, ϕ k M as ρ goes to zero. This implies that (the constant in the first summand is justified by Remark 3.1) Then, in view of the asymptotics (3.5), (3.6) of the function I s/2 , the coefficient d k should be chosen suitably. Thus, taking into account the above considerations, our aim will be to give characterisations for all possible solutions of the extension problem (3.1) in several particular cases. Such characterisations will go beyond [31,3]. Before going through this issue, we will first collect several definitions and properties of Bessel functions.
3.1. Some facts on Bessel functions. Let I ν (z) be the modified Bessel function of first kind given by the formula (see [24,Chapter 5,Section 5.7]) and let K ν be the Macdonald's function of order ν defined by (see also [24, Chapter 5, Section 5.7]) and, for integral ν = n, K n (z) = lim ν→n K ν (z), n = 0, ±1, ±2, . . .. From (3.3) and (3.4) it is clear that there exist constants C 1 , C 2 , c 1 , c 2 > 0 such that Moreover, it is well known (see [24,Chapter 5,Section 5.11]) that and We have the integral representation for the Macdonald's functions, see for instance [24,Chapter 5,(5.10.25)] From (3.8) it can be seen that z s/2 K s/2 (z) is actually a function of z 2 and hence it is even in the z variable.
Remark 3.1. Even more, observe that, after a change of variables, Then, letting z tend to 0 above, we have that z ν K ν (z) tends to the constant 2 ν−1 Γ(ν).

3.2.
Compact Lie groups. We start with some definitions and notations. Some of them were already introduced in Subsection 2.5, but we collect them also here for the sake of the reading. Let K be a connected, simply connected compact Lie group. LetK stand for its unitary dual, viz. the set of all equivalence classes of irreducible unitary representations of K. For each π ∈ K we let d π stand for the dimension of π and χ π the character. For f ∈ L 2 (K), we define its (operator valued) Fourier transform as and we have the Plancherel formula where π(f ) 2 HS = tr(π(f )π(f ) * ) is the Hilbert-Schmidt norm of π(f ). We can write the inversion in two different ways f (k) = π∈K d π tr(π(f )π(k)), or f (k) = π∈K d π f * χ π (k), so that f * χ π (k) = tr(π(f )π(k)). We have and consequently the Plancherel theorem can also be written as For basic facts about representation theory of compact Lie groups we refer to Simon [29] and Hall [20]. Let ∆ K be the Laplacian on K. In this case, we are interested in the extension problem for the Laplacian ∆ K . Let us start with some known facts. If f ∈ L p (K), in view of Theorem 2.4, we can write down a solution of the above problem as where e t∆ K stands for the heat semigroup generated by ∆ K . Indeed, it is known that to each π there corresponds a real number λ π > 0 such that (3.10) By defining the heat kernel the solution of the heat equation associated to ∆ K is given by Then the solution u defined in (3.9) takes the form which can be written as the convolution f * u s,ρ (k), where the kernel u s,ρ is defined by Recalling the expansion of h t (k) in terms of the character χ π we see that The above is precisely the integral representation of the Macdonald function (ρλ π ) s/2 K s/2 (ρλ π ), see (3.8). Thus u s,ρ (k) = π∈K d π (ρλ π ) s/2 K s/2 (ρλ π )χ π (k) and the function solves the extension problem. In view of the asymptotic properties (3.5) and (3.7) of the Macdonald function K s/2 , the series (3.11) converges uniformly as long as f * χ π ∞ has a polynomial growth in terms of λ π . This is certainly the case when, say, f ∈ L p (K), or even when f is a distribution.
Nevertheless, if we do not care about the initial condition, the extension problem admits another solution u 2 (as explained at the beginning of the section) which can be represented as u 2 (k, ρ) = π∈K d π (ρλ π ) s/2 I s/2 (ρλ π )g * χ π (k) (3.12) where I s/2 is the Bessel function of second kind. To see this, let u satisfy the equation Then, in view of (3.10), it follows that u(·, ρ) * χ π (k) satisfies the equation As we already explained, for each fixed k, this equation admits two linearly independent solutions, viz. (ρλ π ) s/2 K s/2 (ρλ π ) and (ρλ π ) s/2 I s/2 (ρλ π ). The first choice leads to solutions of the form u 1 whereas the second choice leads to u 2 . However, as the modified Bessel functions I s/2 (t) have exponential growth as t → ∞, the series (3.12) defining the function u 2 will not converge unless g * χ π has enough decay as λ π → ∞. It is certainly well defined if g has only finitely many non zero components in its Peter-Weyl expansion: that is, g * χ π = 0 for all but finitely many π ∈K. Note also that when u 2 (k, ρ) is well defined, it converges to 0 as ρ → 0.
We are thus led to find conditions on the function g so that g * χ π has enough decay as λ π → ∞. The answer to this question lies in the holomorphic extendability of g to the complexification of the group K.
In order to explain this, we closely follow the notations used in [22, Section 9.2]. Let K C stand for the universal complexification of K, which is a complex Lie group. Let T be a maximal torus in K with Lie algebra t and let W stand for the Weyl group of K with respect to T . Then, we have the classical Cartan decomposition K C = K · exp it · K. In view of this decomposition, domains D ⊂ K C which are biinvariant under the action of K are given by W -invariant closed sets B ⊂ t. Thus, all such domains, called Reinhardt domains, are of the form D B = K · exp iB · K. As K C is a complex Lie group we can talk about holomorphic functions on such Reinhardt domains.
Let π ∈K be an irreducible representation of K on a Hilbert space H π . Then it has a unique extension, denoted by the same symbol, to the complex group K C . Thus the characters χ π of the representation π extend to the whole of K C as holomorphic functions. Hardy spaces associated to Reinhardt domains D B in K C have been studied by Lassalle in [22]. Assuming that B is convex, we introduce the norm  where the operators π(f ), π ∈K satisfy the following condition: In fact the above expression is precisely f 2 B as shown by Lassalle. It easily follows from the following result known as Lassalle-Gutzmer formula.
At this point we would like to know how χ π (exp 2iH) grows as a function of H and d π . To see this we need to recall some results from the representation theory of compact Lie groups, see [19,20,29]. Without getting into details, let us recall that there is a parametrisation ofK in terms of the so called highest weight vectors in the dual of a Cartan subalgebra. Thus there is a one to one correspondence between π ∈K and such highest weight vectors λ. Let the representation π be associated to the weight vector λ. Then it is known that the eigenvalue λ π corresponding to π grows as a constant multiple of |λ| and that the dimension d π of the representation grows polynomially in |λ|. Thus there are constants a, b and C such that: (i) λ π ≥ a|λ|; (ii) d π ≤ C(1 + |λ|) b , see [19,Lemma 6]. Moreover, there exists a constant γ > 0 such that This can be easily proved using the Weyl character formula for χ π , (see [19,Lemma 7]).
From Theorem 3.2 and the above estimate for the characters χ π (exp iH), we infer that whenever f ∈ H 2 (D B ), π(f ) 2 HS and hence f * χ π 2 2 have certain exponential decay (depending on B). In particular, when f ∈ H 2 (K C ) it follows that f * χ π 2 2 decays faster than e −δλπ for any δ > 0. With this information at our disposal, we can prove a characterisation of the solutions of the extension problem.
We consider the extension problem (3.2) for ∆ K with initial condition f , where f is a distribution on K. Note that the initial condition is interpreted in the distribution sense: for every ϕ ∈ C ∞ (K) we have Given distributions f, g, let us define and Q ρ g(k) = π∈K d π (λ π ρ) s/2 I s/2 (λ π ρ)g * χ π (k).
We call P ρ f the generalised Poisson integral of f . For t > 0, let B t := {H ∈ t : |H| < t}. We prove the following. Proof. For any distribution f on K, we note that f * χ π ∞ ≤ c(1 + λ π ) n for a fixed integer n. This follows from the fact that every distribution on K is of finite order. Consequently the series (3.13) defining P ρ f converges uniformly and solves the extension problem. It is also clear that for ϕ ∈ C ∞ (K) we have To see this observe that which is the same as (f, P ρ ϕ). As P ρ ϕ → ϕ in C ∞ (K) as ρ → 0, it follows that (f, P ρ ϕ) converges to (f, ϕ) as ρ → 0.
On the other hand, since we take g from t>0 H 2 (D Bt ), we know from the remarks previous to the statement of the theorem, that Q ρ g is well defined as g * χ π 2 ≤ Ce −δλπ for any δ > 0. Since each of the components (λ π ρ) s/2 I s/2 (λ π ρ)g * χ π satisfies the extension problem, so does their sum Q ρ g. Moreover, as (λ π ρ) s/2 I s/2 (λ π ρ) → 0 as ρ → 0, it follows that Q ρ g → 0 as ρ → 0, again in the sense of distributions. Thus u = P ρ f +Q ρ g solves the extension problem with initial condition f .
Theorem 3.5. Let u(k, ρ) be a solution of the extension problem (3.2) which is of tempered growth (in both variables). Then u = P ρ f for some f ∈ L p (K), 1 < p ≤ ∞ if and only if there exists a constant C > 0 such that (3.14) When p = 1, the same happens if and only if f is a complex Borel measure.
Proof. When f ∈ L p (K), 1 ≤ p ≤ ∞, the estimate (3.14) is immediate from Theorem 2.4 since the heat semigroup e t∆ K is a contraction on L p spaces. Conversely, if (3.14) is assumed, then there exists a subsequence ρ j → 0 and f ∈ L p (K) (f = µ, with µ a complex Borel measure, when p = 1), such that u(k, ρ j ) → f (k) weakly in L p (K), 1 < p < ∞ (u(·, ρ j ) → µ, for p = 1). But then, as we remarked earlier, u has to be of the form P ρ f with the same f . Remark 3.6. When f ∈ L 2 (K) we can say something more about the solution u(k, ρ) = P ρ f (k). In view of Lassalle-Gutzmer formula in Theorem 3.3 it follows that for each ρ the function u(·, ρ) belongs to H 2 (D B δρ ), where B δρ denotes the ball of radius δρ, for any δ < γ where γ is the constant appearing in the estimate |χ π (exp iH)| ≤ d π e γλπ|H| .
Theorem 3.7 (Thangavelu). For any holomorphic function F in the tube domain Ω t , we have the identity where f is the restriction of F to R n .
In Theorem 3.7, L n−1 k (−2t 2 )e t 2 stands for Laguerre functions of type (n − 1). Given a tempered distribution f , we define and if the function g is such that P k g has enough decay we also define We will prove the following characterisation of solutions to the extension problem (3.16). , for some g ∈ ∩ t>0 H 2 (Ω t ).
Proof. Let us suppose that the function u(x, ρ) can be written as P H ρ f (x)+Q H ρ g(x), with the assumptions on f, g as in the statement. If f is a tempered distribution on R n , its Fourier-Hermite coefficients satisfy for some integer m. As a consequence, the expression (3.17) is well defined and it solves the extension problem. On the other hand, from Theorem 3.7, we infer that P k g 2 2 has to decay as (L n−1 k (−2t 2 )) −1 , asymptotically in k. But it is known that L n−1 k (−2t 2 ) ≈ Ce ct(2k+n) 1/2 as k → ∞, for n and t fixed (see [32]), thus P k g 2 decays as Ce −ct(2k+n) 1/2 , for k large enough. Then, the expression (3.18) is also well defined and solves the extension problem. By following analogous reasoning as in the proof of Theorem 3.4, we conclude that u = P H ρ f + Q H ρ g solves the extension problem with initial condition f .
As in the case of compact Lie groups, we have also the following characterisation. The proof is completely analogous, since the heat semigroup e −tH is a contraction on L p spaces (see [33]), so we omit it. Remark 3.10. For the corresponding (generalised) Poisson integrals P H ρ of L 2 functions, an analogous remark to the one for compact Lie groups in Remark 3.6 can be made. Indeed, there is Gutzmer's formula for Hermite expansions available (see [37,Theorem 6.2]). Actually, such Gutzmer's formula is used to prove Theorem 3.7 in [37].
3.4. Special Hermite operator. In this section we study characterisations of solutions of the extension problem for the special Hermite operator. We borrow the notations, facts and definitions from Subsection 2.6.
As in the case of the Hermite operator, we would like to study the most general solutions of the extension problem with the initial condition f a tempered distribution. It is known that a distribution f on R n is tempered if and only if it belongs to the Hermite-Sobolev space W m H (R n ) for some integer m. These spaces are defined by the condition H m f ∈ L 2 (R n ) where H m is defined via spectral theorem when m is negative. The above result simply means that f is tempered if and only if its Hermite coefficients satisfy the estimate |(f, Φ α )| ≤ C(2|α|+n) m for some non-negative integer m. As there is no such characterisation of tempered distributions in terms of their special Hermite coefficients we start with initial conditions f coming from certain analogues of the Sobolev spaces known as twisted Sobolev spaces. For any integer m the space W m L1 (C n ) is defined by the requirement L m 1 f ∈ L 2 (C n ). Again, for m negative, the operator L m 1 is defined via spectral theorem and the condition translates into α,β∈N n (2|β| + m) 2m |(f, Φ α,β )| 2 < ∞.
Note that when f ∈ W m L1 (C n ) we have the estimates α∈N n |(f, Φ α,β )| 2 ≤ C(2|β| + n) −2m (3.19) for any β. Observe that when m is negative, the above is a growth restriction on the special Hermite coefficients of the distribution f.
We consider, for a fixed t > 0, the space of all holomorphic functions on Ω t , that we denote by O(Ω t ). Given F ∈ O(Ω t ), we introduce the norm (observe that we are using the same notation as in the case of the Hermite operator). The special Hermite functions Φ α,β have extensions to C 2n , and so the projections f × ϕ k can be holomorphically extended. Let us consider the generalised Poisson integral P L1 ρ f given by P L1 ρ f (z) = 2 1−s/2 Γ(s/2) ∞ k=0 (ρ √ 2k + n) s/2 K s/2 (ρ √ 2k + n) f × ϕ k (z).
Proof. Again, let us suppose that the function u(z, ρ) can be written as P L1 ρ f (z) + Q L1 ρ g(z), with the assumptions on f, g as in the statement. By the explanations previous to the statement of the theorem, we have that α∈N n |(f, Φ α,β )| 2 ≤ C(2|β| + n) −2m for any β. This implies that f × ϕ k k!(n − 1)! (k + n − 1)! ϕ k (2iy, 2iv) where ϕ k (2iy, 2iv) = L n−1 k (−2(|y| 2 +|v| 2 ))e (|y| 2 +|v| 2 ) . As g ∈ ∩ t>0 H 2 (Ω t ) we deduce that g × ϕ k 2 has to decay as e −t √ 2k+n for large k, by the same reasoning as in the proof of Theorem 3.8. Then, the expression Q L1 ρ g(z) is well defined and solves the extension problem. By following analogous reasoning as in the proof of Theorem 3.4, we conclude that u = P L1 ρ f + Q L1 ρ g solves the extension problem with initial condition f .
Since u(·, ρ) ∈ W m L 1 (C n ), we have the polynomial growth (3.19) in |β| for the infinite sum α∈N n | u α,β (ρ)| 2 . Thus, α∈N n |c α,β | 2 has exponential decay e −ρ √ 2|β|+n for every ρ > 0. Then the function g(z) = β,α∈N n c α,β Φ α,β (z) satisfies g × ϕ k 2 ≤ Ce −t √ 2k+n for every t > 0. This implies that it belongs to the space H 2 (Ω t ) for every t > 0. Remark 3.12. As in the case of compact Lie groups and the Hermite operator, we have also another characterisation as in Theorem 3.5. The proof is completely analogous, and we will omit it. Moreover, an analogous remark to Remark 3.6 also holds in this setting.