A Fourier Coefficients Approach to Hausdorff Dimension in the Heisenberg Group
Fernando Roman-Garcia
math.FA math.MG Primary: 43A05, 43A30, Secondary: 28A75












Fernando Román García, Department of Mathematics, 250 Atgeld Hall, University of Illinois Urbana Champaign,
Urbana, Il 61801, USA. romanga2@illinois.edu





Fourier Coefficients and Hausdorf Dimension in the Heisenberg group]
A Fourier Coefficients Approach to Hausdorff Dimension in the Heisenberg Group





This paper establishes connections between the group-Fourier transform and the geometry of measures in the Heisenberg group. Firstly, it is shown that if the Fourier transform of a compactly supported, finite, Radon measure is square integrable, then the measure must have a square integrable density. If it's Fourier transform is integrable, the the measure must have a continuous density. In addition, an alternative formulation of the Fourier transform on the Heisenberg group is used to show that energies of measures can be computed via integrals on an appropriate frequency space. This in turns opens the possibility of using Fourier methods in the computation of Hausdorff dimension of sets. 



[
    Fernando Román García
    December 30, 2023
=========================





§ INTRODUCTION

In studying the geometric properties of measures in , it is common to study their Fourier transforms. For g∈ L^1(),

    ℱ_E (g)(ξ)=(2π)^-n/2∫ e^-ixξg(x)dx.

Similarly, for a compactly supported, positive, and finite Radon measure, μ, in , its distributional Fourier transform is given by  

    ℱ_E(μ)(ξ):=(2π)^-n/2∫_e^- i x·ξdμ (x).

 As expected, ℱ_E(μ) and μ are closely related. For instance if f is a function in the Schwartz class of , then

    ∫_f(x)dμ(x)=∫_ℱ_E(f)(x)ℱ_E(μ)(x)dx.

These sort of relations between ℱ_E(μ) and μ are key in the application of Fourier theory to the potential-theoretic approach to Hausdorff dimension  introduced by R. Kauffman in <cit.>. This approach establishes a conection between Hausdorff dimension and energies of measures.

Let A⊂ be a Suslin set, and denote by (A) the collection of all positive and finite Radon measures with compact support in A. Then

    A=sup{σ>0:∃ μ∈(A) s.t. I_σ^E(μ)<∞},

where
    I_σ^E(μ)=∫_∫_|x-y|^-σdμ(y)dμ(x).


 This results holds more generally for Suslin subsets of complete metric spaces, by replacing the integrand, |x-y|^-σ, with the distance between x and y, d(x,y)^-σ. In Euclidean space, (<ref>) allows the computation of energies of measures via integrals on the frequency domain. More precisely, there is a constant c (n,σ), depending only on n and σ, such that

    I_σ^E(μ)=c(n,σ)∫_|ξ|^σ-n|ℱ(μ)(ξ)|^2dξ.

 A proof of this result can be found in <cit.>. Another result connecting properties of ℱ_E(μ) to μ itself is the following,
 
Let μ be a compactly supported, positive, finite, Radon measure on .


  * If ℱ_E(μ)∈ L^2(), then μ = gdx with g∈ L^2().

  * If ℱ_E(μ)∈ L^1(), then μ = hdx with h∈𝒞().


 This result has been successfully used in answering several important questions in geometric measure theory in . Marstrand-Mattila projection theorem (<cit.>, <cit.>), dimension of intersections of general sets (<cit.>,<cit.>), and the distance set problem (<cit.>), are a few example of problems where Theorem <ref> has played a crucial role.
A proof of this theorem can be found in <cit.>. 

In this work the aim is to establish a connections between the Fourier transform and the geometry of measures in the Heisenberg group.

The main difficulty of Fourier theory in  is that the Fourier transform of an integrable function is not a complex valued function but an operator-valued map from ∖{0} to , the space of bounded operators on L^2(). Nevertheless, it is possible to obtain analogues of many of the classical results. For instance an appropriate analogues of Plancherel's theorem and the inversion formula for the Schwartz class hold. 

Another important result is the analogue of  (<ref>),

If f is in the Schwartz class of , μ is a compactly supported, positive, finite, Radon measure on , and dϱ(λ)=|λ|^n/(2π)^n+1dλ, then

    ∫_f(p)dμ(p)=∫_^*tr((λ)^*(λ))dϱ(λ).


Here, tr(T) denotes the trace of the operator T, and T^* denotes its adjoint. 
Exploiting some of these properties we obtain the following analogue of Theorem <ref>.

Suppose μ is a compactly supported, positive, finite, Radon measure on . Then,


  * If ∈ L^2(_2), then dμ=f dp with f ∈ L^2().

  * If ∈ L^1(_1), then dμ=g dp with g ∈𝒞(),

where L^p(_p) is an appropriately defined, non-commutative, L^p-space. 

However, one of the consequences of the operator nature of the Fourier transform is that it makes it quite difficult to characterize the image of certain classes of functions. For instance, one would like to know the image of the Schwartz class, (), under the group Fourier transform. This was achieved by D. Geller in <cit.>, where he obtained a characterization in terms of asymptotic series. It is not simple, however, to use this characterization to extend the Fourier transform to distributions. Recently in <cit.>, the authors proposed an alternative frequency space for the Heisenberg group, and with it, an alternative definition of the Fourier transform whereby the Fourier transform of an integrable function is now a continuous function defined on this frequency space. Later,  in <cit.>, the authors were able characterize the image of () under this new “Fourier transform", and extend the definition to tempered distributions. In this work, this approach is used to obtain a Heisenberg group analogue of (<ref>), where now 

    I_σ(μ):=∫_∫_||q^-1p||_^-σdμ(q)dμ(p).

 The result reads,

Let μ∈(), then there is a constant d(n,σ) depending only on n and σ such that,

    I_σ(μ)=d(n,σ)∫_^n𝔎_Q-σ(α,λ)|μ_(ζ)|^2dζ.


Here, Q=2n+2 is the Hausdorff dimension of , ^n denotes the aforementioned frequency space, dζ is an appropriately defined measure, and 𝔎_Q-σ is related to the distributional Fourier transform of K_s(p)=||p||_^-s in the sense of <cit.>.
 
  
 
 
 
 
 
 
 
 
 
 
 
 




§ GROUP FOURIER TRANSFORM IN THE HEISENBERG GROUP

The Heisenberg group is the manifold =^2n×=^n×^n× with the group law (z, t)(z', t')=(z+z', t+t'-1/2ω(z,z')), where ω denotes the standard symplectic form. That is, if z=(x,y), z'=(x', y') then ω(z, z')=x· y'-y· x'. This group law turns  into a locally compact, non-Abelian Lie Group with respect to the usual topology in ^2n+1. The left invariant vector fields are 

    X_j:=∂/∂ x_j-y_j/2∂/∂ t; Y_j:=∂/∂ y_j+x_j/2∂/∂ t,  for  j=1,…,n,


    and, T:=∂/∂ t.

The center of  is the t axis, {(0,t):t∈}. One can check that up to multiplication by a non-zero constant, the Haar measure on  is the (2n+1)-dimensional Lebesgue measure, therefore, for p∈ [0,∞], the space L^p() coincides with the space L^p(^2n+1). Moreover, since the differential structure of  is that of ^2n+1 differentiable functions on  are those that are differentiable in the ^2n+1 sense. In particular, the Schwartz class in , here denoted by 𝒮(), is simply 𝒮(^2n+1). The Koranyi gauge, ||(z,t)||_^4=|z|^4+16t^2, induces a left invariant metric, known as the Koranyi metric, given by d_(p,q)=||q^-1p||_. For any r>0, the map δ_r(z,t)=(rz,r^2t) is a group automorphism and is homogeneous of degree 1 with respect to the left invariant metric d_.  Much more can be said about the structure of the Heisenberg group but much of it will not be of great relevance to us. For a detailed introduction to the Heisenberg group, readers are referred to <cit.>.



As for the general theory of Fourier analysis on groups, in order to introduce the Fourier transform, one first needs a complete description of the representations of the group. The representation theory of  is relatively simple. Denote by  the group of unitary operators on L^2(^n). For each λ∈^*:=∖{0}, the map

    :→𝒰(L^2(^n))

given by 
    (x,y,t)φ(ξ)=e^iλ (t+x·ξ+xy/2)φ(ξ+y),   (φ∈ L^2(^n)),
 
 is a strongly continuous, unitary representation of the Heisenberg group. Moreover, one can check that these representations are irreducible, and  by Stone-von Neumann Theorem (<cit.>, <cit.>), up to unitary equivalence, these are all the representations of  that are non-trivial at the center. Details about this can be found in many places, see for instance <cit.>, or <cit.>.

This allows us to introduce the Fourier transform of integrable functions in . Denote by  the space of bounded operators on L^2(), then the Fourier transform of f∈ L^1() is the operator-valued function 

    (f)(λ)=(λ):^*→,

given by

    (λ)φ=∫_f(p)(p)φ dp.

The integral is in the Bochner sense, that is to say (λ)φ is the function such that for every ψ∈ L^2(),

    ⟨(λ)φ,ψ⟩_L^2()=∫_f(p)⟨(p)φ , ψ⟩_L^2()dp.



It is not hard to see that for each λ, (λ) is indeed bounded. In fact, if we denote by ||·||_op the operator norm in , we have ||(λ)||_op≤ ||f||_1. As expected, the Fourier transform takes convolutions to “products", where now the product is composition of operators. Since neither the group law, nor composition of operators are commutative one has to be consistent with the order in which the functions are placed in the convolution. Here,
 
    f*g(p)=∫_f(pq^-1)g(q)dq,

so that 

    f*g(λ)=(λ)g(λ).

As mentioned before, the group Fourier transform shares many properties with the classical Fourier transform in . In what follows some of this properties will be discussed, many of them follow from the more general theory of Fourier theory on groups but for the sake of completeness Heisenberg-specific proofs are included.


 As one expects from the Fourier transform, depending on the regularity of f, one can obtain more than just boundedness of the operators (λ). To state this formally, I first need to introduce more notation.
 
 
A compact operator T is said to be in the Schatten p-class, denoted S_p(L^2()), if its singular values are in ℓ_p. That is

    tr[(TT^*)^p/2]<∞.

The Schatten p-norm of T∈ is ||T||_S_p=Tr[(TT^*)^p/2]^1/p. Note that S_1 is the trace class and S_2 is the class of Hilbert-Schmidt operators. 

Now, let ϱ be the measure on ^* given by

    dϱ(λ)=|λ|^n/(2π)^n+1,

and denote by L^p(^*,; dϱ) the space of -valued functions on ^* which are ϱ measurable and ∫_*||T(λ)||_S_p^pdϱ(λ)<∞.

The spaces L^p(^*,; dϱ) are examples of non-commutative L^p spaces. In particular, the space L^2(^*,S_2(L^2()); dϱ) is a non-commutative Hilbert space with inner product

    ⟨ T, S ⟩_L^2(S_2)=∫_^*tr(T(λ)S(λ)^*)dϱ(λ).


The simplified notation L^p(S_p) is used to denote these spaces. With these definitions in place, we can continue our discussion by drawing parallels between the Fourier transform in  and the classical Euclidean theory. For instance, one very important property of the classical Fourier transform is Plancherel's theorem. Turns out, there is a non-commutative analogue that holds in .


If f∈ L^1()∩ L^2() then ∈ L^2(_2) with 

    ||f||_2=||||_L^2(_2).

Moreover, if f and g are in L^1()∩ L^2() then 

    ∫_f(p)g(p)dp=∫_^*tr(g(λ)^*(λ)dϱ(λ)





The first thing to note is that (λ) is an integral operator in L^2(), with kernel given by

    T_f^λ(η,ξ)=∫_f^λ(x,η-ξ)e^iλ x(ξ+η/2)dx,

where 
    f^λ(z)=∫_f(z,t)e^iλ tdt.
 Therefore, ||(λ)||_S_2^2=||T_f^λ||^2_L^2(×). For g∈ L^2(^2n), denote by ℱ_E^jg the Euclidean Fourier transform of g in the first j coordinates. Applying change of variables and Euclidean Plancherel theorem, we have


    ||T_f^λ||^2_L^2(^2n)   =∫_∫_(2π)^n/2|ℱ_E^nf^λ(λ(η+ξ)/2,η-ξ)|^2dη dξ
       =(2π/|λ|)^n∫_∫_|f^λ(η, ξ)|^2dη dξ


Recalling the definition of f^λ, it follows by Plancherel theorem in  that 
    ∫|f^λ(z)|^2dλ=2π∫|f(z,t)|^2dt.

Therefore, the claim follows by integrating both sides of (<ref>) with respect to ϱ.
Equation (<ref>) follows by polarizing (<ref>).



Since L^1()∩ L^2() is dense in L^2(), Plancharel theorem extends to all of L^2() making the Fourier transform a Hilbert space isomorphism between L^2() and L^2(_2). 

Another important aspect of the Euclidean Fourier transform is its interaction with the Schwartz class. The space () coincides with (^2n+1). As expected, the regularity of functions in () translate as regularity of their Fourier transform (this will be made much more precise later) and allows the following inversion formula analogous to the Euclidean one. 



 


  * If f∈() then ∈ L^p(_p), for 1≤ p≤∞.

  * If f∈(), then 

    f(p)=∫_^*tr((p)^*(λ))dϱ(λ).









  * Fist, it is clear that f∈ L^2(), so that (λ)∈ S_2. In particular, (λ) is compact. Since for all 1<p≤∞, S_p⊂ S_1, it is enough to show ∈ L^1(_1). That is, ∫_^*tr(|(λ)|^1/2)dϱ(λ)<∞. 
If {e_j}_j∈ℕ is an orthonormal basis of L^2(), and s_j(A) denotes the j^th singular value of A. The singular values satisfy

    ∑_j=1^∞s_j(A)≤∑_i,j=1^∞|⟨ Ae_i , e_j ⟩_L^2()|.

 Let {}_α∈ℕ^n be the orthonormal basis of L^2() comprised of the re-scaled multidimensional Hermite functions (discussed in detail at the beginning of Section <ref>). For (λ), equation (<ref>) re-writes as
 
    ∑_α∈ℕ^ns_α((λ))≤∑_α,β|⟨(λ) , ⟩_L^2()|.

 Since ||(λ)||_S_p=∑_α∈ℕ^ns_α((λ)), it is enough to show that the right hand side of (<ref>) is integrable with respect to ϱ. This follows from Lemma 2.1 in <cit.> which reads,
 
 For any q∈ℕ, there exist N_q∈ℕ and constant C_q>0 such that
 
    [ 1+|λ|(|α|+|β|+n)+|α-β| ]^q|⟨(λ),⟩_L^2()|≤ C_q||f||_N_q,,

 where ||f||_N_q, is the classical family of Schwartz semi-norms,
 
    ||f||_N,():=sup_|α|≤ N||(1+|z|^2+t^2)^N/2∂^α_z,tf||_L^∞().

 
 As mentioned before, it follows directly that
 
    ||||_L^1(_1)   =∫_^*tr(|(λ)|)dϱ(λ)
       ≤∫_^*∑_α,β|⟨(λ),⟩_L^2()| dϱ(λ)<∞

 
 
  * To prove the inversion formula first note that taking p=1 in part (i), we get that (<ref>) is well defined. Since (λ) is an integral operator, it is clear that so is (p)^*(λ). The kernel of  (p)^*(λ) is 
    K_f^λ,p=e^-iλ te^iλ(x y/2-xξ)T_f^λ(η, ξ-y),

where p=(x,y,t). The formula for the trace of integral operators (i.e tr((p)^*(λ))=∫_K_f^λ,p(η, η)dη - see <cit.>), gives

    tr((p)^*(λ))
       =e^-iλ t∫_e^-iλ x( η+y/2)∫_f^λ(u,y)e^iλ u (η-y/2)dudη
       =(2π)^n/|λ|^ne^-iλ tf^λ(x,y).

Here, the last equality follows by using frequency modulation and Fourier inversion formula in Euclidean space, together with the change of variables η→η/λ. Integrating both sides with respect to dϱ(λ) and using Euclidean Fourier inversion in the λ variable, completes the proof.
 


In analogy with Euclidean space, if μ∈() one defines its Fourier transform as the -valued map :^*→ given by

    (λ)φ=∫_(p)φ dμ(p).

This is a specific case of the more general theory of characteristic functions of measures on locally compact groups (see for instance H. Heyer <cit.> and E. Siebert <cit.>). It is not hard to see that (λ) is indeed bounded with ||(λ)||_op≤μ(). Many properties of the Fourier transform of functions also extend to measures. For instance, group convolutions are defined as usual by

    f∗μ(p)=∫_f(pq^-1)dμ(q),

and their Fourier transform is a composition of operators

    f∗μ=(λ)(λ).


In studying the group Fourier transform of measures, it is extremely useful to use convolution approximations. Approximations to the identity in the convolution algebra L^1() coincide with classical approximations to the identity in L^1(). The following properties are easy to check


Let {ψ_ϵ}_ϵ>0 be an approximation to the identity in L^1().


  * ψ_ϵ→δ_0 in the weak sense.

  * If f∈ L^1(), ψ_ϵ∗ f→ f in L^1().

  * If g∈𝒞() is bounded,  ψ_ϵ∗ g→ g point-wise.

  * If μ∈(), ψ_ϵ∗μ→μ in the weak sense. 


Here, δ_0 denotes the Dirac distribution. It is possible, by way of (<ref>), to explicitly compute δ_0. For φ∈ L^2(),

    δ_0(λ)φ=∫_(p)φ dδ_0(p)=(0)φ=φ.

That is, δ_0=I, the identity operator. One expects that the weak convergence of approximations to the identity, ψ_ϵ,  implies some form of convergence of _ϵ to I. This is indeed the case


Let {ψ_ϵ} be an approximation to the identity. Then for each λ,
 ψ_ϵ(λ)→ I, in the strong operator topology.
 
 Note that a corollary of this proposition, and the convolution theorem for measures, is that if μ∈(), the function μ_ϵ=ψ_ϵ*μ satisfies μ_ϵ(λ)→(λ) in the strong operator topology.
 
 Fix λ∈^* and φ∈ L^2(). The aim is to show that 
 
    ||(ψ_ϵ(λ)-I)φ||_L^2()→ 0  as ϵ→0.

 Computing ||(ψ_ϵ(λ)-I)φ||_L^2()^2 yields,
 
    ||(ψ_ϵ(λ)-I)φ||_L^2()^2 =⟨(ψ_ϵ(λ)-I)φ, (ψ_ϵ(λ)-I)φ⟩_L^2()
       = ⟨ψ_ϵ(λ)φ, ψ_ϵ(λ)φ⟩_L^2() - ⟨φ, ψ_ϵ(λ)φ⟩_L^2() - ⟨ψ_ϵ(λ)φ, φ⟩_L^2() +⟨φ, φ⟩_L^2()
       =∫_ψ_ϵ(p)(∫_ψ_ϵ(q)⟨(p)φ, (q)φ⟩_L^2() dq )dp
       -∫_ψ_ϵ(p)⟨(p)φ, φ⟩_L^2() dp-∫_ψ_ϵ(p)⟨φ, (p)φ⟩_L^2() dp+⟨φ, φ⟩_L^2().

Since  is a strongly continuous representation, all the integrands are continuous and bounded therefore, using the fact that ψ_ϵ→δ_0 weakly, one sees that as ϵ→ 0, the right hand side of (<ref>) converges to ⟨φ, φ⟩_L^2()-⟨φ, φ⟩_L^2()-⟨φ, φ⟩_L^2()+⟨φ, φ⟩_L^2()=0. This completes the proof.

As mentioned in the introduction, a quick consequence of the extension of ℱ_E to distributions is that if f∈() and μ∈() then

    ∫_fdμ=∫_ℱ_E(f)ℱ_E(μ).

Since, as of now, there is no satisfactory extension of the group Fourier transform to the space of distributions, the analogous result requires additional work. The following lemma will be useful,


 Let H be a Hilbert space and for each ϵ>0, T_ϵ,T∈ S_1( H). Suppose that T_ϵ→ T in the strong operator topology, then there is a subsequence {T_k}_k∈ℕ⊂{T_ϵ}_ϵ>0 such that tr(T_k)→ tr(T) as k→∞.
 

 
 Let {e_j}_j∈ℕ be a fixed orthonormal basis of H. Since T_ϵ→ T in the strong operator topology, it follows that for each j, |⟨ (T_ϵ-T)e_j,e_j⟩_H|→0 as ϵ→0. Therefore, we can obtain a subsequence {T_k}_k∈ℕ such that for all j,  |⟨ (T_k-T)e_j,e_j ⟩_H|→0 monotonically. Moreover, for each k, both T_k and T are traceable so T_k-T is also traceable. In particular, by  Weyl's additive inequality,
 
    ∑_j=1^∞ |⟨ (T_1-T)e_j,e_j ⟩_H|≤ tr(|T_1-T|)<∞.

Now,
 
    |tr(T_k)-tr(T)|   =|tr(T_k-T)|
       = |∑_ȷ=1^∞⟨ (T_k-T)e_j,e_j ⟩_H|
       ≤∑_j=1^∞ |⟨ (T_k-T)e_j,e_j ⟩_H|.

 So the proof is completed by appealing to the decreasing monotone convergence theorem.
 


Let f∈() and μ∈(), then

    ∫_f(p)dμ(p)=∫_^*tr((λ)(λ)^*)dϱ(λ)

 

Let {ψ_ϵ}_ϵ>0∈𝒞^∞_c() be a compactly supported, non-negative, smooth approximation to the identity, and put μ_ϵ=ψ_ϵ*μ. It is clear that μ_ϵ is a compactly supported smooth function, in particular μ_ϵ∈(). Also, μ_ϵ→μ weakly, and for each λ, μ_ϵ(λ)→(λ) in the strong operator topology. Therefore, (λ)μ_ϵ(λ)^*→(λ)(λ)^* in the strong operator topology.
 Since f∈(), then for each λ∈^*, (λ)∈ S_1(L^2()). It follows that for each ϵ>0, λ∈^*, (λ)μ_ϵ(λ)^*, and (λ)(λ)^* are in S_1(L^2()).
By Lemma <ref>, there is a subsequence {μ_k}_k∈ℕ⊂{μ_ϵ}_ϵ>0 such that


  * μ_k→μ weakly,

  * _k(λ)→(λ) in the strong operator topology, and

  * tr((λ)_k(λ)^*)→ tr((λ)(λ)^*).
  
By Theorem <ref> we have,

    ∫_f(p)dμ(p)   =lim_k→∞∫_f(p)μ_k(p)dp
       =lim_k→∞∫_^*tr((λ)μ_k(λ)^*)d ϱ(λ)

Now, by the additive Weyl's inequality, we have 

    |tr((λ)μ_k(λ)^*)|   ≤ tr(|(λ)μ_k(λ)^*|)
       ≤ ||μ_k||_op||(λ)||_S_1
       ≤ ||ψ_k(λ)||_op||(λ)||_op||(λ)||_S_1
       ≤μ()||(λ)||_S_1.

By Theorem <ref>, since f∈(), ∈ L^1(_1), so by the dominated convergence theorem, the limit can be brought inside the integral on the right hand side of (<ref>) to complete the proof.



With this at hand, we are ready to prove Proposition <ref>, 


prop2
Let μ∈().


  * If ∈ L^2(_2), then dμ=f dp for f ∈ L^2().

  * If ∈ L^1(_1), then dμ=g dp for g ∈𝒞().






  * Since the Fourier transform is an isometric isomorphism between L^2() and L^2(_2), it follows that ∃ f∈ L^2() such that (λ)=(λ) for ϱ-a.e. λ. The aim is to show that this f is the density of μ. As before, let {ψ_ϵ}_ϵ>0⊂𝒞^∞_c be an approximation to the identity and put μ_ϵ=ψ_ϵ*μ, and f_ϵ=ψ_ϵ*f. By the convolution theorem,

    μ_ϵ=ψ_ϵ=ψ_ϵ=f_ϵ, ϱ-almost everywhere ,
 
and since μ_ϵ, f_ϵ∈ L^2(), one has that μ_ϵ=f_ϵ almost everywhere. Let ψ be a continuous bounded function on , then

    ∫_ψ(p)dμ(p)   =lim_ϵ→0∫_ψ(p)μ_ϵ(p)dp
       =lim_ϵ→0∫_ψ(p)f_ϵ(p)dp
       =∫_ψ(p)f(p)dp,

which proves the claim.

  * If ∈ L^1(_1) then, |tr( (p)^*(λ))|≤ tr(| (p)^*(λ)|)≤ tr(|(λ)|). It follows that 

    g(p):=∫_^*tr( (p)^*(λ))d ϱ(λ)

is a well defined continuous function on . As before, the aim is to show that g is the density of μ. To see this, let μ_ϵ be as before. Since μ_ϵ∈(), μ_ϵ(λ)∈ S_1 for each λ. Also, μ_ϵ(λ)→(λ) in the strong operator topology. So by Lemma <ref>, there is a subsequence {μ_k} such that
 
 
  * μ_k∈(),
 
  * μ_k→μ weakly,
 
  * _k(λ)→(λ) in the strong operator topology,
 
  * and tr( (p)^*_k(λ))→ tr( (p)^*(λ)).

By Theorem <ref>,

    μ_k(p)=∫_^*tr( (p)^*_k(λ))d ϱ(λ),

but also, 

    |tr((p)^*_k(λ))|≤||(p)^*||_op||ψ_k(λ)||_optr(|(λ)|)≲ tr(|(λ)|)∈ L^1(S_1).

By the dominated convergence theorem, we have a point-wise limit

    lim_k→∞μ_k(p)   =lim_k→∞∫_^*tr( (p)^*_k(λ))d ϱ(λ)
       =∫_^*tr( (p)^*(λ))d ϱ(λ)=g(p).

Since, for all k∈, ||ψ_k||_L^1()≤ 1, it follows that pointwise convergence, implies weak convergence. That is, for any continuous bounded function ψ,


    ∫_ψ(p)dμ(p)=lim_k→∞∫_ψ(p)μ_k(p)dp=∫_ψ(p)g(p)dp.

This completes the proof.


 
  
 
 
 
 
 
 
 
 
 
 
 
 


§ FOURIER COEFFICIENTS IN 


In the Heisenberg group, energy integrals of measures take the form

    I_σ(μ)=∬||q^-1p||_^-σdμ(q)dμ(p).

The inner integral takes the form 

    μ∗ K_σ(p),

 where “∗" denotes group convolution, and K_σ denotes the Riesz kernel for the Koranyi norm, K_σ(p)=||p||_^-σ for 0≤σ<2n+2. One would like to use the convolution theorem to compute energy integrals in the frequency domain. However, K_σ is not in L^p for any p, therefore first step in such a computation would be to extend the group Fourier transform to tempered distributions. This has been recently done by the authors in <cit.> and <cit.> where they introduced a frequency space for  and, with it, an alternative definition for the Fourier transform in the Heisenberg group.  This section will present this approach and all results that will be relevant to the proof of the energy formula. These will be presented without proof but all the details are contained in the original references.

The main idea behind these Fourier coefficients is, instead of studying the operator , to study its matrix coefficients. The matrix coefficients of an operator, are dependent on the choice of an orthonormal basis, however in this case, it turns out to be convenient to choose a basis comprised of  eigenfunctions for the Euclidean Fourier transform.


For x∈ and k∈, the k-th Hermite polynomial is the polynomial

    ℋ_k(x):=(-1)^k[d^k/dx^k{e^-x^2}e^x^2].

The (normalized) k-Hermite function is 

    ϕ_k(x):=(2^k√(π)k!)^-1/2e^-x^2/2ℋ_k(x).

In higher dimensions, for x∈ and multi-index α∈^n,

    ϕ_α(x):=Π_j=1^nϕ_α_j(x_j).

Finally, the re-scaled Hermite functions are given for α∈^n, λ∈^*, and x∈, by

    (x):=|λ|^n/4ϕ_α(|λ|^1/2x).





 Let =^n×^n×^*, and denote a typical point by ζ=(α,β,λ). For f∈ L^1() let ℱ_(f):→ be the map
 
    ℱ_(f)(ζ)=ℱ_(f)(α,β,λ):=⟨(λ),⟩_L^2().

 For simplicity, ℱ_(f) is interchangeably denoted by . This transform shares many important properties, or at least analogues thereof, with the classical Fourier transform. Some of these properties are what allowed the authors to extend it to tempered distribution by characterizing the image of (). Many of these properties will be of great relevance later in this work when computing energies of measures via integrals on the frequency domain. Because of this, a brief introduction to this Fourier coefficient approach is given next. Many of the proofs are skipped, but readers are referred to the aforementioned references where all the proofs are presented and more details are given.
 
Perhaps the most clear advantage of this approach is that the function  can be written as an integral of f with respect to an appropriate kernel. Specifically, letting Φ:×→ be given by
 
    Φ(p,ζ)=⟨(p),⟩_L^2()=e^iλ t∫_ e^iλ xξ(ξ+y/2)(ξ-y/2)dξ,

a quick computation using Fubini's theorem shows that

    (ζ)=∫_Φ(p,ζ)f(p)dp.
 
It is worth mentioning that the integral ∫_ e^iλ xξ(ξ+y/2)(ξ-y/2)dξ is what is known as the Fourier-Wigner transform of  and . In general, the Fourier-Wigner transform of two functions φ,ψ∈ L^2() is the function on  given by

    V_λ(f,g)(z):=⟨(z,0)f,g⟩_L^2()=∫_e^iλ xξf(ξ+y/2)g(ξ-y/2)dξ.



 This integral formulation of  simplifies many computations. For instance, the inversion formula for () becomes much simpler. Before we state it, the space  must be endowed with a measure. For η:→, let dζ be the measure given by

    ∫_η(ζ)dζ=∑_α∈^n∑_β∈^n∫_^*η(α,β,λ)dρ(λ).

One has, for f∈() ,

    f(p)=ℱ_^-1()(p)=∫_Φ(p,z)(ζ)dζ.

One of the most important properties of the Fourier transform is its effect on convolutions. With this approach the convolution theorem recasts as follows,

    ℱ_(f*g)(α,β,λ)=∑_γ∈^n(γ,β, λ)g_(α,γ,λ)=:·g_(α,β,λ).

Yet another property of the map  is that it turns smoothness into decay. This was precisely the content of Lemma <ref> which we now restate in the notation of this section.


 For any q∈ℕ, there exist N_q∈ℕ and constant C_q>0 such that
 
    [ 1+|λ|(|α|+|β|+n)+|α-β| ]^q|(α,β,λ)|≤ C_q||f||_N_q,,

 where ||f||_N_q, is the classical family of Schwartz semi-norms,
 
    ||f||_N,():=sup_|α|≤ N||(1+|z|^2+t^2)^N/2∂^α_z,tf||_L^∞().


This decay result motivates the definition of the metric  on  defined by 

    (ζ,ζ')=|λ(α+β)-λ'(α'+β')|_ℓ^1+|(α-β)-(α'-β')|_ℓ^1+|λ-λ'|.

The metric  will be relevant later. Now we focus on the characterization of (()), which requires several definitions and results first.


Let η:→ be a function differentiable with respect to λ, and f be a function in (). For j∈, denote by δ_j∈^n the point with a “1" in the j^th coordinate and zeros elsewhere, and set ζ^±_j=(α±δ_j,β±δ_j,λ). Define the following six operators,
 
    Δη(ζ):=1/|λ|(|α+β|+n)η(ζ)+1/|λ|∑_j=1^n(√(α_jβ_j)η(ζ^-_j)+√((α_j+1)(β_j+1))η(ζ^+_j)),


 
    D_λη(ζ):=dη/dλ(ζ)+n/λη(ζ)+1/|λ|∑_j=1^n(√(α_jβ_j)η(ζ^-_j)-√((α_j+1)(β_j+1))η(ζ^+_j) ),



    𝒫f(p):=∫_-∞^t(f(z,s)-f(z,-s))ds



    Σ_0(η):=η(α,β,λ)-(-1)^|α+β|η(α,β,-λ)/λ



    (M^2f)(z,t):=|z|^2f(z,t)



    (M_0f)(z,t):=itf(z,t).



Then, the following holds
 For f∈(),

    (M^2f)=-Δ( f)
       (M_0f)=D_λ( f)
       (𝒫f)=-iΣ_0( f)


In a sense, this is analogous to the fact that the Fourier transform turns decay into smoothness. With this at hand one may define the Schwartz space on  as follows,

η∈() if 


  * For any given (α,β)∈^n, η(α,β,·):^*→ is smooth.

  * For any N∈ the functions Δ^Nη, D_λ^Nη, and Σ_0D_λ^Nη, all decay faster than any power of d_0=|λ|(|α+β|+n)+|α-β|.

The space () is equipped with the family of semi-norms 

    ||η||_N,N',():=sup_ζ∈(1+d_0(ζ))^N[Δ^N'η + D_λ^N'η + Σ_0D_λ^N'η].



The map :()→() is a a continuous isomorphism with continuous inverse given by 
    ^-1η(p)=∫_Φ(p,ζ)η(ζ)dζ.


Perhaps the biggest inconvenience of the metric space (,) is that it is not complete, however, for any f∈ L^1(),  is uniformly continuous on (,). It is therefore natural to extend  to the metric completion of (,) which is denoted by (,). 
The metric space (,) is explicitly given by =∪_0, where _0=_∓×^n, with _∓=(_-)^n∪(_+). For ζ∈, if ζ∈_0 we denote it by ζ=(ẋ,k). The metric d_ is given by


    d_(ζ,ζ')=d_(ζ,ζ'),        if  ζ,ζ'∈
       d_((α,β,λ),(ẋ,k))=|λ(α+β)-ẋ|_ℓ^1+|α-β-k|_ℓ^1+|λ|,        if  (α,β,λ)∈, (ẋ,k)∈_0
       d_((ẋ,k),(ẋ',k'))=|ẋ-ẋ'|_ℓ^1+|k-k'|_ℓ^1,        if  (ẋ,k),(ẋ',k')∈_0.
 
Abusing notation, the extension of  is also denoted by . The space () is defined to be the space of all continuous functions, η, in  such that η⌊_∈(). It is not hard to see that 

    ∫_1_{|λ||α+β|+|α-β|≤ R}(ζ)1_{|λ|≤ϵ}(ζ)dζ≲ R^2nϵ.

Therefore, it is natural to extend dζ to  by defining, for all ψ∈𝒞_c(),

    ∫_ψ(ζ)dζ:=∫_ψ⌊_(ζ)dζ.

The convolution theorem extends as follows,

    (f*g)(ẋ,k)=∑_k'∈^n(ẋ,k-k')g_(ẋ,k').



The space () with the semi-norms (<ref>) is a Frechét space, so It makes sense to talk about its topological dual. This will be the class of tempered distributions on  and will be denoted '(). This allows the following definition.
 
 For T∈'(), (T)∈'() is given by
 
    ⟨ T|η⟩_'()×()=⟨ T| ^τη⟩_'()×().

 Here ^τ:()→() is the map
 
    ^τη(p):=∫_Φ(p,ζ)η(ζ)dζ.

 
 Turns out :'()→'() is a continuous injection.
 
  
 
 
 
 
 
 
 
 
 
 
 
 

 
 

 §.§ Fourier coefficients of measures

 

This distributional definition of  clearly applies to measures in (). The goal of this section is to extend the study of properties of this distributional transform keeping in mind the goal of computing the Fourier transform of K_σ. This is precisely a first step in obtaining a frequency formulation of energy integrals. In addition to the distributional definition of , the definition of  on functions can be extended to () by using either (<ref>) or (<ref>). A quick use of Fubini's theorem shows that, just as with functions, these give equivalent definitions. This gives us a point-wise definition of  for μ∈(),
 
    (ζ)=∫_Φ(p,ζ)dμ(p).


 
 
Let  be as in (<ref>), and let μ denote the distributional Fourier transform  μ as in Definition <ref>. Then, μ= as distributions. That is to say, μ is given by integration against the function .
 
 
 The proof is a straight forward computation using Fubini's theorem. Let η∈(), then
 
    ⟨μ|η⟩   =⟨μ|^τη⟩
       =∫_∫_Φ(p,ζ)η(ζ)dζ dμ(p)
       =∫_η(ζ)∫_Φ(p,ζ)dμ(p)dζ=∫_η(ζ)(ζ)dζ.

 
 The following lemma, which before took some work to prove, is now immediate.
 
 For f∈(), μ∈()
 
    ∫_f(p)dμ(p)=∫_(ζ)(ζ)dζ.

 
 
 Once again the proof is a simple computation involving Fubini's theorem and Theorem <ref>,
 
    ∫_f(p)dμ(p)   =∫_[^-1](p)dμ(p)
       =∫_∫_Φ(p,ζ)(ζ)dζ dμ(p)
       =∫_(ζ)∫_Φ(p,ζ)dμ(p)=∫_(ζ)(ζ)dζ.

 
 The hope is to extend this Lemma to the integral in (<ref>). However, while this applies to f∈(), K_σ*μ is not in the Schwarz class and, a priori, may not even be a tempered distribution. Moreover, as of yet, there is no suitable extension of (<ref>) to distributions. If f,g,h∈() a quick computation yields 
 
    ∫_f*g(p)h(p)dp=∫_g(q)f̃*h(q)dq,

 where f̃(p)=f(-p). This motivates the definition of the convolution of distributions with Schwartz functions. We recall the definition here,
 
 For g∈(), T∈'(), g*T is the distribution given, for all f∈(), by
 
    ⟨ g*T|f ⟩=⟨ T|g̃*f⟩.

 
 
 In a similar way, for η,θ,ψ∈() we compute
 
    ∫_η·θ(ζ)ψ(ζ)dζ   =∫_η·θ(ζ)ψ(ζ)dζ
       =∑_α,β∫_^*∑_γη(γ, β, λ)θ(α, γ,λ)ψ(α,β,λ)dρ(λ)
       =∑_γ,α∫_^*θ(α, γ,λ)∑_βη(γ,β,λ)ψ(α,βλ)dρ(λ)
       =∫_θ(ζ)(η_τ·ψ)(ζ)dζ=∫_θ(ζ)(η_τ·ψ)(ζ)dζ,

 
 where η_τ(α,β,λ)=η(β,α,λ), and η_τ(ẋ,k)=η(ẋ,-k). This motivates the following definition.
 
 For η∈() and Ψ∈'(), η·Ψ is defined as the distribution given, for all θ∈(), by,
 
    ⟨η·Ψ|θ⟩=⟨Ψ|η_τ·θ⟩

 
 
 
 Along side the transform ^τ the following transform is also introduced: ^-τ:()→(), given by 
    ^-τf(ζ)=∫_Φ(p,ζ)f(p)dp.

 With this, we have a total of 4 “Fourier-like" transforms ,^-τ:()→() and ^-1,^τ:()→(). The following relations between these are easily verified.
 
 Let f∈(), η∈() and for any function ϑ:→ denote ϑ_-(α,β,λ)=ϑ(α,β,-λ).
 
 
  * ()^-1=^-1,
 
  * (^τ)^-1=^-τ
 
  * ^-τf=[ f]_-
 
  * ^-1η=^τ(η_-)
 
  * ^-τf̃(ζ)=[ f]_τ(ζ)
  
 
 
 
 
 For f∈(), and T∈'(), (f*T)=·T_ in the sense of distributions.
 
  
 Let f∈(), T∈'() and η∈(). Then, using (5) on Lemma <ref>,
 
    ⟨(f*T)|η⟩   =⟨ f*T|^τη⟩
       =⟨ T|f̃*^τη⟩
       =⟨ T|^τ(^-τf̃·η) ⟩
       =⟨ T|^-τf̃·η⟩
       =⟨ T|[f]_τ·η⟩
       =⟨f· T|η⟩.

This completes the proof.
 
Let's finishing this section by computing  δ_0 in the sense of distributions, where δ_0 is the Dirac distribution at zero. This simple computation was done by the authors on <cit.> but since it will be of great relevance when computing the Fourier transform of K_σ, we do it here as well. For η∈(),

    ⟨δ_0|η⟩   =⟨δ_0|^τη⟩
       =^τη(0)=∫_Φ(0,ζ)η(ζ)dζ
       =∫_∫_(ξ)(ξ)dξη(ζ)dζ.
 
Since {}_α∈^n is an orthonormal basis for L^2() it follows that the inner integral is 1_{α=β}. Therefore,

    ⟨δ_0|η⟩=∫_1_{α=β}(ζ)η(ζ)dζ.

It follows that δ_0=1_{α=β} in the sense of distributions.
 
  
 
 
 
 
 
 
 
 
 
 
 
 
 

 §.§ The Fourier transform of the Koranyi-Riesz kernel

 
 Another improtant property of the classical Fourier transform is its interaction with differential operators. In  one should expect any reasonable definition of the Fourier transform to interact nicely with the subLaplacian operator 
 
    Δ_=∑_j=1^nX_j^2+Y_j^2.
 
Turns out that Δ_ is a  multiplier with Δ_f=|λ|(2|α|+n) f, f∈(). In <cit.>, G.B. Folland showed that there is a constant d=d(n) depending only on n such that the fundamental solution of Δ_ is d(n)^-1K_Q-2. This allows to, quite easily, compute K_Q-2 in the sense of distributions. This computation will be done later in greater generality (i.e. for K_σ, 0<σ<Q) but it turns out that 

 
    K_Q-2(α,β,λ)=d(n)1/|λ|(2|α|+n)1_α=β(α,β,λ).

  
This gives an explicit formula for K_σ for a particular choice of σ. To obtain the more general case of arbitrary 0≤σ≤ Q, one might be tempted to consider a fundamental solution of the fractional subLaplacian Δ_^σ/2, and use a similar computation. One can show that if R_σ is a fundamental solution of Δ_^σ/2, then R_σ=d̃(n,σ)1_α=β/[|λ|(2|α|+n)]^σ/2 in the sense of distributions. However it is not true that R_σ is a constant multiple of K_σ (unless, of course, σ=2). In fact, there is no known explicit formula for R_σ. There has been work done in the direction of realizing R_σ as explicitly as possible, see for instance <cit.>, and more recently <cit.> where a specific case is treated. Instead, it proves useful to consider conformally invariant fractional powers of the subLaplacian. This operator arises naturally in CR geometry as studied for instance in  <cit.>, <cit.>, <cit.> and <cit.>. It also arises in studying the extension problem in , <cit.>, and has been studied in the context of Hardy inequalities in the Heisenberg group, <cit.>. In what follows, this operator will be introduced and studied in the context of . 


The function Φ, introduced before, is of the form e^iλ t|λ|^-n/2(2π)^n/2Θ, where

    Θ(z,ζ)=(|λ|/2π)^n/2∫_e^iλ x·ξ(ξ+y/2)(ξ-y/2)dξ=(|λ|/2π)^n/2⟨(z,0)|⟩,

are the rescaled special Hermite functions. As mentioned before, this is also the Fourieri-Wigner transform of  and . The set {Θ(·,ζ)}_α,β∈^n forms a complete orthonormal basis for L^2(^n). In particular, for f∈()

    f^λ(z)=∑_α∑_β⟨ f^λ|Θ(·, ζ) ⟩Θ(z,ζ).


This expansion has a more compact form in terms of Laguerre functions. For k∈, the k^th Laguerre polynomial of type δ>-1 is 

    L_k^δ(t)e^-tt^δ=1/k!(d/dt)^k(e^-tt^k+δ).

The Laguerre functions are given in terms of L_k^δ by 

    ℓ_k^δ(z):=L_k^δ(1/2|z|^2)e^-1/2|z|^2.

Just as with the Hermite functions, ℓ_k^δ can be adapted to the representation theory of  by a simple rescaling. The rescaled Laguerre function is given by ℓ_k,λ^δ(z)=ℓ_k^δ(√(|λ|)z). This functions are closely related to the functions Θ(z,ζ).


    ∑_|α|=kΘ(z,α,α, λ)=(2π)^-n/2(|λ|)^n/2ℓ_k,λ^n-1(z).


For a proof of this the reader is referred to <cit.> (see in particular 2.3.26). Now consider the convolution

    f*Φ(·,ζ_α)(p)=∫_f(q)Φ(q^-1p,ζ_α)dq,

where ζ_α=(α,α,λ)∈. Since 

    Φ(q^-1p,ζ)=e^iλ(t-s+1/2Im(zw))(2π)^n/2|λ|^-n/2Θ(z-w,ζ),

it follows that ,

    f*Φ(·,ζ_α)(p)=(2π)^n/2|λ|^-n/2e^iλ t∫_^nf^-λ(w)Θ(z-w,ζ_α)e^iλ/2Im(zw)dw=(2π)^n/2|λ|^-n/2e^iλ t(f^-λ*_λΘ(·,ζ_α))(z).

Here, *_λ denotes λ-twisted convolutions, given for g,h∈ L^1(^n) by 

    g*_λh(z)=∫_^ng(w)h(z-w)e^iλ/2Im(zw).
 
Now, Since (p) is in L^2() it follows that (p)=∑_β⟨(p)|⟩. Therefore

    Φ(q^-1p,ζ_α)   =⟨(q^-1)∑_β⟨(p)|⟩|⟩
       =∑_β⟨(p)|⟩⟨(q^-1)|⟩
       =∑_βΦ(p,ζ)Φ(q,ζ).

Which gives

    (f*Φ(·,ζ_α))(p)=∑_β∫_f(q)Φ(q,ζ)dqΦ(p,ζ)=(2π)^n|λ|^-ne^iλ t∑_β⟨ f^-λ|Θ(·,ζ)⟩Θ(z,ζ).

With this, and Theorem <ref>, (<ref>) rewrites as

    f^λ(p)   =|λ|^n/(2π)^n∑_k=0^∞∑_|α|=ke^-iλ t(f*Φ(·,ζ))(p)=|λ|^n/2/(2π)^n/2∑_k=0^∞∑_|α|=k(f^-λ*_λΘ(·,ζ_α))(z)
       =|λ|^n/(2π)^n∑_k=0^∞(f^-λ*_λℓ_k,-λ^n-1)(z)=|λ|^n/(2π)^n∑_k=0^∞(f^λ*_-λℓ_k,λ^n-1)(z).

This relation between the special Hermite expansion and the Laguerre expansion will be instrumental in computing 𝔎_σ. First, conformally invariant fractional powers of the subLaplacian are introduced.

For 0≤σ≤ Q the conformally invariant σ-fractional subLaplacian is the operator ℒ_σ:()→() given by 

    ℒ_σf(p)=∫_-∞^∞∑_k=0^∞(2|λ|)^σ/2Γ(k+Q+σ/4)/Γ(k+Q-σ/4)f^λ*_λℓ_k,λ^n-1(z)e^-iλ tdρ(λ).



This `modified' subLaplacian if of great importance here because its fundamental solution is precisely d(n,σ)^-1K_Q-σ where d(n,σ) is a constant depending only on n and σ. A proof of this can be found in <cit.> (see, specifically, equation (3.10)). One can see that taking σ=2

    (2|λ|)^σ/2Γ(k+Q+σ/4)/Γ(k+Q-σ/4)=2|λ|Γ(k+n/2+1)/Γ(k+n/2)=2|λ|(k+n/2)=|λ|(2k+n).

This highlights the relation between ℒ_σ and Δ_^σ/2. Indeed, Δ_^σ/2=U(σ)ℒ_σ where U(σ) is a bounded operator depending only on σ. In particular, U(2) is the identity operator.



For f∈(), ℒ_σ satisfies 

    ℒ_σf=^-1[(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) f].




From the previous discussion, it follows that ℒ_σf can be written as

    ∫_-∞^∞[∑_α,β(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4)(2π)^n|λ|^-n⟨ f^-λ|Θ(·,ζ)⟩Θ(z,ζ)]e^iλ tdρ(λ).

One can also see that 

    (2π)^n/2|λ|^-n/2⟨ f^-λ|Θ(·,ζ)⟩=∫_f(q)Φ(q,ζ)dq=^-τf(ζ).

By (3) on Lemma <ref> and by the change of variable λ→-λ, (<ref>) becomes 

    ∫_(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4)    f(ζ) (2π)^n/2|λ|^-n/2Θ(z,ζ)e^-iλ tdζ
       =∫_(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) f(ζ) Φ(z,ζ)dζ
       =^-1[(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) f],

as claimed.



For f∈(), and ζ∈, 
    ℒ_σf(ζ)=(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) f(ζ)



Follows trivially from Proposition <ref>.


On the frequency side one has the following relation,

For η∈(),

    ℒ_σ^τη=^τ[(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4)η]



Using (3) on Lemma <ref> and Proposition <ref> one has,

    ℒ_σ^τη   =^-1[(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4)^τη]
       =^-1[(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4)η_-]
       =^τ[(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4)η]



In order to compute K_σ explicitly, Corollary <ref> needs to be extend to distributions. A quick computation using Lemma <ref> shows that for f,g∈()

    ∫_ℒ_σf(p)g(p)dp=∫_f(p)ℒ_σg(p)dp.


Therefore, ℒ_σ is extended to distributions in the usual way. For T∈'(), ℒ_σT∈'() is given, for g∈(), by 

    ⟨ℒ_σT|g⟩=⟨ T|ℒ_σg ⟩.

From here one can check that if T∈'(),

    ℒ_σT=(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) T,

In the '() sense. Indeed,

    ⟨ℒ_σT|η⟩   = ⟨ℒ_σT|^τη⟩
       =⟨ T|ℒ_σ^τη⟩
       =⟨ T|^τ[(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4)η]⟩
       =⟨(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) T| η⟩.

From here one obtains the desired result.



    K_Q-σ=d(n, Q-σ)(2|λ|)^-σ/2Γ(|α|+Q-σ/4)/Γ(|α|+Q+σ/4)1_{α=β},

in the sense of distributions.


Recall that d(σ,Q)^-1K_Q-σ is the fundamental solution of ℒ_σ. Therefore, since

    ℒ_σK_Q-σ=(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) K_Q-σ,

it follows that 

    d(n,Q-σ)δ_0=(2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) K_Q-σ.

Recalling the transform of δ_0 we obtain

    (2|λ|)^σ/2Γ(|α|+Q+σ/4)/Γ(|α|+Q-σ/4) K_Q-σ=d(n,Q-σ)1_{α=β},

in the sense of '(), as claimed.


Hereafter, to simplify notation, I denote 

    𝔎_σ(α,λ)=(2|λ|)^-σ/2Γ(|α|+Q-σ/4)/Γ(|α|+Q+σ/4).

 The group Fourier transform of the Koranyi-Riez kernel, has been known for quite some time. It was originally computed by Cowling and Haagerup in <cit.>. Their computation yields

    K_Q-σ(λ)=d̃_n|λ|^-σ/2Γ(σ/2)/Γ(Q-σ)/4)∑_k=1^∞Γ(k+Q-σ/4)/Γ(k+Q+σ/4)P_k(λ),
 
where d̃_n is a constant, and P_k(λ) is the orthogonal projection of L^2() onto the eigenspace {:|α|=k}. 
A quick, computation renders

    ⟨ P_k(λ),⟩_L^2()=1_{α=β and |α|=k},

which, at least formally, further supports Proposition <ref>.
 
  
 
 
 
 
 
 
 
 
 
 
 
 


 §.§ Application to energy integrals

The explicit expression for K_σ brings us a step closer to the coveted analogue of (<ref>). In fact, it is now possible to compute energies of Schwartz functions as an integral over the frequency space, . For g∈(), the σ- energy of g is defined as the integral

    I_σ(g):=∫_∫_||q^-1p||^-σg(q)g(p)dqdp.


For g∈()

    I_σ(g)=d(n,σ)∫_𝔎_Q-σ(α,λ)|g_(ζ)|^2dζ.





    I_σ(g)   =∫_g*K_σ(p)g(p)dp=⟨ g*K_σ|g ⟩
       =⟨(g*K_σ)| g⟩=⟨ K_σ|( g)_τ·(g)⟩
       =d(n,σ)∫_𝔎_Q-σ(α,λ)1_{α=β}(g_)_τ·g_(ζ)dζ
       =d(n,σ)∑_α∫_^*𝔎_Q-σ(α,λ)(g_)_τ·g_(α,α,λ)dρ(λ)
       =d(n,σ)∑_α∫_^*𝔎_Q-σ(α,λ)∑_γ∈^ng_(α,γ,λ)g_(α,γ,λ)dρ(λ)
       =d(n,σ)∑_α,γ∫_^*𝔎_Q-σ(α,λ)|g_(α,γ,λ)|^2dρ(λ)
       =d(n,σ)∫_𝔎_Q-σ(α,λ)|g_(ζ)|^2dζ.



The goal is to replace g∈() by μ∈(). This can be done with the standard convolution approximation. 

Let μ∈()

    I_σ(μ)=d(n,σ)∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ




Let ψ be a compactly supported, non-negative, smooth function with unit mass, so that defining (p)=ϵ^-Qψ(δ_1/ϵp) makes {}_ϵ>0 a compactly supported, smooth approximation to the identity. Set =μ* so that μ∈𝒞^∞_c() as well. By Lemma <ref>, 

    I_σ()=d(n,σ)∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ.

The idea is to show that I_σ(μ)=lim_ϵ→0I_σ() and also 
    lim_ϵ→0∫_𝔎_Q-σ(α,λ)|(ζ)| ^2dζ=∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ.
 I will first deal with the frequency side, and show the convergence by splitting the proof in 2 cases.

First assume that 

    ∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ=∞.

Since Φ is continuous and bounded, it follows from weak convergence that → point-wise. Therefore, by Fatou's Lemma,

    ∞≤lim inf_ϵ→0∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ,

so the equality is trivial.


Now assume 

    ∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ≤∞.

The following identity will be used: Let H be a Hilbert space and {e_j}_j∈ and orthonormal basis. For an operator A∈ℬ(H),

    ∑_j|⟨ Ae_i|e_j⟩_H|^2=||Ae_i||_H^2.


Since 𝔎_Q-σ is independent of β, it follows that ∑_β|(α,β,λ)|^2 is finite for each fixed α∈^n and ϱ-almost all λ∈^*. Moreover it is integrable with respect to the product measure #×ϱ where # stands for the counting measure on ^n. Recalling (<ref>), 

    ∑_β|(α,β,λ)-   (α,β,λ)|^2
       =∑_β|⟨(μ_ϵ(λ)-(λ)),⟩_L^2()|^2
       =||(μ_ϵ(λ)-(λ))||^2_L^2().

Since μ_ϵ(λ)→(λ) in the strong operator topology, it follows that for each fixed λ∈^* and α∈^n,

    lim_ϵ→0∑_β|(α,β,λ)-(α,β,λ)|^2=0.

In particular,

    lim_ϵ→0∑_β|(α,β,λ)|^2=∑_β|(α,β,λ)|^2.

Furthermore, ∑_β|(α,β,λ)|^2 can be dominated as follows,

    ∑_β|(α,β,λ)|^2   =∑_β|⟨ψ_ϵ(λ)(λ),⟩_L^2()|^2
       =||ψ_ϵ(λ)(λ)||^2_L^2()
       ≤ ||ψ_ϵ(λ)||_op^2||(λ)||_L^2()^2
       =||ψ_ϵ||^2_L^1()∑_β|(α,β,λ)|^2=∑_β|(α,β,λ)|^2.

So, by the dominated convergence theorem,

    lim_ϵ→0∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ
    =∑_α∈^n∫_^*𝔎_Q-σ(α,λ)lim_ϵ→0∑_β|(ζ)|^2dζ=∫_𝔎_Q-σ(α,λ)|(ζ)|^2dζ.


Now the energy on the spatial domain is studied.

Once again, if I_σ(μ)=∞, Fatou's Lemma gives,

    ∞≤lim inf_ϵ→0I_σ(),

so the equality is trivial.


Assume I_σ(μ)<∞. By expanding the convolution ψ_ϵ∗μ, and applying Fubini's theorem,

    I_σ()=∬(∬ ||q^-1p||_^-σψ_ϵ(a^-1p)ψ_ϵ(b^-1q)dpdq)dμ(a)dμ(b).

Using W=δ_1/ϵ(a^-1p) and Z=δ_1/ϵ(b^-1q), the inner integral becomes

    ∬ ||δ_ϵ(Z)b^-1aδ_ϵ(W)||_^-σψ(Z)ψ(W)dZdW.


As ϵ→0, this integral goes to ||b^-1a||^-σ_ as long as a≠ b. Moreover, it shows that 

    ∬ ||q^-1p||_^-σψ_ϵ(a^-1p)ψ_ϵ(b^-1q)dpdq≲ ||b^-1a||^-σ_.

By assumption, ||b^-1a||^-σ_ is integrable, so by the Dominated Convergence Theorem

    lim_ϵ→0I_σ()=lim_ϵ→0∬(∬ ||q^-1p||_^-σψ_ϵ(a^-1p)ψ_ϵ(b^-1q)dpdq)dμ(a)dμ(b)
    =∬(lim_ϵ→0∬ ||q^-1p||_^-σψ_ϵ(a^-1p)ψ_ϵ(b^-1q)dpdq)dμ(a)dμ(b)
    =∬ ||a^-1b||_^-σdμ(a)dμ(b)=I_σ(μ).

This completes the proof.





 





§ ACKNOWLEDGEMENTS

The author would like to thank Professor Jeremy Tyson for his advising and guidance during this project. The author would also like to thank Professor Marius Junge for many helpful discussions and insights related to this project.  


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