Heat Kernels for a Class of Hybrid Evolution Equations

The aim of this paper is to construct (explicit) heat kernels for some hybrid evolution equations which arise in physics, conformal geometry and subelliptic PDEs. Hybrid means that the relevant partial differential operator appears in the form L1+L2−∂t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathscr{L}}_{1} + {\mathscr{L}}_{2} - \partial _{t}$\end{document}, but the variables cannot be decoupled. As a consequence, the relative heat kernel cannot be obtained as the product of the heat kernels of the operators L1−∂t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathscr{L}}_{1} - \partial _{t}$\end{document} and L2−∂t\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathscr{L}}_{2} - \partial _{t}$\end{document}. Our approach is new and ultimately rests on the generalised Ornstein-Uhlenbeck operators in the opening of Hörmander’s 1967 groundbreaking paper on hypoellipticity.

Consider a second order partial differential operator L and the heat equation L u−∂ t u = 0 associated with it.Following a well-established tradition by heat kernel we mean a function p(x, ξ, t) such that for any ξ the function p(•, ξ, •) is a solution of the heat equation, and p(x, •, t) −→ δ x in the distributional sense as t → 0 + .The aim of this paper is to construct explicit heat kernels for some hybrid evolution equations which arise in diverse frameworks such as e.g.sub-Riemannian geometry and problems from the applied sciences that are modelled by some classes of subelliptic equations.By hybrid we mean that the relevant partial differential operator appears in the form L 1 + L 2 − ∂ t , but the variables cannot be decoupled.Consequently, the relative heat kernel cannot be simply obtained as the product of the heat kernels of the operators L 1 − ∂ t and L 2 − ∂ t .Our approach is completely self-contained, elementary, and it is purely based on PDE methods whose final objective is to emphasise the so far unexplored connection of the relevant class of hybrid equations with the generalised operators of Ornstein-Uhlenbeck type in the opening of Hörmander's groundbreaking 1967 work [29].It is worth mentioning here that as a by-product we obtain a simple proof of the well-known (non-hybrid) cases of the heat operator in a stratified nilpotent Lie group of step two and of the Baouendi-Grushin operator (see respectively Sections 4 and 3 below).
To motivate our results we next discuss some prototypical examples which fall within the scope of our approach.We begin with an example from conformal CR geometry.In recent years the study of the so-called extension operators has received increasing attention from workers in analysis and geometry especially in connection with certain conformally invariant nonlocal operators.A typical situation of interest is represented by the Heisenberg group H n ∼ = C n × R with real coordinates (z, σ)1 and horizontal Laplacian (1.1) In their seminal paper [18] Frank et al. have introduced the following extension problem: given where the fractional parameter s ∈ (0, 1).The term y2 4 ∂ σσ U has a geometric meaning whose explanation comes from the equivalence between (1.2) and the scattering eigenvalue problem in complex hyperbolic space.The second order time-independent PDE in (1.2) is a notable example of the type of hybrid equations that are the object of interest of the present paper.To clarify this aspect we observe that if we formally think of w as a generic point in the space with fractal dimension R 2(1−s) , and we let y = |w| denote its "distance" to the origin, then the PDE in (1.2) can be interpreted as the action of the differential operator ∆ w + |w| 2  4 ∂ σσ +L on functions having spherical symmetry in w.If we consider the heat equation associated with such operator, it is immediate to recognise that in such equation the variables (w, σ) ∈ R 2(1−s) × R and g = (z, σ) ∈ H n cannot be decoupled since the variable σ appears in both the limiting operators ∆ w + |w| 2 4 ∂ σσ − ∂ t and L − ∂ t (see (1.1)).In Section 4 we will show that the heat kernel with pole at the origin associated with (1.3) is given by q (s) ((z, σ), t, y) = 2 (4πt) We emphasise that (1.2) is dramatically different from the extension problem à la Caffarelli-Silvestre (1.5) in which the geometric term y 2 4 ∂ σσ U is missing.The evolution PDE associated with (1.5) is and it should be clear to the reader that this is not of hybrid type since its fundamental solution q (s) ((z, σ), t, y) = (4πt) −(1−s) e − y 2 4t p((z, σ), t) is indeed the product of the fundamental solutions of the two heat operators ∆ w − ∂ t and L − ∂ t (the reader should note that we have used a superscript (s) to distinguish such heat kernel from that in (1.4), for which we have used a subscript (s)).Formula (1.4) (see also the more general case treated in Theorem 4.1 below) plays a critical role in the analysis of conformal properties of a certain pseudodifferential operator L s which arises as the Dirichlet-to-Neumann map of (1.2), and we refer the interested reader to the works [41], [38], [42], [24], [25] for more insights into this aspect.Another significant model of the class of equations encompassed by the present paper is the following: where w, y ∈ R n , σ ∈ R k and t > 0. The equation (1.6) is a hybrid between the time-dependent Baouendi-Grushin operator in There exists no available treatment of (1.6) in the literature, but our approach produces the following explicit heat kernel (with pole at a generic point (w 0 , σ 0 , y I n e 2πt|λ| (2πt|λ|−sinh(2πt|λ|))+(cosh(2πt|λ|)−1)(e 2πt|λ| −1) Formula (1.7) is a special case of the more general Theorem 3.6 below, to which we refer the reader.We now briefly discuss the organisation of the paper.In Section 2 we recall the class (2.1) of generalised Ornstein-Uhlenbeck operators in the opening of Hörmander's cited paper [29], and for completeness provide a short proof of Proposition 2.1 since this result constitutes the backbone of the present work.Section 3 introduces the hybrid class (3.1), of which the equation (1.6) discussed above is a prototypical representative.Besides its own interest, such section is instrumental to the rest of the paper.In the subsection 3.1 we solve the Cauchy problem (3.4) for a generalised harmonic oscillator.The main result is Proposition 3.2 that establishes a generalisation of the classical formula of Mehler.In Section 3.2 we use this result to derive the heat kernel for the Baouendi-Grushin operator, see (3.19) in Theorem 3.4.In Theorem 3.6 we finally construct the heat kernel for the class of hybrid equations in (3.1).Section 4 represents the more geometric part of the paper.There we construct the heat kernel for the conformal extension problem (4.1).The latter represents a time-dependent generalisation to arbitrary groups of Heisenberg type of the above discussed conformal extension problem (1.2) from [18].The main result of the section is Theorem 4.1.To prove it we follow a pattern similar to that in Section 3. We first construct the heat kernel for a generalised harmonic oscillator with a complex drift.This step serves as a building block in the proof of the main Theorem 4.1.In the process, and as a by-poduct of our approach, we also provide a new elementary proof of the famous formula of Hulanicki-Gaveau-Cygan for the heat kernel on a Carnot group of step two, see Theorem 4.6.

The generalised Ornstein-Uhlenbeck operators of Hörmander
In this section we recall a well-known explicit heat kernel that constitutes the essential ingredient of the present work.Consider the class of differential equations in R N × (0, ∞), Here, the N × N matrices Q and B have real, constant coefficients, and moreover A basic feature of the operator K is the invariance with respect to the following non-Abelian group law in [35].We emphasise that the evolution equation K u = A u − ∂ t u = 0 encompasses operators that are very different in nature.Besides of course the classical heat equation (Q = I N and [39] (Q = I N and B = −I N ), but also the very degenerate equation of Kolmogorov from the kinetic theory of as well as the degenerate Ornstein-Uhlenbeck equation which arises in the Smoluchowski-Kramers approximation of Brownian motion with friction, see [9] In [29] Hörmander proved that (2.1) is hypoelliptic if and only if its covariance matrix satisfies the following Kalman condition for one (and therefore every) t > 0 (2.2) The hypothesis (2.2) will henceforth be assumed in this section.Under such assumption we note that t → tK(t) is strictly increasing in the sense of quadratic forms: one has in fact for any It follows that Therefore, the matrix is well-defined, and of course it is symmetric and nonnegative definite.Formally, K −1 ∞ is the inverse of the matrix ∞ 0 e sB Qe sB ⋆ ds, but it is well-known that the latter is well-defined if and only if all the eigenvalues of B have strictly negative real part (see, e.g., [10,Proposition 2.3]).On the other hand K −1 ∞ is well-defined for any choice of Q, B satisfying (2.2), even if it possibly has a non-trivial kernel.
To introduce the main result of this section we next recall the time-dependent intertwined pseudo-distance The behaviour for large t of tK(t) and m t (•, •) has been instrumental in our previous works [21,22] in establishing several functional inequalities related to the differential operator A .
Proposition 2.1.The heat kernel of (2.1) is given by More precisely, for any f ∈ C(R N ) such that Proof.The proof of (2.4) is known and fairly elementary.Denoting by ξ the dual variable of z, and letting û(ξ, t) = R N e −2πi ξ,z u(z, t)dz, then on the Fourier transform side the Cauchy problem reduces to solving (2.7) Now (2.7) can be easily solved via the method of characteristics.Fixing (ξ, t) ∈ R N × (0, ∞), one considers g(s) = û(e sB ⋆ ξ, s + t).This function, in turn, solves Recalling (2.2), we see that g(s) is given by g(s) = û(ξ, t)e −4π 2 s K(s)ξ,ξ e −s tr B .
Since g(−t) = f (e −tB ⋆ ξ) and K(−t) = e −tB K(t)e −tB * , this implies the remarkable formula The representation formula (2.6) follows from (2.8) by taking the inverse Fourier transform and straightforward manipulations.It is proved in [16,Theorem 1.4] that, if |f (z)| ≤ Ce c|z| 2 for some positive constants c, C, then the function u = P t f is solution to K u = 0 in a suitable strip R N × (0, T ) and it attains the initial datum f .We need to prove that, given f satisfying (2.5), the function u = P t f is well defined for every t > 0 and it defines in fact a solution of the Cauchy problem in the whole R N × (0, ∞).To see this, for any (z, t) ∈ R N × (0, ∞) we write Exploiting (2.5) we thus infer Property (2.3) ensures, for every fixed t > 0, the existence of µ t > 0 such that Inserting this information in (2.9) we deduce that By arguing in a similar way one can compute the derivatives of u at any point (z, t) ∈ R N ×(0, ∞) by exchanging the order of derivation and integration: this shows that u is solution and completes the proof of the theorem.
It is easy to see that for the heat equation (Q = I N and B = O N ) the matrix tK(t) = tI N and therefore K −1 ∞ = O N .The matrix K −1 ∞ is the null matrix also for the Kolmogorov equation Given a N ×N matrix C with real coefficients, the notation j(C) will denote the matrix identified by the power series of the function j.It is worth noting that j(C) is invertible with inverse matrix given by j(C) where j is as in (2.10).Since the previous identity can be rewritten as follows we then obtain (2.12) (tK(t)) −1 = 1 t e tD j(2tD)e tD .
The Smoluchowski-Kramers equation the opening of the section falls within the class considered in the following example.
Proof.As a first step we observe that with Q and B given above, where the lower indices indicate the dimensions of the various zero matrices, the Kalman condition (2.2) is satisfied.Since the Moreover, the commutator between V i and V 0 is given by From this relation and the fact that Im (B 0 ) = R n 1 we deduce that the vector fields V 0 , V 1 , ...V n satisfy Hörmander's finite rank condition on the Lie algebra in [29] and therefore the operator K is hypoelliptic.As we have recalled, this is equivalent to saying that (2.2) hold.
Next, we compute and , where By means of known formulas for the inverse of a partitioned matrix (see, e.g., [30, Section 0.7.3]), we obtain We now notice that, as t → ∞, we have the limiting relations which establishes (2.13).

Baouendi met Kolmogorov
In this section we discuss a first interesting class of hybrid evolution equations which, remarkably, is directly amenable to the setting of Proposition 2.1 by means of partial Fourier transform and a suitable exponential transformation, see (3.30) below.The reader should bear in mind that the work in this section is also instrumental to the remainder of the paper.Consider an invertible and symmetric n × n matrix Q 1 .Let n 1 ≤ n, and fix also a n 1 × n matrix B 0 having maximum rank n 1 .We denote the relevant variables , and the term ∆ w f appears in both L 1 and L 2 .Before proceeding we emphasise that the hybrid PDE in (3.1) encompasses equations as diverse as the parabolic Baouendi-Grushin equation in see [1], [27,28], and the already mentioned degenerate Kolmogorov equation in see [34].The former of these two limiting cases is obtained by taking n 1 = 0 and Q 1 = I n in (3.1), whereas the latter corresponds to taking k = 0, n = n 1 , and B 0 = I n in (3.1).To ease the reader's understanding we first discuss in detail our approach to constructing the fundamental solution of (3.2) since this allows to present some of the ideas in a significant, yet simpler model.This will be accomplished in the next two Subsections 3.1 and 3.2, the former of which contains a self-contained construction of the Mehler fundamental solution for the generalised harmonic oscillator in (3.4) below by reducing such operator to a special case of Proposition 2.1.We mention that when the matrix D is a multiple of the identity such fundamental solution is wellknown and we could have simply lifted its expression from the literature, see Remark 3.3 below.
In line with the declared self-contained spirit of the present paper, our objective is to show that Proposition 3.2 below can be derived from Proposition 2.1 by elementary considerations.

The harmonic oscillator aka Ornstein-Uhlenbeck.
In what follows given a number n ∈ N we denote by D ∈ M n×n (R) a matrix such that D = D ⋆ , D ≥ 0. We consider the Cauchy problem for the generalised harmonic oscillator where v 0 is suitably chosen.We have the following key lemma.
Lemma 3.1.Suppose that the functions v and w are connected by the transformation Dz,z +t tr D) w(z, t).Then, v is a solution to the PDE in (3.4) if and only if w is a solution to the following equation of Ornstein-Uhlenbeck type ∆w − 2 Dz, ∇w − w t = 0.
Proof.We compute The equation (3.6) proves the lemma.
The next proposition is the main result of this subsection.Proposition 3.2 (generalised Mehler formula).Let M be given by the following formula with j as in (2.10).Then, for any v 0 ∈ C(R n ) such that Proof.In view of Lemma 3.1 we see that if v solves the Cauchy problem (3.4), then w = e To solve (3.9) we intend to apply Proposition 2.1 with N = n, Q = I n and B = −2D.Keeping Example 2.2 in mind, we know from (2.12) and (2.11) that with this choice we have Therefore, the initial datum in (3.9) is equal to and, since v 0 is continuous and satisfies (3.8), it nicely fits the assumption of Proposition 2.1.According to (2.4)-(2.6) the solution of (3.9) is thus given by where we have let This shows (3.7).We have finally proved Proposition 3.2.
In what follows we will use the following alternative expression of (3.7) In particular, by performing the change of variable Formula (3.16) will be useful in the proof of Theorem 3.4 below.
Remark 3.3.The reader may find it interesting to compare (3.7) with the classical 1866 Mehler formula for the harmonic oscillator ∆u − ω|x| 2 u − u t = 0, see e.g.[7, Section 4.2], This formula follows immediately from (3.7) by taking D = √ ωI n in its expression.

3.2.
The heat kernel of the Baouendi-Grushin operator.In his 1967 Ph.D. Dissertation [1] under the supervision of B. Malgrange, S. Baouendi first studied the Dirichlet problem in L 2 for a class of degenerate elliptic operators that includes the following model where (w, σ) ∈ R n × R k .At that time M. Vishik was visiting Malgrange, who discussed with him the thesis project of Baouendi.Vishik subsequently asked Malgrange permission to suggest to his own Ph.D. student, Grushin, to work on some questions related to the hypoellipticity of operators modelled on (3.17), see [27,28].This is how the operator (3.17) became known as the Baouendi-Grushin operator.This operator is also important since it is connected to harmonic functions with special symmetries in a group of Heisenberg type G.We notice, in this respect, that there is no global group law underlying (3.17), but the operator is invariant with respect to standard translations (w, σ) → (w, σ + σ ′ ) along the manifold of degeneracy M = {0} × R k .We consider the Cauchy problem The next result provides an explicit heat kernel for (3.17).
Then for every f ∈ S (R n+k ) the function Proof.We indicate with û(w, λ, t) = R k e −2πi λ,σ u(w, σ, t)dσ the partial Fourier transform of u with respect to the variable σ ∈ R k , with dual variable λ ∈ R k .Applying such Fourier transform to (3.18), for any fixed λ ∈ R k we obtain This is a Cauchy problem for a harmonic oscillator such as (3.4) above, with matrix D = D(λ) = π|λ|I n .From Proposition 3.2 we know that the solution of such problem is given by the formula where M λ (w, w ′ , t) is Mehler's fundamental solution given by (3.20) see Remark 3.3.We know that û((w, λ), t) −→ t→0 + f (w, λ) in the pointwise sense.We will now show that, for every fixed w ∈ R n , such convergence also holds in L 1 (R k , dλ).To see this we write Applying (3.16) with D = π|λ|I n we easily obtain From this identity we immediately see that From (i) and (ii) we infer by dominated convergence theorem that On the other hand, by applying (3.15) with D = π|λ|I n and performing the change of variables 2πt|λ| , we deduce where the last limiting relation can be again justified via dominated convergence theorem since f ∈ S and therefore f (w ′ , λ) is continuous at (w, λ) and it can be bounded by an integrable function uniformly in w ′ .This proves that (3.21) lim If we now take the inverse Fourier transform of û((w, λ), t), we find the following representation for the solution of problem (3.18) We stress that (3.21) ensures uniform, and therefore pointwise convergence of u((w, σ), t) to f (w, σ) as t → 0 + .From (3.22) it is clear that the heat kernel of the parabolic Baouendi-Grushin equation in (3.18) is thus given by where M λ (w, w ′ , t) is as in (3.20).Changing variable λ → 2πtλ in the integral over R k , we finally obtain (3.19) from (3.23).
We mention the works [12,2,11] for various derivations and integral representations of the heat kernel for (3.17) when k = 1.For related discussions about the fundamental solutions of more general (time-independent) Baouendi-Grushin operators we refer the reader to [19,5,3].
3.3.Back to the hybrid equation.In this subsection we finally solve the Cauchy problem (3.1).In preparation for our main result, Theorem 3.6 below, we introduce for every λ ∈ R k the (n + n 1 ) × (n + n 1 ) matrices For We next establish a lemma that will play a crucial role in the proof of Theorem 3.6.
Lemma 3.5.For any t > 0 and λ ∈ R k we have .
Proof.The case λ = 0 is trivial since tK λ (t) is still a positive definite matrix and K −1 λ,∞ is the null matrix.We can thus assume λ = 0. Keeping in mind the explicit form of e −tB λ (see Example 2.3 with O n 1 ×n 1 is a (n + n 1 ) × (n + n 1 ) matrix of rank n.We can then exploit the formula for the inverse of a small-rank adjustment (see, e.g., [30, Section 0.7.4]), which is sometimes referred to as the Sherman-Morrison-Woodbury formula, to infer that where we have used the fact that the first block K 11 (t) in tK λ (t) is equal to

3). By exploiting the simple inequality
which implies the desired conclusion.
Proof.As before, our first step is to take the Fourier transform of (3.1) with respect to the variable σ, with dual variable λ ∈ R k .If we let Our second step is to make the following change of dependent variable v λ → u λ , where the two functions are linked by the relation This step represents a generalised version of (3.5) in Lemma 3.1.After some straightforward computations one recognises that in terms of the function u λ the problem (3.29) becomes in Remarkably, the PDE in (3.31) can be cast in the form (2.1), where now N = n + n 1 , and the matrices Q and B = B λ are given by (3.24).The covariance matrix is given by the positive definite matrix K λ (t) in (3.25).By (3.26) we can rewrite the initial datum in (3.31) as f0 (w, λ, y).
As we intend to take the inverse Fourier transform of v λ (w, y, t) with respect to λ, we want to understand the behaviour of v λ (w, y, t) with respect to this variable.Since f 0 belongs to the Schwartz class, we know that f0 (w 0 , λ, y 0 ) decays faster than any polynomial in λ in a uniform way with respect to (w 0 , y 0 ).Our objective is thus to analyse the function p λ ((w, y), (w 0 , y 0 ), t)e 1 2 π|λ| √ Q 1 w 0 ,w 0 dw 0 dy 0 .
Using (3.26) and the explicit expression of p λ (X, X 0 , t) in (2.4), we write where in the last equality we have used the change of variables X 0 → Z, where and the fact that R n+n 1 e −|Z| 2 dZ = π n+n 1 2 . At this point a small miracle happens since from Lemma 3.5 we have, for every X ∈ R n+n 1 , λ ∈ R k , and t > 0, that e tB λ X, e tB λ X .
Inserting this information in (3.32) we obtain where in the last inequality we have exploited (2.3) to obtain I − 1 2 (tK λ (t)) ≥ 1 2 I. Moreover we also have (3.34)lim The limiting behaviour in (3.34) can be checked by using the change of variables X 0 → Z where Z in the definition of I X (λ, t), as this gives Since e tB λ X + (4tK λ (t)) 1 2 Z → X as t → 0 + , we easily obtain (3.34) from the above identity.Hence, by exploiting (3.33) and (3.34), we can argue as in the proof of (3.21) in Theorem 3.4 to deduce that Keeping in mind that v λ (w, y, t) = f (w, λ, y, t), we are now ready to take the inverse Fourier transform, obtaining √ Q 1 w 0 ,w 0 e −2πi σ 0 ,λ f 0 (w 0 , σ 0 , y 0 )dσ 0 dw 0 dy 0 dλ.
Since f 0 ∈ S and thanks to (3.33), f (w, σ, y, t) is a well-defined and smooth function and it coincides with the expression stated in (3.28).Proceeding verbatim as in the proof of Theorem 3.4 and using (3.35), we also see that f is solution of the Cauchy problem (3.1).We conclude that the function h(•, •, •) defined by (3.27) does provide the heat kernel.We stress that, for any (w, σ, y), (w 0 , σ 0 , y 0 ) ∈ R n × R k × R n 1 and t > 0, h((w, σ, y), (w 0 , σ 0 , y 0 ), t) is well-defined (and smooth) since, by arguing as in (3.32)-(3.33),we have where in the last inequality we have used that tr √ Q 1 > 0 and the fact that (det tK λ (t)) −1 grows at most polynomially with respect to λ (see the explicit form of tK λ (t) in Example 2.3 with . This finishes the proof of the theorem.

A class of heat kernels from conformal CR geometry
In this section we construct the heat kernel of a class of hybrid evolution equations that play an important role in conformal CR geometry.In Section 1 we have already discussed the extension problem (1.2) in the Heisenberg group H n in the seminal work of Frank et al. [18].More in general, we now consider a Lie group of Heisenberg type G with logarithmic coordinates (z, σ), Our present objective is the computation of the heat kernel of the evolution PDE in (4.1), i.e. of the equation defined in The following is our main result.
Theorem 4.1.Let G be a group of Heisenberg type.For every 0 < s < 1 the heat kernel with pole at the origin of the operator L (s) − ∂ t in (4.2) is given by We emphasise that although the heat kernel (4.3), and its intertwined counterpart obtained by replacing s with −s, have played a central role in our recent works [24] and [25], see also the earlier papers by Roncal and Thangavelu [41,42], their derivation was not given in the cited sources and it appears for the first time in the present paper.
As in the case of H n (which constitutes the k = 1 case of our treatment), a key observation here is that the PDE in (4.2) is the restriction of the equation to functions depending on the variable y = |w|, where w belongs to the space with fractal dimension R 2 (1−s) .The link between (4.4) and the PDE in (4.2) is readily seen by observing that, if y = |w|, then on a function u(w) = ψ(y) we have ∆ w u = ∂ yy ψ + 1−2s y ∂ y ψ.Another remark is that (4.4) is of hybrid type since the variable σ appears in both equations In the special case of the Heisenberg group H n one has m = 2n, k = 1, and (4.6) gives back the famous formula independently found by Hulanicki [31] and Gaveau [26].We mention here that in [17] Folland proved the existence of the heat kernel in any Carnot group, but of course in such generality one does not have an explicit representation such as (4.6).
Remark 4.2.The reader should note that if in (4.3) we formally set s = 1 and y = 0 we perfectly recover the Hulanicki-Gaveau formula (4.6) when the pole (ζ, τ ) = (0, 0)! Similarly to what we did in Section 3, in the present section we first provide in Theorem 4.6 a totally self-contained and elementary proof of the construction of the heat kernel for the limiting case (4.5).We do this not just for groups of Heisenberg type, but in the more general framework of a Carnot group G of step two.Of course the result per se is not new, as Cygan established it in [15], but our proof is.Although the relevant PDE (4.5) is not hybrid in the sense specified in the opening of this paper, the motivation for including here the construction of its heat kernel is twofold: (i) on one hand it allows to present our approach to Theorem 4.1 in a significant, yet simplified setting; (ii) on the other hand we feel that our self-contained proof will be of interest to workers in analysis and PDEs who are not directly familiar with those important and deep tools, such as e.g. group representation theory, Laguerre calculus, complex Hamiltonians or a priori ansatz, which in one form or another have entered the previous related works such as [31,26,15,4,40,33,6,13,36,8,37].
4.1.The generalised harmonic oscillator with a complex drift.In what follows given a number n ∈ N we denote by S ∈ M n×n (R) a skew-symmetric matrix, i.e., we assume S ⋆ = −S.We intend to solve the Cauchy problem for the following generalised harmonic oscillator with a complex drift where ṽ0 is suitably chosen.We will need the following lemma that allows to eliminate the complex drift from (4.7).
Lemma 4.3.Suppose that v and ṽ are connected by the relation Then, ṽ is a solution to the PDE in (4.7) if and only if v is a solution to the equation Proof.Let ṽ be a solution to the PDE in (4.7).We note that by the skew-symmetry of S we know that e −2itS ⋆ e −2itS = I n .
Using this observation the reader can verify by a direct computation that ∆v(z, t) = ∆ṽ(e −2itS z, t).
It is also clear from (4.8) that Combining the latter two equations we easily reach the desired conclusion.
Returning to the Cauchy problem (4.7) the following is the main result of this subsection.
the function Proof.It is clear that using Lemma 4.3 the Cauchy problem (4.7) is transformed into for the function v defined by (4.8).If we now define D = √ S ⋆ S = √ −S 2 , then clearly D ≥ 0, D ⋆ = D, and |Sz| 2 = |Dz| 2 .We can thus re-write (4.11) as follows According to Proposition 3.2 the function and M (z, ζ, t) as in (3.7), solves the latter problem.Undoing (4.8) we have proved that the function defined by (4.12) ṽ(z, t) = R n M (e 2itS z, ζ, t)ṽ 0 (ζ)dζ solves (4.7).We next want to further simplify the expression (4.12).Keeping in mind the explicit expression (3.7) for M , we have We now observe that, since S commutes with any even analytic function of D = √ −S 2 , we have in particular (4.14) j(2tD) cosh(2tD)e 2itS z, e 2itS z = j(2tD) cosh(2tD)z, z , as well as We thus find from (4.15) Replacing now (4.14) and (4.16) in (4.13), we finally obtain This concludes the proof.
It is clear from the previous proof that the kernel Q(z, ζ, t) is equal to M (e 2itS z, ζ, t) with M given by (3.7) and with D = √ S ⋆ S. Using the commutation property in (4.14) we then deduce from (3.16) that This equation will be used in the proof of Theorem 4.6 below.
Remark 4.5.Notice that Ker A(λ) = Ker J(λ) can have positive dimension.Nevertheless, the injectivity of J ensures that A(λ) is not the null endomorphism for every λ = 0.Moreover, being J(λ) skew-symmetric, the dimension of the range of A(λ) has to be at least two.Since the linearity of J allows us to write A(λ) = |λ| A(λ/|λ|), one can deduce that there exists k 0 > 0 such that A(λ) has at least two eigenvalues bigger than k 0 |λ|.This implies that Although we will not make an explicit use of this fact, we note that (4.23) implies the following commutation relation Given a function f ∈ C 1 we will indicate by ∇ H f = (X 1 f, ..., X m f ) its horizontal gradient, and set . The horizontal Laplacian generated by the orthonormal basis {e 1 , ..., e m } of V 1 is the second-order differential operator on G defined by where X 1 , ..., X m are given by (4.23).A computation gives (4.24) where ∆ z represents the standard Laplacian in the variable z = (z 1 , ..., z m ), and For f ∈ S (recall that we are thinking of G ∼ = R m × R k ) we now consider the Cauchy problem The aim of this section is to provide a new simple proof of the following classical result.
It is appropriate to mention that although the integral representations in the above cited literature appear in different forms with respect to (4.27), they are in fact equivalent to it.The advantage of our presentation is that it is particularly transparent and is easily applicable for instance to delicate questions in geometric measure theory treated in our work [23].
Proof of Theorem 4.6.Fix f ∈ S .To solve (4.26) we start from the expression (4.24) of the horizontal Laplacian, keeping also (4.25) in mind.We identify G with R m × R k and, denoting for the partial Fourier transform of u with respect to the central variable σ ∈ R k , we obtain from (4.24) Furthermore, we obtain from (4.25) Using the identities (4.29), (4.30) we can thus write (4.28) in the form where for λ ∈ R k fixed, we have let ṽ(z, t) = û(z, λ, t).To solve problem (4.31) we now apply Proposition 4.4 with the choice of the skew-symmetric matrix S = πJ(λ), so that the symmetric matrix D = √ −S 2 is presently given by D = π A(λ) = π −J(λ) 2 (we warn the reader that the role of the dimension n in Proposition 4.4 is now taken by the dimension m of the first layer V 1 ).Formula (4.10) allows to conclude that û with the kernel Q as in (4.9).We can now argue as in the proof of Theorem 3.4.First, by exploiting (4.17), we can ensure that û(z, λ, t) → f (z, λ) in L 1 (R k , dλ) as t → 0 + .Then, we can conclude that the function solves the Cauchy problem (4.26).The desired heat kernel on the group G is thus given by where we have used the expression (4.9).Making the change of variable (−2πtλ) → λ and exploiting the linearity and the skew-symmetry of the mapping J(•), we finally obtain From (4.22) it is easy to verify that the integral in (4.32) is finite, and the kernel q((z, σ), (ζ, τ ), t) is a smooth function.This completes the proof of Theorem 4.6.
For the sake of completeness we close this brief subsection by showing how to recover the Gaveau-Hulanicki formula (4.6) from Theorem 4.6.We recall that a Carnot group of step two G is said of Heisenberg type if for every λ ∈ V 2 the mapping A(λ) in (4.21) satisfies A(λ) = −J(λ) 2 = |λ| 2 I m .Therefore, A(λ) = |λ|I m , and we have j( A(λ)) = j(|λ|I m ) = j(|λ|)I m , and also We thus obtain from (4.27) which coincides with the expression recalled in (4.6).

4.3.
Proof of Theorem 4.1.In this section we finally turn to the proof of Theorem 4.1.As we have already observed, we begin from the crucial observation that, in a group of Heisenberg type G, the relevant heat equation associated with (4.2) is (4.4), with the variable w running in the space with fractal dimension R 2 (1−s) .Henceforth, to continue the analysis we proceed formally and treat the number 2(1 − s) as if it were an integer.In this way we will arrive to a specific candidate for a heat kernel in the form (4.3).Only at that point we will rigorously justify our computations and complete the proof.We begin by observing that the assumption J(λ) 2 = −|λ| 2 I m implies in particular that J(ε ℓ )z, J(ε ℓ ′ )z = |z| 2 δ ℓℓ ′ for every z ∈ V 1 and every ℓ, ℓ ′ ∈ {1, ..., k} (see, e.g, [20, Prop.2.9]).Inserting this information in (4.24) we conclude that in a group of Heisenberg type the horizontal Laplacian is given by (4.33) Combining (4.4) and (4.33), it is clear that we presently want to solve the Cauchy problem in (1−s) .Proceeding as in the Section 4.2 we now take a partial Fourier transform of the problem (4.34) with respect to the variable σ ∈ R k .Denoting by Û (z, λ, w, t) = R k e −2πi λ,σ U (z, σ, w, t)dσ, and using the identity (4.30), we obtain from (4.34) At this point we make critical use of the elimination of the complex drift transformation introduced in Lemma 4.3.For λ ∈ R k , we define V (z, w, t) = Û (e −2πitJ(λ) z, λ, w, t).
We leave it to the reader to verify that, similarly to Lemma 4.3, the function V now solves the problem According to Proposition 3.2, if we denote by M the kernel in (3.7) with the choice of D = π|λ|I m+2(1−s) , we deduce that V is given by By taking the inverse Fourier transform we then have From (4.35) we can extract the following heat kernel for the problem (4.34) By arguing as in (4.13)-(4.16)(where we have deduced the expression of Q) and similarly to (4.32), we obtain from the explicit expression of M that If we set w ′ = 0 into (4.36), and we keep (4.20) in mind, as well as the definition of q (s) (•, •, •) in (4.3), we easily realize that The previous identities, together with the already discussed relation between the equations (4.4) and (4.2), make the function q (s) ((z, σ), t, y) the candidate heat kernel of L (s) − ∂ t with pole at the origin for any value of the fractional parameter s ∈ (0, 1).We now rigorously prove this fact, thus establishing Theorem 4.1.
On the one hand, a straightforward computation shows that q (s) ((z, σ), t, y) does solve the equation (4.2) in {t > 0}.We are thus left with understanding in which sense the kernel q (s) ((z, σ), t, y) approaches the delta-function δ (0,0,0) as t → 0 + .With this in mind we introduce the measure This is a natural measure for the operator L (s) since it is easy to check that L (s) is symmetric (i.e.formally self-adjoint) with respect to dµ.Moreover, the renormalising constant 2π 1−s Γ(1−s) has been chosen in such a way that the following lemma holds true.Lemma 4.7.For every t > 0 we have R m ×R k ×(0,∞) q (s) ((z, σ), t, y)dµ = 1.
It is easy to see that, for any fixed t > 0, K M q (s) ((z, σ), t, y) dµ where in the last step we have applied the inversion theorem for the Fourier transform and the fact that f t (0) = 1.This concludes the proof of (4.37).

t 2 I n 12 t 3
I n .Instead, for the Ornstein-Uhlenbeck equation (Q = I N and B = −I N ) the matrix tK(t) = 1 2 (1 − e −2t )I N and therefore K −1 ∞ = 2I N .In the following two examples we discuss two situations that will be useful in the remainder of the present work.Henceforth in this paper we indicate by j : R → R the real analytic function defined by (2.10) j(τ ) = τ sinh(τ ).

Example 2 . 3 (
Degenerate Ornstein-Uhlenbeck).Let N = n + n 1 with n, n 1 ∈ N and n 1 ≤ n.Consider a symmetric and positive definite n × n matrix D 1 and a n 1 × n matrix B 0 with rank n 1 .For the operator K in (2.1) corresponding to the choice
every t > 0 we next consider the covariance matrix associated with Q and B λ If we keep in mind Example 2.3 with the choice D 1 = π|λ| √ Q 1 , we know that the matrix K λ (t) is positive definite for every t > 0, i.e. it satisfies the Kalman condition (2.2) above.
denotes a horizontal Laplacian in G (see (4.33) below), then in this more general framework the parabolic counterpart of the extension problem (1.2) is as follows: given 4.2.Gaveau, Hulanicki and Cygan met Ornstein and Uhlenbeck.Henceforth, we denote by G a Carnot group of step two and we let g = V 1 ⊕V 2 indicate its Lie algebra, with inner product •, • .Recall that the step two assumption means that [ we know that the map J is injective.Via the Baker-Campbell-Hausdorff formula