Feeling the heat in a group of Heisenberg type

Abstract

In this paper we use the heat equation in a group of Heisenberg type $\mathbb{G}$ to provide a unified treatment of the two very different extension problems for the time independent pseudo-differential operators ${\mathcal{L}}^s$ and ${\mathcal{L}}_s$, $0<s\le 1$. Here, ${\mathcal{L}}^s$ is the fractional power of the horizontal Laplacian, and ${\mathcal{L}}_s$ is the conformal fractional power of the horizontal Laplacian on $\mathbb{G}$. One of our main objective is compute explicitly the fundamental solutions of these nonlocal operators by a new approach exclusively based on partial differential equations and semigroup methods. When $s=1$ our results recapture the famous fundamental solution found by Folland and generalised by Kaplan.

Introduction

In Theorem 2 of his 1973 note [17] Folland proved the following remarkable result:

Theorem

The fundamental solution with pole at the group identity of the horizontal Laplacian $-\mathcal{L}$ in the Heisenberg group ${\mathbb{H}}^n$ is given by

$$\mathcal{E}(z,\sigma )=C(n){(|z{|}^4+16{\sigma }^2)}^{-\frac{n}{2}},$$

where $C(n)>0$ is a suitable explicit constant.

Here, we have indicated with $(z,\sigma )\in {\mathbb{R}}^{2n+1}$ the real coordinates of a point in ${\mathbb{H}}^n$. We also note that the normalisation constants in the above expression of $\mathcal{E}(z,\sigma )$ are not the same as those in [17], where a different group law was adopted. In this paper we always use the group law dictated by the Baker-Campbell-Hausdorff formula, see the opening of Section 2 below.

The above theorem has played a pivotal role in the development of analysis in ${\mathbb{H}}^n$. One of the main reasons lies in the interpretation of this Lie group as the boundary of the Siegel upper half-space in ${\mathbb{C}}^{n+1}$, see in this connection the seminal work [19]. But (1.1) has also entered in problems with a geometric flavour, such as the theory of conformal and quasiconformal mappings, the CR Yamabe problem and that of the best constants in the Hardy-Littlewood-Sobolev inequality on ${\mathbb{H}}^n$, see the celebrated works of Korányi and Reimann [45], Jerison and Lee [37] and Frank and Lieb [21]. We also mention that Folland's theorem was generalised by Kaplan to all groups of Heisenberg type in [41, Theorem 2] (see also the earlier work [42, Theorem 1] where the same result was proved for groups of Iwasawa type). One notable aspect of (1.1) is the resemblance with the fundamental solution of −Δ which for $n\ge 3$ is given by $c(n)|x{|}^{2-n}$. To see this, denote by $Q=2n+2$ the homogeneous dimension of ${\mathbb{H}}^n$ attached to the non-isotropic group dilations ${\delta }_{\lambda }(z,\sigma )=(\lambda z,{\lambda }^2\sigma )$, then we can rewrite $\mathcal{E}(z,\sigma )=C(n)N{(z,\sigma )}^{2-Q}$, where $N(z,\sigma )={(|z{|}^4+16{\sigma }^2)}^{1/4}$ is the so called gauge function on ${\mathbb{H}}^n$.³ Since it appears evident that there is no unique choice of a homogeneous gauge (in principle, all properly normalised functions such as ${(|z{|}^{4k}+{\sigma }^{2k})}^{1/4k}$, $k\in \mathbb{N}$, might be deemed as reasonable choices), one is left with wondering how does one a priori know that the function $N(z,\sigma )$ is the natural choice to try?

As a by-product of the results in this note we provide an answer to this question by resorting to the one object which occupies a central position in analysis and geometry: the ubiquitous heat equation. More precisely, suppose one does not know a priori the magic gauge function $N(z,\sigma )$ in ${\mathbb{H}}^n$, or more in general in a group of Heisenberg type $\mathbb{G}$. Corollary 1.3 below shows that, by running the heat flow on $\mathbb{G}$, one is naturally lead into such function. As it will be clear from the subsequent discussion, this result can be viewed as the limiting value ($s=1$) where the results for two different families of nonlocal operators merge: the former can be defined in an arbitrary stratified nilpotent Lie group, aka a Carnot group, and can be almost entirely analysed by semigroup methods; the latter has instead its roots in conformal CR geometry. As a consequence, its natural geometric framework is not a general Carnot group, but rather the Heisenberg group ${\mathbb{H}}^n$ (which is the prototypical CR manifold), and it has so far been studied by a combination of different ideas which include scattering and non-commutative harmonic analysis. In accordance with the notation adopted by Frank and Lieb in their cited work [21], if $\mathcal{L}$ is a horizontal Laplacian on $\mathbb{G}$ as in (2.8) below, we agree to indicate ${\mathcal{L}}^s={(-\mathcal{L})}^s$, whereas ${\mathcal{L}}_s$ will denote the conformal fractional power of the horizontal Laplacian on $\mathbb{G}$. The ambient for the results in this paper will be that of groups of Heisenberg type (for the relevant notion and the main properties of such groups see Section 2 below). Henceforth, we will adhere to the convention of using the superscript s for any quantity which has to do with ${\mathcal{L}}^s$, whereas we will use the subscript s for anything that has to do with ${\mathcal{L}}_s$. Thus, for instance, the fundamental solution of ${\mathcal{L}}^s$ is denoted by ${\mathcal{E}}^{(s)}(g)$, whereas we use ${\mathcal{E}}_{(s)}(g)$ for that of ${\mathcal{L}}_s$.

The main objective of the present work is to show that, notwithstanding their substantial differences, these two classes of nonlocal operators can be treated in a unified way by a systematic use of the heat equation and suitable modifications of the latter. To describe our framework consider first a general Carnot group $\mathbb{G}$ with a fixed horizontal Laplacian $\mathcal{L}$. Throughout this paper we indicate by $P_{t}u(g)=e^{-t\mathcal{L}}u(g)={\int }_{\mathbb{G}}p(g,g^{\prime },t)u(g^{\prime })d{g}^{\prime }$ the heat semigroup constructed by Folland in [18]. We recall that such semigroup is stochastically complete, i.e., $P_{t}1=1$. The semigroup $P_t$ is all that is needed to study the fractional powers ${\mathcal{L}}^s$, for $0<s<1$. We emphasise that we define the action of this nonlocal operator on a function $u\in C_0^{\infty }(\mathbb{G})$ by the well-known formula of Balakrishnan [2],

$${\mathcal{L}}^{s}u(g)=-\frac{s}{\mathrm{\Gamma }(1-s)}\underset{0}{\overset{\infty }{\int }}\frac{1}{t^{1+s}}(P_{t}u(g)-u(g))dt.$$

With (1.2) in hands, we next consider the Riesz potentials defined by the formula

$${\mathcal{I}}^{(2s)}u(g)=\frac{1}{\mathrm{\Gamma }(s)}\underset{0}{\overset{\infty }{\int }}{t}^{s-1}{P}_{t}u(g)dt,0<s<1,$$

see [18]. It is easy to prove, see Proposition 4.1 below, that

$${\mathcal{I}}^{(2s)}\circ {\mathcal{L}}^s={\mathcal{L}}^s\circ {\mathcal{I}}^{(2s)}=I.$$

A direct important consequence of (1.4) is that the kernel

$${\mathcal{E}}^{(s)}(g)\stackrel{def}{=}\frac{1}{\mathrm{\Gamma }(s)}\underset{0}{\overset{\infty }{\int }}{t}^{s-1}p(g,t)dt$$

of the operator ${\mathcal{I}}^{(2s)}$ constitutes the fundamental solution of the nonlocal operator ${\mathcal{L}}^s$ with pole at the group identity. In Theorem 5.1 below, we provide an explicit integral expression for such kernel in the setting of groups of Heisenberg type. In discrepancy with the above mentioned Corollary 1.3, such result shows in particular that in the logarithmic coordinates $g=(z,\sigma )\in \mathbb{G}$ one has ${\mathcal{E}}^{(s)}(z,\sigma )=\mathrm{\Phi }(|z{|}^4,|\sigma {|}^2)$, but gauge symmetry breaks down when $0<s<1$. However, in the limit as $s\nearrow 1$ the conformal geometry of the group appears, in the sense that with further work it is possible to recover Corollary 1.3 from Theorem 5.1.

This leads us to introduce the second pseudo-differential operator ${\mathcal{L}}_s$. Following the work by Branson, Fontana and Morpurgo [6, (1.33)], the nonlocal operator ${\mathcal{L}}_s$ can be defined in the Heisenberg group ${\mathbb{H}}^n$, with $T={\partial }_{\sigma }$, via the spectral formula

$${\mathcal{L}}_s=2^s|T{|}^s\frac{\mathrm{\Gamma }(-\frac{1}{2}\mathcal{L}|T{|}^{-1}+\frac{1+s}{2})}{\mathrm{\Gamma }(-\frac{1}{2}\mathcal{L}|T{|}^{-1}+\frac{1-s}{2})}.$$

Formula (1.6) is the counterpart of the well-known representation ${(-\mathrm{\Delta })}^{s}u={\mathcal{F}}^{-1}{(2\pi |\xi |)}^{2s}\hat{u})$, see [52, Chapter 5], except that, as it will soon be clear, now matters are much more involved. More in general, in a group of Heisenberg type $\mathbb{G}$, with logarithmic coordinates $g=(z,\sigma )\in \mathbb{G}$, where σ is the vertical variable, the pseudo-differential operator ${\mathcal{L}}_s$ is defined by the following generalisation of (1.6)

$${\mathcal{L}}_s=2^s{(-{\mathrm{\Delta }}_{\sigma })}^{s/2}\frac{\mathrm{\Gamma }(-\frac{1}{2}\mathcal{L}{(-{\mathrm{\Delta }}_{\sigma })}^{-1/2}+\frac{1+s}{2})}{\mathrm{\Gamma }(-\frac{1}{2}\mathcal{L}{(-{\mathrm{\Delta }}_{\sigma })}^{-1/2}+\frac{1-s}{2})},$$

see [51].

From our perspective, the unfavourable aspect of either definitions (1.6) or (1.7) is that if one wants to study the nonlocal operators ${\mathcal{L}}_s$ starting from them, then one is immediately led into the fairly elaborate computational aspects connected with non-commutative Fourier analysis on the group $\mathbb{G}$, thus losing sight of the remarkable flexibility of the heat equation offered by (1.2). Since the present work is about the heat flow, instead of (1.7) we will henceforth adopt the following definition which, at least formally, seems identical to (1.2),

$${\mathcal{L}}_{s}u(g)=-\frac{s}{\mathrm{\Gamma }(1-s)}\underset{0}{\overset{\infty }{\int }}\frac{1}{t^{1+s}}[P_{(-s),t}u(g)-u(g)]dt,$$

where $0<s<1$, and $u\in C_0^{\infty }(\mathbb{G})$. In (1.8) we have indicated with $P_{(-s),t}$ a linear operator on $L^p(\mathbb{G})$ that is associated with a modified heat equation and whose origin will be explained in detail in Section 3, but see also the discussion leading to (1.17) below. The equivalence between formulas (1.7) and (1.8) was established by Roncal and Thangavelu in Proposition 4.1 of their remarkable paper [50] on optimal Hardy inequalities, see also the companion work [51] in which they generalised their results to groups of Heisenberg type.

Having defined ${\mathcal{L}}_s$ via (1.8), for $0<s\le 1$ we now introduce two modified heat flows on $\mathbb{G}$. We consider the kernel

$${\mathcal{K}}_{(s)}((z,\sigma ),t)=\frac{2^k}{{\left(4\pi t\right)}^{\frac{m}{2}+k}}\underset{{\mathbb{R}}^k}{\int }{e}^{-\frac{i}{t}〈\sigma ,\lambda 〉}{\left(\frac{|\lambda |}{\sinh |\lambda |}\right)}^{\frac{m}{2}+1-s}{e}^{-\frac{|z{|}^2}{4t}\frac{|\lambda |}{\tanh |\lambda |}}d\lambda ,$$

and we denote by ${\mathcal{K}}_{(-s)}((z,\sigma ),t)$ the function obtained by changing s into −s in (1.9). We note that, in the local case $s=1$, the kernel ${\mathcal{K}}_{(s)}$ coincides with the Gaveau-Hulanicki-Cygan heat kernel $p((z,\sigma ),t)$, see definition (2.12) below. If with a slight abuse of notation we let

$${\mathcal{K}}_{(\pm s)}(g,g^{\prime },t)={\mathcal{K}}_{(\pm s)}(g^{-1}\circ g^{\prime },t),$$

then we consider the two linear operators on $L^p(\mathbb{G})$ defined by the formula

$$P_{(\pm s),t}u(g)=\underset{\mathbb{G}}{\int }{\mathcal{K}}_{(\pm s)}(g,g^{\prime },t)u(g^{\prime })d{g}^{\prime }.$$

We note explicitly that $P_{(-s),t}$ is the Roncal-Thangavelu operator that appears in (1.8) (as a help to the reader we mention that in [50, Formula (2.18)] they denote by ${\mathcal{K}}_t^s$ what we indicate with ${\mathcal{K}}_{(-s),t}$).

At this point, using the operator $P_{(s),t}$ we introduce the counterpart of the Riesz operator (1.3) above

$${\mathcal{I}}_{(2s)}u(g)=\frac{1}{\mathrm{\Gamma }(s)}\underset{0}{\overset{\infty }{\int }}{t}^{s-1}{P}_{(s),t}u(g)dt.$$

We stress that, unlike (1.2) and (1.3), which are both defined using the same semigroup $P_t$, in (1.8) and (1.11) two different modified heat operators appear. We are now in a position to state our first main result which represents the conformal counterpart of (1.4).

Theorem 1.1

For every $0<s<1$ and $u\in C_0^{\infty }(\mathbb{G})$ one has

$$({\mathcal{I}}_{(2s)}\circ {\mathcal{L}}_s)u=({\mathcal{L}}_s\circ {\mathcal{I}}_{(2s)})u=u.$$

We emphasise that our proof of Theorem 1.1, which is given in Section 4, is based on some lemmas of independent interest which are inspired to semigroup methods. In particular, in Lemma 4.2 we establish a key representation formula for the group convolution of the intertwined kernels ${\mathcal{K}}_{(s)}(\cdot ,t)$ and ${\mathcal{K}}_{(-s)}(\cdot ,\tau )$. We also mention Lemma 4.6, which establishes a remarkable cancellation property. In the following theorem, which is our second main result, we compute the kernel of the operator (1.11) explicitly.

Theorem 1.2

Let $\mathbb{G}$ be a group of Heisenberg type. The following statements hold:

• With ${\mathcal{K}}_{(s)}$ defined by (1.9), for any $0<s\le 1$ one has

  $${\mathcal{E}}_{(s)}(z,\sigma )\stackrel{def}{=}\frac{1}{\mathit{\Gamma }(s)}\underset{0}{\overset{\infty }{\int }}{t}^{s-1}{\mathcal{K}}_{(s)}((z,\sigma ),t)dt=\frac{C_{(s)}(m,k)}{N{(z,\sigma )}^{Q-2s}},$$

  where $Q=m+2k$ is the homogeneous dimension of $\mathbb{G}$, $N(z,\sigma )={(|z{|}^4+16|\sigma {|}^2)}^{1/4}$ is the natural gauge, and we have let

  $$C_{(s)}(m,k)=\frac{2^{\frac{m}{2}+2k-3s-1}\mathit{\Gamma }(\frac{1}{2}(\frac{m}{2}+1-s))\mathit{\Gamma }(\frac{1}{2}(\frac{m}{2}+k-s))}{{\pi }^{\frac{m+k+1}{2}}\mathit{\Gamma }(s)}.$$

• The distribution ${\mathcal{E}}_{(s)}\in C^{\infty }(\mathbb{G}\setminus \{e\})\cap L_{loc}^1(\mathbb{G})$, and it provides a fundamental solution of ${\mathcal{L}}_s$ with pole at the group identity $e\in \mathbb{G}$ and vanishing at infinity.

The reader should note the notable similarity between (1.12) and the well-known result, see e.g. [24, Theorem 8.4], stating that the function $E_s(x)=\frac{c(n,s)}{|x{|}^{n-2s}}$ is the fundamental solution with pole at $x=0$ of the nonlocal operator ${(-\mathrm{\Delta })}^s$. Theorem 1.2 provides a first example of the remarkable connection between the heat equation in a group of Heisenberg type and its conformal geometry. We mention that (1.12) was first obtained with a completely different approach, based on the Fourier analysis on groups, by Roncal and Thangavelu in [50, Section 3]. For instance, in order to invert the operator ${\mathcal{L}}_s$ they rely on the deep formula of Cowling-Haagerup [11] which gives the group Fourier transform of the kernels ${\mathcal{E}}_{(-s)}$ obtained by changing s into −s in (1.12). Instead, we deduce the invertibility of ${\mathcal{L}}_s$ directly from our Theorem 1.1, and then prove (1.12), (1.13) independently. Another difference is that we treat all groups of Heisenberg type at once, whereas in [50] the authors first establish the Heisenberg group case, and then in [51] they use partial Radon transform and several facts from Lie theory to ultimately reduce matters to the case of ${\mathbb{H}}^n$.

Concerning the question in the opening of this paper, at this moment it seems appropriate to state separately the special case $s=1$ of Theorem 1.2. As we have already mentioned, the next result provides a heat equation proof of Folland's formula (1.1) and its generalisation in the cited paper [41] by Kaplan.

Corollary 1.3 Discovering the gauge from the heat

Let $\mathbb{G}$ be a group of Heisenberg type. Then, the fundamental solution of $-\mathcal{L}$ is given by

$$\underset{0}{\overset{\infty }{\int }}p((z,\sigma ),t)dt=\frac{2^{\frac{m}{2}+2k-2}\mathit{\Gamma }(\frac{m}{4})\mathit{\Gamma }(\frac{1}{2}(\frac{m}{2}+k-1))}{{\pi }^{\frac{m+k+1}{2}}}{(|z{|}^4+16|\sigma {|}^2)}^{-\frac{1}{2}(\frac{m}{2}+k-1)}.$$

As we have said, the present work is purely based on the analysis of various heat kernels using techniques inspired by pde's and semigroup theory. The common starting point of our analysis are the parabolic extension problems associated with both nonlocal operators ${\mathcal{L}}^s$ and ${\mathcal{L}}_s$. While we defer a detailed discussion to Section 3, to provide the reader with some perspective here we confine ourselves to recall such problems. Given a function $u\in C_0^{\infty }(\mathbb{G}\times {\mathbb{R}}_t)$, the extension problem for ${({\partial }_t-\mathcal{L})}^s$ consists in finding $U\in C^{\infty }(\mathbb{G}\times {\mathbb{R}}_t\times {\mathbb{R}}_y^{+})$ such that

$$\{\begin{array}{c}{\mathfrak{P}}^{(s)}U\stackrel{def}{=}\frac{{\partial }^{2}U}{\partial y^2}+\frac{1-2s}{y}\frac{\partial U}{\partial y}+\mathcal{L}U-\frac{\partial U}{\partial t}=0,\hfill \\ U(g,t,0)=u(g,t).\hfill \end{array}$$

Here, we have let $g=(z,\sigma )\in \mathbb{G}$, and we have denoted by $y>0$ the extension variable. The conformal counterpart of (1.14) is formulated in a similar way, but the problem is substantially different. Given a function $u\in C_0^{\infty }(\mathbb{G}\times {\mathbb{R}}_t)$, find a function $U\in C^{\infty }(\mathbb{G}\times {\mathbb{R}}_t\times {\mathbb{R}}_y^{+})$ such that

$$\{\begin{array}{c}{\mathfrak{P}}_{(s)}U\stackrel{def}{=}\frac{{\partial }^{2}U}{\partial y^2}+\frac{1-2s}{y}\frac{\partial U}{\partial y}+\frac{y^2}{4}{\mathrm{\Delta }}_{\sigma }U+\mathcal{L}U-\frac{\partial U}{\partial t}=0,\text{in}\mathbb{G}\times {\mathbb{R}}_t\times {\mathbb{R}}_y^{+},\hfill \\ U(g,t,0)=u(g,t).\hfill \end{array}$$

The presence of the differential operator $\frac{y^2}{4}{\mathrm{\Delta }}_{\sigma }$ is what makes (1.15) so diverse from (1.14), but it is also what gives a geometric meaning to (1.15). To justify this statement we recall that the fundamental solution of the operator ${\mathfrak{P}}^{(s)}$ in (1.14) is given by

$$q^{(s)}(g,g^{\prime },t,y)\stackrel{def}{=}{g}^{(s)}(y,t)p(g,g^{\prime },t),$$

where $p(g,g^{\prime },t)$ is the heat kernel in $\mathbb{G}$, and we have let $g^{(s)}(y,t)={(4\pi t)}^{-(1-s)}{e}^{-\frac{y^2}{4t}}$ denote the heat kernel in the space with fractal dimension ${\mathbb{R}}_y^{2(1-s)}\times {\mathbb{R}}_t^{+}$. On the other hand, the fundamental solution of the operator ${\mathfrak{P}}_{(s)}$ in (1.15) is given by the function

$$q_{(s)}((z,\sigma ),t,y)=\frac{2^k}{{(4\pi t)}^{\frac{m}{2}+k+1-s}}\underset{{\mathbb{R}}^k}{\int }{e}^{-\frac{i}{t}〈\sigma ,\lambda 〉}{\left(\frac{|\lambda |}{\sinh |\lambda |}\right)}^{\frac{m}{2}+1-s}{e}^{-\frac{|z{|}^2+y^2}{4t}\frac{|\lambda |}{\tanh |\lambda |}}d\lambda .$$

The origin of this function is in the fact that ${\mathfrak{P}}_{(s)}$ is to be viewed as a parabolic Baouendi-Grushin operator (see (3.18)) in the space with fractal dimension ${\mathbb{R}}^{m+2(1-s)}\times {\mathbb{R}}^k\times (0,\infty )$ and whose fundamental solution can be explicitly computed, see Proposition 3.1. Once this is recognised, then the connection between the kernel (1.9) of the operators $P_{(s),t}$ in the conformal Riesz operator (1.11) and the function (1.16) is given by the formula

$${\mathcal{K}}_{(s)}((z,\sigma ),t)={(4\pi t)}^{1-s}{q}_{(s)}((z,\sigma ),t,0).$$

In other words, ${\mathcal{K}}_{(s)}$ is an appropriately rescaled restriction to the thin space $y=0$ of the Baouendi-Grushin kernel $q_{(s)}$ in (1.16).

To unravel the conformal geometry in (1.16) we now note that, from general principles, we know that

$${\mathfrak{e}}_{(s)}((z,\sigma ,y)\stackrel{def}{=}\underset{0}{\overset{\infty }{\int }}{q}_{(s)}((z,\sigma ),t,y)dt$$

is a fundamental solution with pole at the origin of the time-independent part of ${\mathfrak{P}}_{(s)}$, i.e., the conformal extension operator

$${\mathfrak{L}}_{(s)}=\frac{{\partial }^2}{\partial y^2}+\frac{1-2s}{y}\frac{\partial }{\partial y}+\frac{y^2}{4}{\mathrm{\Delta }}_{\sigma }+\mathcal{L}.$$

This observation leads us to state the following result.

Theorem 1.4

Let $0<s\le 1$. In any group of Heisenberg type $\mathbb{G}$, the distribution in the thick space $\mathbb{G}\times {\mathbb{R}}_y^{+}$ defined by (1.18) is given by

$${\mathfrak{e}}_{(s)}((z,\sigma ),y)=\frac{\mathit{\Gamma }(s)}{{(4\pi )}^{1-s}}{C}_{(s)}(m,k){({(|z{|}^2+y^2)}^2+16|\sigma {|}^2)}^{-\frac{1}{2}(\frac{m}{2}+k-s)},$$

where $C_{(s)}(m,k)$ is the constant in (1.13). An equation similar to (1.20) holds if we replace s with −s, provided that $\mathit{\Gamma }(s)$ is replaced by $|\mathit{\Gamma }(-s)|$.

Remark 1.5

The reader should note that, changing s into −s in the constant $C_{(s)}(m,k)$, we obtain from (1.20) exactly the same number in formula (1.9) in [51, Theorem 1.2]. This can be easily recognised by using in their formula Legendre's duplication property for the gamma function recalled in (5.18) below.

In closing, we mention that in the Heisenberg group ${\mathbb{H}}^n$ the time-independent extension problem for the operator (1.19) (see (3.15) below) was first introduced and solved by Frank, Gonzalez, Monticelli and Tan in [20] using harmonic analysis and scattering theory. The same extension problem was also studied by Möllers, Orsted and Zhang in [48]. Exploiting the conformal invariance of the differential operators they used unitary representation theory of reductive Lie groups to solve the problem. A point of view different from these authors was taken up by Roncal and Thangavelu in their cited works [50], [51] on optimal Hardy inequalities. In these papers the authors use the parabolic extension problem (1.15) and Fourier analysis on groups to establish an explicit Poisson representation formula for the solution of the time-independent problem (3.15). One of the purposes of the present work is to further develop the point of view in [50], [51] from a different semigroup perspective with the objective to unify the treatment of the very diverse nonlocal operators ${\mathcal{L}}^s$ and ${\mathcal{L}}_s$.

A brief discussion about the organisation of our work seems in order. In Section 2 we collect various known facts that will be needed in the main body of the paper. In Section 3 we describe in detail the evolutive extension problems for ${\mathcal{L}}^s$ and ${\mathcal{L}}_s$ from a unifying perspective. It is there that we introduce the fundamental solution $q_{(s)}$ in (1.16), its companion $q_{(-s)}$, and the intertwined operators $P_{(\pm s),t}$ in (1.10). Section 4 is entirely devoted to proving Theorem 1.1. In Section 5 we prove Theorem 1.2, Theorem 1.4.

Acknowledgment: We are grateful to M. Cowling, G. Folland, A. Korányi, C. Morpurgo and S. Thangavelu for providing us with some interesting historical overviews during the preparation of this work. Special thanks go to S. Thangavelu for his insightful discussions of his joint papers with L. Roncal.

We also thank the anonymous referee for his/her very careful reading of the original manuscript and constructive comments that have contributed to improve the presentation of the paper.