2512.23830

# Fractal Mehler kernels and nonlinear geometric flows **Authors**: Nicola Garofalo > School of Mathematical and Statistical Sciences Arizona State University 2020 Mathematical subject classification: 35K08, 35R11, 53C18, 35K55 ## Abstract In this paper we introduce a two-parameter family of Mehler kernels and connect them to a class of Baouendi-Grushin flows in fractal dimension. We also highlight a link with a geometric fully nonlinear equation and formulate two questions. Key words and phrases: Mehler kernels. Conformal invariance. Fundamental solutions. Nonlinear flows Contents 1. 1 Introduction 1. 1.1 Generalized Mehler kernels 1. 2 Proof of Theorem 1.1 1. 3 Proof of Theorem 1.3 1. 4 Connections with nonlinear equations ## 1. Introduction The heat equation in a stratified nilpotent Lie group $\mathbb{G}$ is the $L^{2}$ gradient flow $$ \frac{\partial f}{\partial t}=-\frac{\partial\mathscr{E}_{2}(f)}{\partial f} $$ of the energy $$ \mathscr{E}_{2}(f)=\frac{1}{2}\int_{\mathbb{G}}|\nabla_{H}f|^{2}, $$ where $|\nabla_{H}f|^{2}=\sum_{i=1}^{m}(X_{i}f)^{2}$ is the left-invariant carré du champ associated with a basis of the horizontal (bracket generating) layer of the Lie algebra. The discovery of explicit heat kernel formulas in nilpotent Lie groups endowed with special symmetries has played a central role in harmonic analysis, geometric analysis, and the theory of subelliptic operators. In their seminal and independent works [32, 35], Gaveau and Hulanicki established the representation for the heat kernel in any Lie group $\mathbb{G}$ of Heisenberg type $$ \displaystyle G_{2}((z,\sigma),t)=\frac{2^{k}}{(4\pi t)^{\frac{m}{2}+k}}\int_{\mathbb{R}^{k}}e^{-\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\frac{m}{2}}e^{-\frac{|z|^{2}}{4t}\frac{|\lambda|}{\tanh|\lambda|}}\,d\lambda. $$ Cygan subsequently extended the representation (1.1) to the broader class of stratified nilpotent Lie groups of step two; see [14]. Throughout this paper, the symbol $k$ will always refer to the “vertical” dimension of $\mathbb{G}$ If the Lie algebra $\mathfrak{g}$ of $\mathbb{G}$ decomposes as $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{v}$ , with $[\mathfrak{h},\mathfrak{h}]=\mathfrak{v}$ and $[\mathfrak{h},\mathfrak{v}]=\{0\}$ , we denote by $m=\dim\mathfrak{h}$ and $k=\dim\mathfrak{v}$ . Identifying $\mathfrak{h}\cong\mathbb{R}^{m}$ and $\mathfrak{v}\cong\mathbb{R}^{k}$ , the pair $(z,\sigma)$ represents the logarithmic coordinates of the group element $\exp(z,\sigma)\in\mathbb{G}$ .. The prototypical example of Lie groups of Heisenberg type is the $2n+1$ -dimensional Heisenberg group $\mathbb{H}^{n}$ , for which $k=1$ ; however, beyond $\mathbb{H}^{n}$ , there exists a rich and natural family of such groups. Notably, in the Iwasawa decomposition $\mathbb{G}=KAN$ of a simple Lie group of real rank one, the nilpotent component $N$ is always a group of Heisenberg type; see [40], [42], and [13]. We recall that such Lie groups can be described as the boundary at infinity of the real, complex, quaternionic or octonionic hyperbolic space, see [52, Sec. 9.3] and also [55, Sec. 10]. The natural family of dilations in $\mathbb{G}$ , associated with the step two stratification of its Lie algebra, is given by $$ \delta_{\ell}(z,\sigma)=(\ell z,\ell^{2}\sigma), $$ from which it follows that the homogeneous dimension of the group is $Q=m+2k$ . This is consistent with the structure of (1.1), which shows that the heat kernel $G_{2}$ is homogeneous of degree $\kappa=-Q$ with respect to the heat dilations $$ \delta^{(h)}_{\ell}((z,\sigma),t)=(\delta_{\ell}(z,\sigma),\ell^{2}t), $$ see also [22, (3.2) in Theor. 3.1]. A remarkable feature of the Fourier integral (1.1) is that it reveals the intrinsic Korányi-Folland gauge Such function first appeared in [45], see also [42, 43]. $$ N(z,\sigma)=(|z|^{4}+16|\sigma|^{2})^{1/4} $$ which appears in the explicit fundamental solution of the horizontal Laplacian $\Delta_{H}=\sum_{j=1}^{m}X_{j}^{2}$ on $\mathbb{G}$ , first obtained by Folland [21] in the Heisenberg group $\mathbb{H}^{n}$ (see also [23]), and subsequently generalized by Kaplan [39] to all groups of Heisenberg type. By the word “reveals” in the above comment we mean the following: in the well-known Euclidean heat kernel $g(x,t)=(4\pi t)^{-\frac{n}{2}}e^{-\frac{|x|^{2}}{4t}}$ , the distance function $|x|$ appears explicitly. In the sub-Riemannian heat kernel (1.1) there exists no hint of the function (1.2). Despite this, we have the following result, which is a by-product of the main conformal theorem in the work [27] (see Cor. 1.3): $$ \int_{0}^{\infty}G_{2}((z,\sigma),t)\,dt=C(m,k)\ N(z,\sigma)^{-(m+2k-2)}, $$ where $$ C(m,k)=2^{\frac{m}{2}+2k-2}\Gamma(\frac{m}{4})\Gamma(\frac{1}{2}(\frac{m}{2}+k-1))\pi^{-\frac{m+k+1}{2}}. $$ As a consequence of (1.3), if we had no a priori knowledge of the important function (1.2), then by running the heat flow (1.1), we would be forced to discover it. ### 1.1. Generalized Mehler kernels The first objective of this paper is to introduce the following two-parameter extension of (1.1): $$ \displaystyle G_{\alpha,\beta}((z,\sigma),t)=\frac{2^{k}}{(4\pi t)^{\beta}}\int_{\mathbb{R}^{k}}e^{-\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\alpha}e^{-\frac{|z|^{2}}{4t}\frac{|\lambda|}{\tanh|\lambda|}}\,d\lambda,\qquad\beta>\alpha>0. $$ One should note that the kernel $G_{\alpha,\beta}$ remains spherically symmetric in both $z$ and $\sigma$ , and it is homogeneous of degree $\kappa=-2\beta$ with respect to the above mentioned heat dilations $\delta^{(h)}_{\ell}((z,\sigma),t)$ . Indeed, one checks immediately from (1.4) that $$ \delta^{(h)}_{\ell}G_{\alpha,\beta}=\ell^{-2\beta}\,G_{\alpha,\beta}. $$ In Theorem 1.1 and Proposition 1.2 we will prove that, up to a computable universal constant, (1.4) represents the heat kernel, with pole at the origin, of a degenerate parabolic operator on $\mathbb{R}^{+}_{r}\times\mathbb{R}^{k}_{\sigma}\times\mathbb{R}^{+}_{t}$ which arises naturally from a reflected Bessel process of fractal dimension $m_{\alpha}=2\alpha$ . More precisely, consider the degenerate evolution operator $$ \mathfrak{P}_{\alpha,k}=\partial_{t}-\mathscr{L}_{\alpha,k}=\partial_{t}-\partial_{rr}-\frac{2\alpha-1}{r}\partial_{r}-\frac{r^{2}}{4}\Delta_{\sigma}, $$ acting on $(r,\sigma,t)\in\mathbb{R}^{+}_{r}\times\mathbb{R}^{k}_{\sigma}\times\mathbb{R}^{+}_{t}$ . We call this operator a fractal Baouendi-Grushin flow since, when $m=2\alpha\in\mathbb{N}$ , it represents the action of the operator in $\mathbb{R}^{m}_{z}\times\mathbb{R}^{k}_{\sigma}\times\mathbb{R}^{+}_{t}$ $$ \partial_{t}-\Delta_{z}-\frac{|z|^{2}}{4}\Delta_{\sigma} $$ on functions $u((r,\sigma),t)$ , with $r=|z|$ . In (1.5) the Bessel operator $$ \mathscr{B}_{r}^{(2\alpha-1)}=\partial_{rr}+\frac{2\alpha-1}{r}\partial_{r} $$ is classical and plays a central role in analysis, geometry, and stochastic processes with radial symmetries; see [64, 65, 66, 53, 61, 49, 37, 41, 8, 30, 31]. The operator $\mathscr{B}_{r}^{(2\alpha-1)}+\Delta_{\sigma}$ is also known as the Weinstein operator, but the novelty here is the degenerate factor $\frac{r^{2}}{4}$ in front of $\Delta_{\sigma}$ . Such factor makes the analysis more delicate, but also gives a special geometric significance to (1.5). As it will be clear from Theorems 1.1 and 1.3, the broader perspective of this paper is closely connected to the geometric framework in [7], and especially to that in the “extension” work [25], see also the subsequent papers [57, 51, 50, 58, 24, 28, 29]. We mention here that, instead of $\Delta_{\sigma}$ , we can also consider the situation of a Baouendi-Grushin flow associated with two fractional Bessel processes intertwined as in $$ \partial_{t}-\mathscr{B}_{r}^{(2\alpha-1)}-\frac{r^{2}}{4}\left[\partial_{ss}+\frac{k-1}{s}\partial_{s}\right]=\partial_{t}-\mathscr{B}_{r}^{(2\alpha-1)}-\frac{r^{2}}{4}\mathscr{B}_{s}^{(k-1)},\ \ \ \ \alpha>0,k>0. $$ For this class, we have results corresponding to the ones listed below, but for the sake of exposition, we have preferred to stick with (1.5) and avoid additional technical aspects. Our first main result identifies the heat kernel for the Cauchy problem $$ \mathfrak{P}_{\alpha,k}u=0,\qquad u((r,\sigma),0)=\varphi(r,\sigma), $$ subject to the reflected condition $$ \lim_{r\to 0^{+}}r^{2\alpha-1}\partial_{r}u((r,\sigma),t)=0. $$ In what follows, we will denote by $$ \Sigma^{+}=\{\varphi\in C^{\infty}(\mathbb{R}^{+}\times\mathbb{R}^{k})\mid\gamma_{\ell,m,\beta}(\varphi)<\infty,\ \forall\ell,m\in\mathbb{N}_{0},\ \forall\beta\in\mathbb{N}_{0}^{k}\}, $$ where we have let $$ \gamma_{\ell,m,\beta}(\varphi):=\underset{(r,\sigma)\in\mathbb{R}^{+}\times\mathbb{R}^{k}}{\sup}\left|r^{\ell}\left(\frac{1}{r}\partial_{r}\right)^{m}\partial^{\beta}_{\sigma}\varphi(r,\sigma)\right|<\infty. $$ The family $\gamma_{\ell,m,\beta}$ is a countable collection of seminorms which generates a Frechet space topology on $C^{\infty}(\mathbb{R}^{+}\times\mathbb{R}^{k})$ . Note that since $\gamma_{0,1,0}(\varphi)<\infty$ , we have $|\partial_{r}\varphi(r,\sigma)|\leq\gamma_{0,1}(\varphi)r$ for any $r>0$ and any $\sigma\in\mathbb{R}^{k}$ . This guarantees in particular that any function $\varphi\in\Sigma^{+}$ satisfies the Neumann condition (1.7) when $\alpha>0$ . **Theorem 1.1** *Let $\varphi\in\Sigma^{+}$ . The unique solution of (1.6)–(1.7) is $$ u((r,\sigma),t)=\int_{0}^{\infty}\!\int_{\mathbb{R}^{k}}\mathscr{K}_{\alpha,k}((r,\sigma),(\rho,\sigma^{\prime}),t)\,\varphi(\rho,\sigma^{\prime})\,d\sigma^{\prime}\,\rho^{2\alpha-1}d\rho, $$ where the generalized Mehler kernel is $$ \displaystyle\mathscr{K}_{\alpha,k}((r,\sigma),(\rho,\sigma^{\prime}),t)=\frac{(r\rho)^{1-\alpha}}{\pi^{k}(2t)^{k+1}}\int_{\mathbb{R}^{k}}e^{-\frac{i}{t}\langle\sigma^{\prime}-\sigma,\lambda\rangle}\frac{|\lambda|}{\sinh|\lambda|}e^{-\frac{|\lambda|}{\tanh|\lambda|}\frac{r^{2}+\rho^{2}}{4t}}I_{\alpha-1}\!\left(\frac{|\lambda|\rho r}{2t\sinh|\lambda|}\right)\,d\lambda. $$* In (1.8) we have indicated with $I_{\nu}(z)$ the modified Bessel function of order $\nu$ . The connection between (1.4) and the heat kernel (1.8) of $\mathfrak{P}_{\alpha,k}$ is made explicit in the following proposition. **Proposition 1.2** *For every $(r,\sigma)\in\mathbb{R}^{+}\times\mathbb{R}^{k}$ , $$ \mathscr{K}_{\alpha,k}((r,\sigma),(0,0),t)=\lim_{(\rho,\sigma^{\prime})\to(0,0)}\mathscr{K}_{\alpha,k}((r,\sigma),(\rho,\sigma^{\prime}),t)=\frac{2\pi^{\alpha}}{\Gamma(\alpha)}\,G^{\star}_{\alpha,\alpha+k}((r,\sigma),t), $$ where $$ \displaystyle G^{\star}_{\alpha,\beta}((r,\sigma),t):=\frac{2^{k}}{(4\pi t)^{\beta}}\int_{\mathbb{R}^{k}}e^{-\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\alpha}e^{-\frac{r^{2}}{4t}\frac{|\lambda|}{\tanh|\lambda|}}\,d\lambda,\qquad\beta>\alpha>0. $$* Since from (1.4), (1.10), we see that $G_{\alpha,\beta}((z,\sigma),t)=G^{\star}_{\alpha,\beta}((|z|,\sigma),t)$ , the limit relation (1.9) indicates that the parameter choice $\beta=\alpha+k$ is forced by the homogeneity of degree $2$ of the operator $\mathfrak{P}_{\alpha,k}$ . In fact, this seemingly natural dimensional restriction conceals a deeper analytic structure, intimately related to the Gegenbauer’s identity (3.3), Kummer’s transformation formula (3.6) for the Gauss hypergeometric function, and to the Bateman’s identity (3.9). From a geometric perspective, this hidden phenomenon manifests itself in the uniqueness of the parameter $\beta=\alpha+k$ , which is the sole value for which the following conformal theorem can possibly hold. **Theorem 1.3** *For every $\alpha>0$ , and with $N(z,\sigma)$ as in (1.2), we have $$ \mathscr{E}_{\alpha,\alpha+k}(z,\sigma):=\int_{0}^{\infty}G_{\alpha,\alpha+k}((z,\sigma),t)\,dt=\frac{2^{\alpha+2k-4}\Gamma(\frac{\alpha}{2})\Gamma(\frac{\alpha+k-1}{2})}{\pi^{\frac{2\alpha+k+1}{2}}}N(z,\sigma)^{-(2\alpha+2k-2)}. $$* Comparing (1.11) with (1.3) shows that $m_{\alpha}=2\alpha$ plays the role of a fractal dimension associated with the variable $r\in\mathbb{R}^{+}$ . The plan of the paper is as follows. In Section 2 we prove Theorem 1.1 and Proposition 1.2. Section 3 is devoted to the proof of Theorem 1.3. In the final Section 4 we highlight an interesting link of our results with the geometric fully nonlinear equation (4.11), and we formulate two questions which we feel are of interest in connection with geometric flows in sub-Riemannian geometry. ## 2. Proof of Theorem 1.1 In this section we prove Theorem 1.1. Given $\alpha>0$ and $k\in\mathbb{N}$ , , we consider the operator $$ \mathscr{L}_{\alpha,k}=\partial_{rr}+\frac{2\alpha-1}{r}\partial_{r}+\frac{r^{2}}{4}\Delta_{\sigma}, $$ where $r>0$ and $\sigma\in\mathbb{R}^{k}$ . Henceforth, we denote by $$ \mathscr{B}_{x}^{(a)}u=\partial_{xx}u+\frac{a}{x}\partial_{x}u=x^{-a}\partial_{x}(x^{a}\partial_{x}) $$ the Bessel operator on the half-line $\mathbb{R}^{+}$ . As it is well-known, with its associated invariant measure $d\omega_{a}(x)=x^{a}dx$ , such operator plays a central role in analysis and geometry, particularly in the study of partial differential equations and stochastic processes in which symmetries are involved, see [64, 65, 66, 53, 61, 49, 37, 41]. Using (2.2), we can write (2.1) as follows $$ \mathscr{L}_{\alpha,k}=\mathscr{B}_{r}^{(2\alpha-1)}+\frac{r^{2}}{4}\Delta_{\sigma}. $$ Since (2.3) is translation-invariant in $\sigma\in\mathbb{R}^{k}$ , to solve the Cauchy problem $$ \partial_{t}u-\mathscr{L}_{\alpha,k}u=0,\ \ \ \ \ u((r,\sigma),0)=\varphi(r,\sigma), $$ we apply a partial Fourier transform with respect to this variable, $$ \hat{u}((r,\lambda),t)=\int_{\mathbb{R}^{k}}e^{-2\pi i\langle\lambda,\sigma\rangle}u((r,\sigma),t)d\sigma. $$ If for any fixed $\lambda\in\mathbb{R}^{k}\setminus\{0\}$ , we let $$ v(r,t)=\hat{u}((r,\lambda),t), $$ then (2.4) is converted into a Cauchy problem for the harmonic oscillator associated with the Bessel process $\mathscr{B}_{r}^{(2\alpha-1)}$ , $$ \partial_{t}v-\mathscr{B}_{r}^{(2\alpha-1)}v+\pi^{2}|\lambda|^{2}r^{2}v=0,\ \ \ \ \ v(r,0)=\hat{\varphi}(r,\lambda). $$ Since we are interested in reflected Brownian motion, we also impose the Neumann condition (1.7). The heat kernel for (2.5) can be obtained using the spectral analysis of the operator $\mathscr{B}_{r}^{(2\alpha-1)}v-\pi^{2}|\lambda|^{2}r^{2}$ . This however would lead to a lengthy introduction to the relevant calculus. We instead follow an alternative route, purely PDE based. We will need the following well-known fact; see [54, 4, 59, 6, 62]. **Proposition 2.1** *The solution of the Cauchy problem for the Ornstein-Uhlenbeck operator in $\mathbb{R}^{m}\times(0,\infty)$ $$ \begin{cases}u_{t}-\Delta u+2\omega\langle x,\nabla u\rangle=0,\ \ \ \ \ \omega>0,\\ u(x,0)=\psi(x),\end{cases} $$ is given by the classical Mehler formula $$ \displaystyle u(x,t) \displaystyle=(4\pi)^{-\frac{m}{2}}e^{mt\omega}\left(\frac{2\omega}{\sinh(2t\omega)}\right)^{\frac{m}{2}} \displaystyle\times\int_{\mathbb{R}^{m}}\exp\left(-\frac{\omega}{2\sinh(2t\omega)}|e^{t\omega}y-e^{-t\omega}x|^{2}\right)\psi(y)dy. $$* To solve (2.5), (1.7), we convert such problem into one for the Ornstein-Uhlenbeck operator by means of the following. **Lemma 2.2** *Let $\Phi\in C(\mathbb{R}^{m+1})$ and $h\in C^{2}(\mathbb{R}^{m+1})$ be connected by the following nonlinear equation $$ h_{t}-\mathscr{B}_{r}^{(2\alpha-1)}h+(\partial_{r}h)^{2}=\Phi. $$ Then $v$ solves the partial differential equation $$ v_{t}-\mathscr{B}_{r}^{(2\alpha-1)}v+\Phi v=0 $$ if and only if $f$ defined by the transformation $$ v(r,t)=e^{-h(r,t)}f(r,t), $$ solves the equation $$ f_{t}-\mathscr{B}_{r}^{(2\alpha-1)}f+2\partial_{r}h\partial_{r}f=0. $$* * Proof* From (2.10) we have $$ v_{t}=-fe^{-h}h_{t}+e^{-h}f_{t}, $$ and | | $\displaystyle\mathscr{B}_{r}^{(2\alpha-1)}v$ | $\displaystyle=f\mathscr{B}_{r}^{(2\alpha-1)}(e^{-h})+e^{-h}\mathscr{B}_{r}^{(2\alpha-1)}f+2\partial_{r}(e^{-h})\partial_{r}f$ | | | --- | --- | --- | --- | for the function $v$ we have $$ \partial_{t}v-\mathscr{B}_{r}^{(2\alpha-1)}v+v[\partial_{t}h-\mathscr{B}_{r}^{(2\alpha-1)}h+(\partial_{r}h)^{2}]=e^{-h}[\partial_{t}f-\mathscr{B}_{r}^{(2\alpha-1)}f+2\partial_{r}h\partial_{r}f]. $$ The desired conclusion immediately follows from this identity. ∎ For the harmonic oscillator PDE $$ \partial_{t}v-\mathscr{B}_{r}^{(2\alpha-1)}v+\pi^{2}|\lambda|^{2}r^{2}v=0, $$ it is clear that we must take $\Phi(r,t)=\omega^{2}r^{2}$ , with $\omega=\pi|\lambda|>0$ . With this right-hand side, we try the ansatz $h(r,t)=\frac{A}{2}r^{2}+Ct$ . Such $h$ solves (2.8) if and only if $A=\omega$ , and $C=(1+a)\omega$ . This gives $h(r,t)=\frac{\omega}{2}r^{2}+(1+a)\omega t$ . From Lemma 2.2 we infer that for every $\lambda\in\mathbb{R}^{k}\setminus\{0\}$ the function $f(r,t)=e^{\frac{\pi|\lambda|}{2}r^{2}+(1+a)\pi|\lambda|t}v(r,t)$ solves the problem $$ f_{t}-\mathscr{B}_{r}^{(2\alpha-1)}f+2\omega r\partial_{r}f=0,\ \ \ \ \ f(r,0)=e^{\frac{\omega}{2}r^{2}}\hat{\varphi}(r,\lambda),\ \ \ \ \underset{r\to 0^{+}}{\lim}r^{2\alpha-1}\partial_{r}f(r,t)=0. $$ If $\psi(y)=\Psi(|y|)$ , we obtain from (2.7) $$ \displaystyle u(x,t) \displaystyle=(4\pi)^{-\frac{m}{2}}e^{mt\omega}\left(\frac{2\omega}{\sinh(2t\omega)}\right)^{\frac{m}{2}}\times\int_{0}^{\infty}\Psi(\rho)\exp\left(-\frac{\omega}{2\sinh(2t\omega)}[e^{2t\omega}\rho^{2}+e^{-2t\omega}|x|^{2}]\right) \displaystyle\times\int_{\mathbb{S}^{m-1}}e^{\frac{\omega\rho|x|}{\sinh(2t\omega)}\langle\frac{x}{|x|},y\rangle}d\sigma(y)\rho^{m-1}d\rho. $$ Using Bochner’s argument ([5, Theor. 40 on p. 69]), and the Poisson representation of the modified Bessel function $I_{\nu}$ , one has for $z>0$ and any $\xi\in\mathbb{S}^{m-1}$ $$ \int_{\mathbb{S}^{m-1}}\exp\left\{z\langle\xi,y\rangle\right\}d\sigma(y)=(2\pi)^{\frac{m}{2}}z^{1-\frac{m}{2}}I_{\frac{m}{2}-1}(z), $$ see e.g. p.2 in [26]. Applying this formula with $z=\frac{\omega\rho|x|}{\sinh(2t\omega)}$ , we find $$ \int_{\mathbb{S}^{m-1}}e^{\frac{\omega\rho|x|}{\sinh(2t\omega)}\langle\frac{x}{|x|},y\rangle}d\sigma(y)=(2\pi)^{\frac{m}{2}}\left(\frac{\omega\rho|x|}{\sinh(2t\omega)}\right)^{1-\frac{m}{2}}I_{\frac{m}{2}-1}(\frac{\omega\rho|x|}{\sinh(2t\omega)}). $$ Substituting this identity in (2.13), we obtain $$ \displaystyle u(x,t) \displaystyle=e^{mt\omega}\left(\frac{\omega}{\sinh(2t\omega)}\right)^{\frac{m}{2}}\left(\frac{\omega\rho|x|}{\sinh(2t\omega)}\right)^{1-\frac{m}{2}}I_{\frac{m}{2}-1}(\frac{\omega\rho|x|}{\sinh(2t\omega)}) \displaystyle\times\exp\left(-\frac{\omega e^{-2t\omega}|x|^{2}}{2\sinh(2t\omega)}\right)\int_{0}^{\infty}\Psi(\rho)\exp\left(-\frac{\omega e^{2t\omega}\rho^{2}}{2\sinh(2t\omega)}\right)\rho^{m-1}d\rho. $$ If we now keep in mind that $m=2\alpha$ , $r=|x|$ , $\omega=\pi|\lambda|$ , and $\Psi(\rho)=e^{\frac{\omega}{2}\rho^{2}}\hat{\varphi}(\rho,\lambda)$ , from (2.14) we obtain for the solution of (2.12) $$ \displaystyle f(r,t) \displaystyle=e^{2\alpha t\pi|\lambda|}\left(\frac{\pi|\lambda|}{\sinh(2t\pi|\lambda|)}\right)^{\alpha}\left(\frac{\pi|\lambda|\rho r}{\sinh(2t\pi|\lambda|)}\right)^{1-\alpha}I_{\alpha-1}(\frac{\pi|\lambda|\rho r}{\sinh(2t\pi|\lambda|)}) \displaystyle\times\exp\left(-\frac{\pi|\lambda|e^{-2t\pi|\lambda|}r^{2}}{2\sinh(2t\pi|\lambda|)}\right)\int_{0}^{\infty}e^{\frac{\pi|\lambda|}{2}\rho^{2}}\hat{\varphi}(\rho,\lambda)\exp\left(-\frac{\pi|\lambda|e^{2t\pi|\lambda|}\rho^{2}}{2\sinh(2t\pi|\lambda|)}\right)\rho^{2\alpha-1}d\rho \displaystyle=e^{2\alpha t\pi|\lambda|}r^{1-\alpha}\frac{\pi|\lambda|}{\sinh(2t\pi|\lambda|)}\exp\left(-\frac{2\pi|\lambda|e^{-2t\pi|\lambda|}r^{2}}{4\sinh(2t\pi|\lambda|)}\right) \displaystyle\times\int_{0}^{\infty}e^{\frac{\pi|\lambda|}{2}\rho^{2}}\hat{\varphi}(\rho,\lambda)\exp\left(-\frac{2\pi|\lambda|e^{2t\pi|\lambda|}\rho^{2}}{4\sinh(2t\pi|\lambda|)}\right)I_{\alpha-1}(\frac{\pi|\lambda|\rho r}{\sinh(2t\pi|\lambda|)})\rho^{\alpha}d\rho. $$ Going back to (2.5), keeping in mind that the connection between $v$ and $f$ is given by $$ f(r,t)=e^{\frac{\pi|\lambda|}{2}r^{2}+2\alpha\pi|\lambda|t}v(r,t), $$ and that $v(r,t)=\hat{u}((r,\lambda),t)$ , we obtain from (2.15) $$ \displaystyle\hat{u}((r,\lambda),t) \displaystyle=e^{-\frac{2\pi|\lambda|}{4}r^{2}}r^{1-\alpha}\frac{\pi|\lambda|}{\sinh(2t\pi|\lambda|)}\exp\left(-\frac{2\pi|\lambda|e^{-2t\pi|\lambda|}r^{2}}{4\sinh(2t\pi|\lambda|)}\right) \displaystyle\times\int_{0}^{\infty}e^{\frac{2\pi|\lambda|}{4}\rho^{2}}\hat{\varphi}(\rho,\lambda)\exp\left(-\frac{2\pi|\lambda|e^{2t\pi|\lambda|}\rho^{2}}{4\sinh(2t\pi|\lambda|)}\right)I_{\alpha-1}(\frac{\pi|\lambda|\rho r}{\sinh(2t\pi|\lambda|)})\rho^{\alpha}d\rho. $$ Using the identity $$ 1+\frac{e^{-2t\pi|\lambda|}}{\sinh(2t\pi|\lambda|)}=\frac{1}{\tanh(2t\pi|\lambda|)}, $$ we can rewrite (2.16) in the following more compact form $$ \displaystyle\hat{u}((r,\lambda),t) \displaystyle=e^{-\frac{2\pi|\lambda|}{\tanh(2t\pi|\lambda|)}\frac{r^{2}}{4}}r^{1-\alpha}\frac{\pi|\lambda|}{\sinh(2t\pi|\lambda|)} \displaystyle\times\int_{0}^{\infty}\rho^{\alpha}e^{-\frac{2\pi|\lambda|}{\tanh(2t\pi|\lambda|)}\frac{\rho^{2}}{4}}I_{\alpha-1}(\frac{\pi|\lambda|\rho r}{\sinh(2t\pi|\lambda|)})\hat{\varphi}(\rho,\lambda)d\rho $$ Taking the inverse Fourier transform with respect to the variable $\lambda\in\mathbb{R}^{k}$ in (2.17), we find $$ \displaystyle u((r,\sigma),t) \displaystyle=r^{1-\alpha}\int_{\mathbb{R}^{k}}e^{2\pi i\langle\sigma,\lambda\rangle}e^{-\frac{2\pi|\lambda|}{\tanh(2t\pi|\lambda|)}\frac{r^{2}}{4}}\frac{\pi|\lambda|}{\sinh(2t\pi|\lambda|)} \displaystyle\times\int_{0}^{\infty}\rho^{\alpha}e^{-\frac{2\pi|\lambda|}{\tanh(2t\pi|\lambda|)}\frac{\rho^{2}}{4}}I_{\alpha-1}(\frac{\pi|\lambda|\rho r}{\sinh(2t\pi|\lambda|)})\hat{\varphi}(\rho,\lambda)d\rho d\lambda \displaystyle=r^{1-\alpha}\int_{\mathbb{R}^{k}}e^{2\pi i\langle\sigma,\lambda\rangle}e^{-\frac{2\pi|\lambda|}{\tanh(2t\pi|\lambda|)}\frac{r^{2}}{4}}\frac{\pi|\lambda|}{\sinh(2t\pi|\lambda|)} \displaystyle\times\int_{0}^{\infty}\rho^{\alpha}e^{-\frac{2\pi|\lambda|}{\tanh(2t\pi|\lambda|)}\frac{\rho^{2}}{4}}I_{\alpha-1}(\frac{\pi|\lambda|\rho r}{\sinh(2t\pi|\lambda|)})d\lambda\int_{\mathbb{R}^{k}}e^{-2\pi i\langle\lambda,\sigma^{\prime}\rangle}\varphi(\rho,\sigma^{\prime})d\sigma^{\prime}d\rho. $$ Reordering integrals, we obtain from (2.18) $$ \displaystyle u((r,\sigma),t) \displaystyle=\int_{0}^{\infty}\int_{\mathbb{R}^{k}}\mathscr{K}_{\alpha,k}((r,\sigma),(\rho,\sigma^{\prime}),t)\varphi(\rho,\sigma^{\prime})d\sigma^{\prime}\rho^{2\alpha-1}d\rho $$ where | | $\displaystyle\mathscr{K}_{\alpha,k}((r,\sigma),(\rho,\sigma^{\prime}),t)$ | $\displaystyle=(r\rho)^{1-\alpha}\int_{\mathbb{R}^{k}}e^{-2\pi i\langle\sigma^{\prime}-\sigma,\lambda\rangle}\frac{\pi|\lambda|}{\sinh(2t\pi|\lambda|)}$ | | | --- | --- | --- | --- | Setting $\lambda^{\prime}=2\pi t\lambda$ , so that $|\lambda^{\prime}|=2\pi t|\lambda|$ , and $d\lambda^{\prime}=(2\pi t)^{k}d\lambda$ , we finally obtain (1.8), thus completing the proof of Theorem 1.1. We next turn to the * Proof of Proposition1.2* Throughout, we fix $\alpha>0$ , $k\in\mathbb{N}$ , $r>0$ , $t>0$ and $\sigma\in\mathbb{R}^{k}$ . If we define $$ \Phi_{\rho,\sigma^{\prime}}(\lambda)=\frac{(r\rho)^{1-\alpha}}{\pi^{k}(2t)^{k+1}}e^{-\frac{i}{t}\langle\sigma^{\prime}-\sigma,\lambda\rangle}\frac{|\lambda|}{\sinh|\lambda|}e^{-\frac{|\lambda|}{\tanh|\lambda|}\frac{r^{2}+\rho^{2}}{4t}}I_{\alpha-1}\!\left(\frac{|\lambda|\rho r}{2t\sinh|\lambda|}\right), $$ then from (1.8) we have $$ \mathscr{K}_{\alpha,k}((r,\sigma),(\rho,\sigma^{\prime}),t)=\int_{\mathbb{R}^{k}}\Phi_{\rho,\sigma^{\prime}}(\lambda)d\lambda. $$ If we let $$ s_{\rho}(\lambda):=\frac{|\lambda|\rho r}{2t\sinh|\lambda|}, $$ then since $s_{\rho}(\lambda)\to 0$ as $\rho\to 0$ , from the power series representation of $I_{\nu}$ , see e.g. [46, (5.7.1) on p. 108] $$ I_{\nu}(z)=\sum_{k=0}^{\infty}\frac{(z/2)^{\nu+2k}}{k!\Gamma(\nu+k+1)}, $$ we see that, with $\nu=\alpha-1>-1$ , we have | | $\displaystyle(r\rho)^{1-\alpha}I_{\alpha-1}(s_{\rho}(\lambda))=\left(\frac{2t\sinh|\lambda|}{|\lambda|}\right)^{1-\alpha}s_{\rho}(\lambda)^{1-\alpha}I_{\alpha-1}(s_{\rho}(\lambda))\underset{\rho\to 0^{+}}{\longrightarrow}\ \frac{2^{1-\alpha}}{\Gamma(\alpha)}\left(\frac{2t\sinh|\lambda|}{|\lambda|}\right)^{1-\alpha}.$ | | | --- | --- | --- | This implies for each fixed $\lambda\in\mathbb{R}^{k}$ $$ \lim_{(\rho,\sigma^{\prime})\to(0,0)}\Phi_{\rho,\sigma^{\prime}}(\lambda)=\frac{1}{\Gamma(\alpha)(4t)^{\alpha-1}}e^{\,\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\alpha}e^{-\frac{|\lambda|}{\tanh|\lambda|}\frac{r^{2}}{4t}}\frac{1}{\pi^{k}(2t)^{k+1}}. $$ Therefore, if we can interchange the limit with the integral, we infer that $$ \lim_{(\rho,\sigma^{\prime})\to(0,0)}\mathscr{K}_{\alpha,k}((r,\sigma),(\rho,\sigma^{\prime}),t)=\frac{2\pi^{\alpha}}{\Gamma(\alpha)}\frac{2^{k}}{(4\pi t)^{\alpha+k}}\int_{\mathbb{R}^{k}}e^{\,\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\alpha}e^{-\frac{|\lambda|}{\tanh|\lambda|}\frac{r^{2}}{4t}}d\lambda. $$ Comparing with (1.10), we conclude that (1.9) holds. We now fix $\delta\leq 1$ , $\rho_{0}>0$ such that $$ \frac{\rho_{0}r}{2t}\leq 1, $$ and define $Q_{0}:=(0,\rho_{0}]\times B_{\delta}(0)$ . To finish the proof, we only need to produce a dominating function $M(\lambda)\geq 0$ such that $\int_{\mathbb{R}^{k}}M(\lambda)\,d\lambda<\infty$ , and for which for all $(\rho,\sigma^{\prime})\in Q_{0}$ , we have $$ \left|\Phi_{\rho,\sigma^{\prime}}(\lambda)\right|\leq M(\lambda),\qquad\text{for a.e. }\lambda\in\mathbb{R}^{k}. $$ Keeping in mnd that $\frac{s}{\sinh s}\leq 1$ for $s\geq 0$ , we have from (2.22) $$ 0\leq s_{\rho}(\lambda)\leq\frac{\rho r}{2t}\leq 1,\ \ \ \ \ \text{when}\ 0\leq\rho\leq\rho_{0}. $$ Since by the power series representation of $I_{\nu}$ we have $$ 0\leq I_{\nu}(s)\leq C_{\nu}s^{\nu},\ \ \ \ \ 0\leq s\leq 1, $$ we infer that for $0\leq\rho\leq\rho_{0}$ $$ I_{\alpha-1}\left(\frac{|\lambda|\rho r}{2t\sinh|\lambda|}\right)\leq C_{\alpha}\left(\frac{|\lambda|\rho r}{2t\sinh|\lambda|}\right)^{\alpha-1}. $$ Noting that $\frac{s}{\tanh s}\geq 1$ for every $s\geq 0$ , we conclude that for every $(\rho,\sigma^{\prime})\in Q_{0}$ and every $\lambda\in\mathbb{R}^{k}$ , we have $$ |\Phi_{\rho,\sigma^{\prime}}(\lambda)|\leq\frac{C_{\alpha}}{\pi^{k}(2t)^{k+\alpha}}e^{-\frac{r^{2}}{4t}}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\alpha}:=M(\lambda)\in L^{1}(\mathbb{R}^{k}). $$ This completes the proof. ∎ ## 3. Proof of Theorem 1.3 We say that a function is spherically symmetric in $\mathbb{R}^{k}$ if $f(x)=f^{\star}(|x|)$ , for some $f^{\star}$ . When $k=1$ , we simply intend that $f$ is even, and we let $f^{\star}(r)=f(r)$ for $\geq 0$ . We begin by recalling the following classical formula due to Bochner, see [5, Theor. 40 on p. 69]. **Proposition 3.1** *Assume that $k\geq 1$ . Then, for any $\xi\in\mathbb{R}^{k}$ one has $$ \hat{f}(\xi)=\frac{2\pi}{|\xi|^{\frac{k}{2}-1}}\int_{0}^{\infty}r^{\frac{k}{2}}f^{\star}(r)J_{\frac{k}{2}-1}(2\pi|\xi|r)dr. $$* To see that the above formula continues to be valid when $k=1$ , one needs to use the well-known identity $$ J_{-1/2}(z)=\sqrt{\frac{2}{\pi z}}\cos z, $$ see [46, (5.8.2) on p.111]. We then turn to the proof of Theorem 1.3. From (1.11), using Cavalieri’s principle, we find | | $\displaystyle\mathscr{E}_{\alpha,\beta}(z,\sigma)=2^{k}(4\pi)^{-\beta}\int_{0}^{\infty}\frac{1}{t^{\beta-1}}\int_{\mathbb{R}^{k}}e^{-\frac{i}{t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\alpha}e^{-\frac{|z|^{2}}{4t}\frac{|\lambda|}{\tanh|\lambda|}}d\lambda\frac{dt}{t}$ | | | --- | --- | --- | where we have let $$ f(\lambda)=\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\alpha}e^{-\frac{t|z|^{2}}{4}\frac{|\lambda|}{\tanh|\lambda|}}. $$ Invoking Proposition 3.1, we find $$ \hat{f}(\frac{t\sigma}{2\pi})=\frac{(2\pi)^{\frac{k}{2}}}{t^{\frac{k}{2}-1}|\sigma|^{\frac{k}{2}-1}}\int_{0}^{\infty}r^{\frac{k}{2}}\left(\frac{r}{\sinh r}\right)^{\alpha}e^{-\frac{t|z|^{2}}{4}\frac{r}{\tanh r}}J_{\frac{k}{2}-1}(t|\sigma|r)dr. $$ Substituting this identity in the above expression of $\mathscr{E}_{\alpha,\beta}(z,\sigma)$ , and exchanging the order of integration, we find $$ \displaystyle\mathscr{E}_{\alpha,\beta}(z,\sigma)=\frac{2^{k}(4\pi)^{-\beta}(2\pi)^{\frac{k}{2}}}{|\sigma|^{\frac{k}{2}-1}}\int_{0}^{\infty}r^{\frac{k}{2}}\left(\frac{r}{\sinh r}\right)^{\alpha}\int_{0}^{\infty}t^{\beta-\frac{k}{2}-1}e^{-\frac{t|z|^{2}}{4}\frac{r}{\tanh r}}J_{\frac{k}{2}-1}(t|\sigma|r)dtdr. $$ We next recall the following classical formula due to Gegenbauer, see (3) on p. 385 in [63] $$ \int_{0}^{\infty}t^{\mu-1}e^{-at}J_{\nu}(bt)dt=\frac{2^{-\nu}b^{\nu}\Gamma(\nu+\mu)}{\Gamma(\nu+1)(a^{2}+b^{2})^{\frac{\nu+\mu}{2}}}F\left(\frac{\nu+\mu}{2},\frac{1-\mu+\nu}{2};\nu+1;\frac{b^{2}}{a^{2}+b^{2}}\right), $$ provided that $$ \Re(\nu+\mu)>0,\ \Re(a+ib)>0,\ \Re(a-ib)>0. $$ We apply (3.3) with the choice $$ \nu=\frac{k}{2}-1,\ \ \ \mu=\beta-\frac{k}{2},\ \ \ a=\frac{|z|^{2}}{4}\frac{r}{\tanh r},\ \ \ b=r|\sigma|. $$ This gives $$ \nu+\mu=\beta-1>0,\ \ \ \ 1-\mu+\nu=k-\beta. $$ Notice that $$ a^{2}+b^{2}=\frac{r^{2}}{16\tanh^{2}r}(|z|^{4}+16|\sigma|^{2}\tanh^{2}r),\ \ \ \frac{\beta^{2}}{\alpha^{2}+\beta^{2}}=\frac{16|\sigma|^{2}\tanh^{2}r}{|z|^{4}+16|\sigma|^{2}\tanh^{2}r}. $$ We obtain | | $\displaystyle\int_{0}^{\infty}t^{\beta-\frac{k}{2}-1}e^{-\frac{t|z|^{2}}{4}\frac{r}{\tanh r}}J_{\frac{k}{2}-1}(t|\sigma|r)dt$ | | | --- | --- | --- | Substituting this formula in (3.2), and at this point using the crucial assumption $\boxed{\beta=\alpha+k}$ , we see that the powers of $r$ cancel, and we find $$ \displaystyle\mathscr{E}_{\alpha,\beta}(z,\sigma)=\frac{2^{1+\frac{k}{2}}(4\pi)^{-\beta}(2\pi)^{\frac{k}{2}}\Gamma(\beta-1)4^{\beta-1}}{\Gamma(\frac{k}{2})}\int_{0}^{\infty}r^{k+\alpha-\beta}\left(\frac{1}{\sinh r}\right)^{\alpha}(\tanh r)^{\beta-1} \displaystyle\times\frac{1}{(|z|^{4}+16|\sigma|^{2}\tanh^{2}r)^{\frac{\beta-1}{2}}}F\left(\frac{\beta-1}{2},\frac{k-\beta}{2};\frac{k}{2};\frac{16|\sigma|^{2}\tanh^{2}r}{|z|^{4}+16|\sigma|^{2}\tanh^{2}r}\right)dr \displaystyle=\frac{2^{k-1}\pi^{\frac{k}{2}-\beta}\Gamma(\beta-1)}{\Gamma(\frac{k}{2})}\int_{0}^{\infty}\left(\frac{1}{\sinh r}\right)^{\alpha}(\tanh r)^{\alpha}(\tanh r)^{\beta-\alpha-1} \displaystyle\times\frac{1}{(|z|^{4}+16|\sigma|^{2}\tanh^{2}r)^{\frac{\beta-1}{2}}}F\left(\frac{\beta-1}{2},\frac{k-\beta}{2};\frac{k}{2};\frac{16|\sigma|^{2}\tanh^{2}r}{|z|^{4}+16|\sigma|^{2}\tanh^{2}r}\right)dr \displaystyle=\frac{2^{k-2}\pi^{\frac{k}{2}-\beta}\Gamma(\beta-1)}{\Gamma(\frac{k}{2})}\int_{0}^{\infty}\left(\frac{1}{\cosh^{2}r}\right)^{\frac{\alpha}{2}-1}(\tanh^{2}r)^{\frac{\beta-\alpha}{2}-1}\frac{1}{(|z|^{4}+16|\sigma|^{2}\tanh^{2}r)^{\frac{\beta-1}{2}}} \displaystyle\times F\left(\frac{\beta-1}{2},\frac{k-\beta}{2};\frac{k}{2};\frac{16|\sigma|^{2}\tanh^{2}r}{|z|^{4}+16|\sigma|^{2}\tanh^{2}r}\right)\frac{2\tanh r}{\cosh^{2}r}dr \displaystyle=\frac{2^{k-2}\pi^{\frac{k}{2}-\beta}\Gamma(\beta-1)}{\Gamma(\frac{k}{2})}\int_{0}^{\infty}\left(1-\tanh^{2}r\right)^{\frac{\alpha}{2}-1}(\tanh^{2}r)^{\frac{\beta-\alpha}{2}-1}\frac{1}{(|z|^{4}+16|\sigma|^{2}\tanh^{2}r)^{\frac{\beta-1}{2}}} \displaystyle\times F\left(\frac{\beta-1}{2},\frac{k-\beta}{2};\frac{k}{2};\frac{16|\sigma|^{2}\tanh^{2}r}{|z|^{4}+16|\sigma|^{2}\tanh^{2}r}\right)\frac{2\tanh r}{\cosh^{2}r}dr. $$ In the integral in the right-hand side of (3.5) we now make the change of variable $y=\tanh^{2}r$ , for which $dy=\frac{2\tanh r}{\cosh^{2}r}dr$ , obtaining $$ \displaystyle\mathscr{E}_{\alpha,\beta}(z,\sigma)=\frac{2^{k-2}\pi^{\frac{k}{2}-\beta}\Gamma(\beta-1)}{\Gamma(\frac{k}{2})}\int_{0}^{1}\left(1-y\right)^{\frac{\alpha}{2}-1}y^{\frac{\beta-\alpha}{2}-1}\frac{1}{(|z|^{4}+16|\sigma|^{2}y)^{\frac{\beta-1}{2}}} \displaystyle\times F\left(\frac{\beta-1}{2},\frac{k-\beta}{2};\frac{k}{2};\frac{16|\sigma|^{2}y}{|z|^{4}+16|\sigma|^{2}y}\right)dy. $$ Next, we recall the following Kummer’s relation concerning how the hypergeometric function $F$ changes under linear transformations (see formula (3) on p. 105 in [16], or also (9.5.1) on p. 247 in [46]), $$ F(a,b;c;u)=(1-u)^{-a}F\left(a,c-b;c;\frac{u}{u-1}\right),\ \ \ \ \ \ u\not=1,\ |\arg(1-u)|<\pi, $$ valid for any $a,b,c\in\mathbb{C}$ , with $c\not=0,-1,-2,...$ We use (3.6) with the choices $$ a=\frac{\beta-1}{2},\ \ \ c=\frac{k}{2},\ \ \ c-b=\frac{k-\beta}{2},\ \ \ \frac{u}{u-1}=\frac{16|\sigma|^{2}y}{|z|^{4}+16|\sigma|^{2}y}. $$ Notice that, with these choices, we have $$ b=\frac{\beta}{2}, $$ and that $$ u=-\frac{16|\sigma|^{2}}{|z|^{4}}y,\ \ \ \ \ 1-u=\frac{|z|^{4}+16|\sigma|^{2}y}{|z|^{4}}. $$ We thus find from (3.6) $$ \displaystyle F\left(\frac{\beta-1}{2},\frac{k-\beta}{2};\frac{k}{2};\frac{16|\sigma|^{2}y}{|z|^{4}+16|\sigma|^{2}y}\right)=\frac{(|z|^{4}+16|\sigma|^{2}y)^{\frac{\beta-1}{2}}}{|z|^{2(\beta-1)}}F\left(\frac{\beta-1}{2},\frac{\beta}{2};\frac{k}{2};-\frac{16|\sigma|^{2}}{|z|^{4}}y\right). $$ We now use (3.7) in (3.5), obtaining $$ \displaystyle\mathscr{E}_{\alpha,\beta}(z,\sigma)=\frac{2^{k-2}\pi^{\frac{k}{2}-\beta}\Gamma(\beta-1)}{\Gamma(\frac{k}{2})|z|^{2(\beta-1)}}\int_{0}^{1}y^{\frac{\beta-\alpha}{2}-1}\left(1-y\right)^{\frac{\alpha}{2}-1}F\left(\frac{\beta-1}{2},\frac{\beta}{2};\frac{k}{2};-\frac{16|\sigma|^{2}}{|z|^{4}}y\right)dy $$ We now use the following formula due to H. Bateman (see formula (2) on p. 78 in [16], and also Problem 6. on p. 277 in [46]) $$ \int_{0}^{1}y^{c-1}(1-y)^{\gamma-c-1}F(a,b;c;\delta y)dy=\frac{\Gamma(c)\Gamma(\gamma-c)}{\Gamma(\gamma)}F(a,b;\gamma;\delta), $$ provided that $\Re\gamma>\Re c>0$ , $\delta\not=1$ and $|\arg(1-\delta)|<\pi$ . To apply (3.9) we must choose $$ a=\frac{\beta-1}{2},\ \ \ b=\frac{\beta}{2},\ \ \ c=\frac{k}{2}>0,\ \ \ \delta=-\frac{16|\sigma|^{2}}{|z|^{4}}. $$ Moreover, we must make sure that $c=\frac{k}{2}$ , but this is true since $\beta-\alpha=k$ and comparing (3.9) with (3.8) we see that $c=\frac{\beta-\alpha}{2}$ . The same comparison shows that we must have $\gamma-c=\frac{\alpha}{2}$ , which gives $$ \gamma=c+\frac{\alpha}{2}=\frac{\alpha+k}{2}=\frac{\beta}{2}. $$ Applying (3.9) we thus find $$ \int_{0}^{1}y^{\frac{\beta-\alpha}{2}-1}\left(1-y\right)^{\frac{\alpha}{2}-1}F\left(\frac{\beta-1}{2},\frac{\beta}{2};\frac{k}{2};-\frac{16|\sigma|^{2}}{|z|^{4}}y\right)dy=\frac{\Gamma(\frac{k}{2})\Gamma(\frac{\alpha}{2})}{\Gamma(\frac{\beta}{2})}F(\frac{\beta-1}{2},\frac{\beta}{2};\frac{\beta}{2};-\frac{16|\sigma|^{2}}{|z|^{4}}). $$ We see now that a miracle has happened, in the sense that the hypergeometric function is in the special form $$ F(a,b;b;-\delta)=\ _{1}F_{0}(a;-\delta)=(1+\delta)^{-a}, $$ see formula (4) on p. 101 in [16]. We thus find | | $\displaystyle F(\frac{\beta-1}{2},\frac{\beta}{2};\frac{\beta}{2};-\frac{16|\sigma|^{2}}{|z|^{4}})=\left(1+\frac{16|\sigma|^{2}}{|z|^{4}}\right)^{-\frac{\beta-1}{2}}=\frac{|z|^{2(\beta-1)}}{\left(|z|^{4}+16|\sigma|^{2}\right)^{2(\beta-1)}}.$ | | | --- | --- | --- | Substituting in the above, we find | | | $\displaystyle\mathscr{E}_{\alpha,\beta}(z,\sigma)=\frac{2^{k-2}\pi^{\frac{k}{2}-\beta}\Gamma(\frac{\alpha}{2})\Gamma(\beta-1)}{\Gamma(\frac{\beta}{2})}\frac{1}{\left(|z|^{4}+16|\sigma|^{2}\right)^{\frac{2(\beta-1)}{4}}}.$ | | | --- | --- | --- | --- | Using Legendre’s duplication formula for the gamma function we have $$ \frac{\Gamma(\beta-1)}{\Gamma(\frac{\beta}{2})}=\frac{2^{\beta-1}\Gamma(\frac{\beta+1}{2})}{\sqrt{\pi}(\beta-1)}. $$ Substituting in the previous formula, and using (1.2), we finally obtain | | | $\displaystyle\mathscr{E}_{\alpha,\beta}(z,\sigma)=\frac{2^{k-2}\pi^{\frac{k}{2}-\beta}\Gamma(\frac{\alpha}{2})\Gamma(\beta-1)}{\Gamma(\frac{\beta}{2})}\frac{1}{\left(|z|^{4}+16|\sigma|^{2}\right)^{\frac{2(\beta-1)}{4}}}.$ | | | --- | --- | --- | --- | Finally, keeping in mind that $\beta=\alpha+k$ , and using again Legendre’s formula, which gives $$ \frac{\Gamma(\beta-1)}{\Gamma(\frac{\beta}{2})}=\frac{\Gamma(\alpha+k-1)}{\Gamma(\frac{\alpha+k}{2})}=\pi^{-1/2}2^{\alpha+k-2}\Gamma(\frac{\alpha+k-1}{2}), $$ we finally obtain | | $\displaystyle\mathscr{E}_{\alpha,\alpha+k}(z,\sigma)=\frac{2^{\alpha+2k-4}\Gamma(\frac{\alpha}{2})\Gamma(\frac{\alpha+k-1}{2})}{\pi^{\frac{2\alpha+k+1}{2}}}N(z,\sigma)^{-(2\alpha+2k-2)}.$ | | | --- | --- | --- | This establishes (1.11), thus completing the proof of Theorem 1.3. ## 4. Connections with nonlinear equations In this final section we point out an intriguing connection between the above linear results and a nonlinear geometric flow that presently remains unexplored. The discussion that follows is intended only as a preliminary indication of possible directions and its main motivation is to lead to the open Problems 1 and 2 below. Consider in $\mathbb{R}^{n}$ the $p$ -energy functional $$ J_{p}(u)=\frac{1}{p}\int_{\mathbb{R}^{n}}|\nabla u|^{p}\,dx, $$ whose Euler–Lagrange equation is the nonlinear $p$ -Laplace equation $$ \Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)=0. $$ Its fundamental solution is the well-known power law $$ e_{p}(x)=C_{n,p}\,|x|^{-\frac{n-p}{p-1}}, $$ which reduces to the classical Newtonian potential when $p=2$ . The parabolic version $\partial_{t}u=\Delta_{p}u$ of (4.1) is strongly asymmetric in space and time, and does not correspond to a geometric flow. A more appropriate evolution is the normalized $p$ -Laplacian, $$ \partial_{t}u=|\nabla u|^{2-p}\Delta_{p}u. $$ At points of non-degeneracy of the gradient such equation can be equivalently written $$ \partial_{t}u=\Delta u+(p-2)|\nabla u|^{-2}\Delta_{\infty}u, $$ which connects in the limit $p\to 1$ to the mean curvature flow studied in the celebrated works [17], [18], [19], [20], [60, 12, 9, 33, 36]. The equation (4.3) enjoys a rich theory of viscosity solutions [1, 15] and reveals deep connections with stochastic tug-of-war games [56, 47]. The a priori estimates in [2] and the breakthrough regularity theory in [38] further highlight its geometric nature. In connection with Riemannian geometry, we mention the work in preparation [3] where a Li-Yau theory and an optimal Harnack inequality have been developed for the flow (4.3) in the Riemannian case. One remarkable aspect of the fully nonlinear operator (4.3), which also constitutes its main link to the present work, is that on functions which are spherically symmetric in the space variable, it acts as a linear operator in a fractal dimension. The key observation is that when $u(x,t)=f(r,t)$ , $r=|x|$ , then $$ |\nabla u|^{-2}\Delta_{\infty}u=f_{rr}, $$ so that, on such functions, the fully nonlinear equation (4.3) reduces to the linear $$ f_{t}=(p-1)f_{rr}+\frac{n-1}{r}f_{r}. $$ Rescaling time, $f(r,t)=g(r,(p-1)t)$ , shows that $f$ is governed by the heat flow in a space $\mathbb{R}^{\kappa}\times(0,\infty)$ of “fractal dimension” $$ \kappa=\frac{n+p-2}{p-1}. $$ This led the authors of [1] to discover the following notable solution of (4.3): $$ g_{p}(x,t)=t^{-\frac{n+p-2}{2(p-1)}}\exp\bigg(-\frac{|x|^{2}}{4(p-1)t}\bigg). $$ We stress that (4.7) satisfies $$ \int_{\mathbb{R}^{n}}g_{p}(x,t)\,dx=(4\pi(p-1))^{\frac{n}{2}}t^{\frac{(n-1)(p-2)}{2(p-1)}}, $$ and therefore it is not a fundamental solution, except when $p=2$ or $n=1$ . Despite this, one still has the conformal property $$ \int_{0}^{\infty}g_{p}(x,t)\,dt=c_{n,p}\,|x|^{-\frac{n-p}{p-1}}, $$ which mirrors the linear case $p=2$ in view of (4.2). These considerations led us to ask whether the intertwining between linear and nonlinear structures observed in Euclidean space extends to the sub-Riemannian setting of the present study. For instance, in view of the cited works [1, 3], it would be quite interesting to know the answer to the following question. **Problem 1** *Does a counterpart $G_{p}((z,\sigma),t)$ of the prototypical solution (4.7) exists in a group of Heisenberg type?* Answering this question is a nontrivial task since in sub-Riemannian geometry there is no counterpart of the phenomenon (4.4). We show below that Theorems 1.1 and 1.3 offer a possible candidate. We recall that, in these Lie groups, the foundational work of Korányi-Reimann [43, 44], Mostow [52] and Margulis-Mostow [48] have underscored the role of nonlinear equations such as $$ \Delta_{H,p}f=\operatorname{div}_{H}(|\nabla_{H}f|^{p-2}\nabla_{H}f)=0. $$ A basic result (see [11, Theor. 2.1] and independently [34] for $p=Q$ ) shows that the Korányi gauge continues to govern nonlinear potential theory: $$ E_{p}(z,\sigma)=\begin{cases}C_{p}\,N(z,\sigma)^{-\frac{Q-p}{p-1}},&p\neq Q,\\[6.0pt] C_{p}\,\log N(z,\sigma),&p=Q.\end{cases} $$ **Problem 2** *In the Heisenberg group $\mathbb{H}^{n}$ we propose to study the following counterpart of the geometric flow (4.3) $$ \partial_{t}u=\Delta_{H}u+(p-2)|\nabla_{H}u|^{-2}\Delta_{H,\infty}u,\ \ \ \ \ \ 1<p<Q=2n+2, $$ where we have let $\Delta_{H,\infty}u=\frac{1}{2}\langle\nabla_{H}u,\nabla_{H}(|\nabla_{H}u|^{2})\rangle$ . More specifically, we propose to show existence and uniqueness of viscosity solutions of the Cauchy problem for (4.11), as well as the optimal regularity of their horizontal gradient.* Formally, (4.11) converges as $p\to 1$ to the horizontal mean curvature flow studied in [10], but the situation is now much more complex than its Euclidean counterpart. The analysis of (4.11) is presently lacking and it would lead to various interesting developments. In connection with Problem 1, and motivated by the Euclidean formula (4.7) and Theorem 1.3, in the Heisenberg group $\mathbb{H}^{n}$ we introduce the following Fourier integral $$ \displaystyle G_{p}((z,\sigma),t) \displaystyle=t^{-\frac{Q+p-2}{2(p-1)}}\int_{\mathbb{R}}e^{-\frac{i}{(p-1)t}\langle\sigma,\lambda\rangle}\left(\frac{|\lambda|}{\sinh|\lambda|}\right)^{\frac{Q-p}{2(p-1)}}e^{-\frac{|z|^{2}}{4(p-1)t}\frac{|\lambda|}{\tanh|\lambda|}}\,d\lambda,\ \ \ 1<p<Q. $$ We emphasize that, when $p=2$ , the function (4.12) reduces to a constant multiple of the classical Gaveau-Hulanicki kernel (1.1). Concerning (4.12) we have the following consequence of our Theorem 1.3. **Proposition 4.1** *For any $n\in\mathbb{N}$ and $1<p<Q=2n+2$ , there exists an explicit universal constant $C_{n,p}>0$ such that $$ \int_{0}^{\infty}G_{p}((z,\sigma),t)\,dt=C_{n,p}\,N(z,\sigma)^{-\frac{Q-p}{p-1}}. $$* * Proof* The proof of Proposition 4.1 follows by taking $\alpha=\frac{Q-p}{2(p-1)}$ in Theorem 1.3, and noting that, since for the Heisenberg group $\mathbb{H}^{n}$ one has $k=1$ , we presently have $$ \beta=\alpha+k=\frac{Q-p}{2(p-1)}+1=\frac{Q+p-2}{2(p-1)}. $$ This observation shows that, up to the time scaling $t\to(p-1)t$ , the function (4.12) is exactly a $G_{\alpha,\alpha+k}$ as in Theorem 1.3, with $k=1$ and $\alpha$ as above. ∎ Thus in $\mathbb{H}^{n}$ the nonlinear potential (4.10) arises from the time integral of the kernel (4.12), just as in the Euclidean identity (4.8). This parallel strongly suggests a deeper relationship between the kernel $G_{p}$ and the fully nonlinear operator (4.9), and Proposition 4.1 seems to indicate in the function (4.12) the appropriate sub-Riemannian analogue of the prototypical solution (4.7). However, the actual verification that $G_{p}$ solves (4.11) involves complex computations and it is presently unresolved. We mention in closing that, for both Problems 1 and 2, we have some interesting work in progress. ## References - [1] A. Banerjee & N. Garofalo, Gradient bounds and monotonicity of the energy for some nonlinear singular diffusion equations. Indiana Univ. Math. J. 62 (2013), no. 2, 699-736. - [2] A. Banerjee & N. Garofalo, Modica type gradient estimates for an inhomogeneous variant of the normalized $p$ -Laplacian evolution. Nonlinear Anal. 121 (2015), 458-468. - [3] A. Banerjee, N. Garofalo & H. Mahmoudian, Li-Yau and Harnack estimates for some fully nonlinear geometric flows. Work in preparation. - [4] N. Berline, E. Getzler & M. Vergne, Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 298. Springer-Verlag, Berlin, 1992. - [5] S. Bochner & K. Chandrasekharan, Fourier Transforms, Annals of Mathematics Studies, no. 19, Princeton University Press, Princeton, N. J.; Oxford University Press, London, 1949. ix+219 pp. - [6] V. I. Bogachev, Ornstein-Uhlenbeck operators and semigroups. Uspekhi Mat. Nauk 73 (2018), no. 2, 3-74. Russian Math. Surveys 73 (2018), no. 2, 191-260. - [7] T. P. Branson, G. Ólafsson & B. Orsted, Spectrum generating operators and intertwining operators for representations induced from a maximal parabolic subgroup. J. Funct. Anal. 135 (1996), no. 1, 163-205. - [8] L. Caffarelli & L. Silvestre, An extension problem related to the fractional Laplacian. Comm. Partial Differential Equations 32 (2007), no. 7-9, 1245-1260. - [9] L. Caffarelli & L. Wang, A Harnack inequality approach to the interior regularity of parabolic equations, Indiana University Math. J. 42 (1993), 159-165. - [10] L. Capogna & G. Citti, Generalized mean curvature flow in Carnot groups. Comm. Partial Differential Equations 34 (2009), no. 7-9, 937-956. - [11] L. Capogna, D. Danielli & N. Garofalo, Capacitary estimates and the local behavior of solutions of nonlinear subelliptic equations. Amer. J. Math. 118 (1996), no. 6, 1153-1196. - [12] Y.-G. Chen, Y. Giga & S. Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, J. Differential Geom. 33 (1991), 749-786. - [13] M. Cowling, A. H. Dooley, A. Korányi & F. Ricci, $H$ -type groups and Iwasawa decompositions. Adv. Math. 87 (1991), no. 1, 1-41. - [14] J. Cygan, Heat kernels for class $2$ nilpotent groups. Studia Math. 64 (1979), no. 3, 227-238. - [15] K. Does, An evolution equation involving the normalized $p$ -Laplacian. Commun. Pure Appl. Anal. 10 (2011), no. 1, 361-396. - [16] A. Erdélyi, W. Magnus, F. Oberhettinger & F. G. Tricomi, Higher transcendental functions. Vol. I. Based on notes left by Harry Bateman. With a preface by Mina Rees. With a foreword by E. C. Watson. Reprint of the 1953 original. Robert E. Krieger Publishing Co., Inc., Melbourne, Fla., 1981. xiii+302 pp. - [17] L. C. Evans & J. Spruck, Motions of level sets by mean curvature, 1, J. Differential Geom. 33 (1991), 635-681. - [18] L. C. Evans & J. Spruck, Motions of level sets by mean curvature, 2, Trans. Amer. Math. Soc. 330 (1992), 321-332. - [19] L. C. Evans & J. Spruck, Motions of level sets by mean curvature, 3, J. Geom. Anal. 2 (1992), 121-150. - [20] L. C. Evans & J. Spruck, Motions of level sets by mean curvature, 4, J. Geom. Anal. 5 (1995), 77-114. - [21] G. B. Folland, A fundamental solution for a subelliptic operator. Bull. Amer. Math. Soc. 79 (1973), 373-376. - [22] G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Mat. 13 (1975), no. 2, 161-207. - [23] G. B. Folland & E. M. Stein, Estimates for the $\bar{\partial}_{b}$ complex and analysis on the Heisenberg group. Comm. Pure Appl. Math. 27 (1974), 429-522. - [24] J. Frahm, B. Orsted & G. Zhang, An extension problem related to the fractional Branson-Gover operators. J. Funct. Anal. 278 (2020), no. 5, 108395, 33 pp. - [25] R. L. Frank, M. del Mar González, D. Monticelli & J. Tan, An extension problem for the $CR$ fractional Laplacian. Adv. Math. 270 (2015), 97-137. - [26] N. Garofalo, Two classical properties of the Bessel quotient $I_{\nu+1}/I_{\nu}$ and their implications in pde’s. Contemp. Math., 748 American Mathematical Society, [Providence], RI, 2020, 57-97. - [27] N. Garofalo & G. Tralli, Feeling the heat in a group of Heisenberg type. Adv. Math. 381 (2021), Paper No. 107635, 42 pp. - [28] N. Garofalo & G. Tralli, A heat equation approach to intertwining. J. Anal. Math. 149 (2023), no. 1, 113-134. - [29] N. Garofalo & G. Tralli, Heat kernels for a class of hybrid evolution equations. Potential Anal. 59 (2023), no. 2, 823-856. - [30] N. Garofalo & D. Vassilev, Overdetermined problems in groups of Heisenberg type: conjectures and partial results. J. Funct. Anal. 287 (2024), no. 10, Paper No. 110588, 46 pp. - [31] N. Garofalo & D. Vassilev, Overdetermined fractional Serrin problem. ArXiv:2509.02742 - [32] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimées sous elliptiques sur certains groupes nilpotents. Acta Math. 139 (1977), no. 1-2, 95-153. - [33] Y. Giga, S. Goto, H. Ishii & M. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J. 40 1991, 443-470. - [34] J. Heinonen & I. Holopainen, Quasiregular maps on Carnot groups. J. Geom. Anal. 7 (1997), no. 1, 109-148. - [35] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian field and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group. Studia Math. 56 (1976), no. 2, 165-173. - [36] T. Ilmanen, P. Sternberg & W. Ziemer, Equilibrium solutions to generalized motion by mean curvature, J. Geom. Anal 8 (1998), 845-858. - [37] K. Ito & H. P. Jr. McKean, Diffusion processes and their sample paths. Die Grundlehren der mathematischen Wissenschaften, Band 125 Springer-Verlag, Berlin-New York, 1974, xv+321 pp. - [38] T. Jin & L. Silvestre, Hölder gradient estimates for parabolic homogeneous $p$ -Laplacian equations. J. Math. Pures Appl. (9) 108 (2017), no. 1, 63-87. - [39] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Amer. Math. Soc. 258 (1980), no. 1, 147-153. - [40] A. Kaplan, Riemannian nilmanifolds attached to Clifford modules. Geom. Dedicata 11 (1981), no. 2, 127-136. - [41] S. Karlin & H. M. Taylor, A second course in stochastic processes. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1981, xviii+542 pp. - [42] A. Korányi, Geometric properties of Heisenberg-type groups. Adv. in Math. 56 (1985), no. 1, 28-38. - [43] A. Korányi & H. M. Reimann, Quasiconformal mappings on the Heisenberg group. Invent. Math. 80 (1985), no. 2, 309-338. - [44] A. Korányi & H. M. Reimann, Foundations for the theory of quasiconformal mappings on the Heisenberg group. Adv. Math. 111 (1995), no. 1, 1–87. - [45] A. Korányi & S. Vági, Singular integrals on homogeneous spaces and some problems of classical analysis. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 25 (1971), no. 4, 575-648. - [46] N. N. Lebedev, Special functions and their applications. Revised edition, translated from the Russian and edited by R. A. Silverman. Unabridged and corrected republication. Dover Publications, Inc., New York, 1972. xii+308 pp. - [47] J. J. Manfredi, M. Parviainen & J. D. Rossi, An asymptotic mean value characterization for a class of nonlinear parabolic equations related to tug-of-war games. SIAM J. Math. Anal. 42 (2010), no. 5, 2058-2081. - [48] G. A. Margulis & G. D. Mostow, The differential of a quasiconformal mapping on a Carnot-Carathéodory space, Geom. and Func. Anal., 5, no. 2 (1995), 402–433. - [49] S. A. Molchanov & E. Ostrovskii, Symmetric stable processes as traces of degenerate diffusion processes, Theor. Probability Appl. 14 (1969), 128-131. - [50] J. Möllers, B. Orsted & G. Zhang, On boundary value problems for some conformally invariant differential operators. Comm. Partial Differential Equations 41 (2016), no. 4, 609-643. - [51] J. Möllers, B. Orsted & G. Zhang, Invariant differential operators on $H$ -type groups and discrete components in restrictions of complementary series of rank one semisimple groups. J. Geom. Anal. 26 (2016), no. 1, 118-142. - [52] G. D. Mostow, Strong Rigidity of Locally Symmetric Spaces, Princeton University Press, Princeton, New Jersey (1973). - [53] B. Muckenhoupt & E. M. Stein, Classical expansions and their relation to conjugate harmonic functions. Trans. Amer. Math. Soc. 118 (1965), 17-92. - [54] L. S. Ornstein & G. E. Uhlenbeck, On the theory of the Brownian motion. I. Phys. Rev. (2) 36 (1930), 823-841. - [55] P. Pansu, Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. (2) 129 (1989), no. 1, 1–60. - [56] Y. Peres & S. Sheffield, Tug-of-war with noise: a game-theoretic view of the $p$ -Laplacian. Duke Math. J. 145 (2008), no. 1, 91-120. - [57] L. Roncal & S. Thangavelu, Hardy’s inequality for fractional powers of the sublaplacian on the Heisenberg group. Adv. Math. 302 (2016), 106-158. - [58] L. Roncal & S. Thangavelu, An extension problem and trace Hardy inequality for the sublaplacian on $H$ -type groups. Int. Math. Res. Not. IMRN (2020), no. 14, 4238-4294. - [59] M. Sanz-Solé, Malliavin calculus, with applications to stochastic partial differential equations. Fundam. Sci. EPFL Press, Lausanne; distributed by CRC Press, Boca Raton, FL, 2005, viii+162 pp. - [60] P. Sternberg & W. Ziemer, Generalized motion by curvature with a Dirichlet condition, J. Differential Equations 114 (1994), 580-600. - [61] G. Talenti, Soluzioni a simmetria assiale di equazioni ellittiche. Ann. Mat. Pura Appl. (4) 73 (1966), 127-158. - [62] W. Urbina-Romero, Gaussian harmonic analysis. Springer Monogr. Math. Springer, Cham, 2019, xix+477 pp. - [63] G. N. Watson, A treatise on the theory of Bessel functions. Reprint of the second (1944) edition. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995. viii+804 pp. - [64] A. Weinstein, Discontinuous integrals and generalized potential theory. Trans. Amer. Math. Soc. 63 (1948), 342-354. - [65] A. Weinstein, On the wave equation and the equation of Euler-Poisson. McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954, pp. 137-147. - [66] A. Weinstein, On a singular differential operator. Ann. Mat. Pura Appl. (4) 49 (1960), 359-365.