# Pointwise convergence of fractional powers of Hermite type operators

G. Flores, G. Garrigós, T. Signes, B. Viviani  G. Flores, CIEM-CONICET, FaMAF,
Universidad Nacional de Córdoba, Av. Medina Allende s/n, Ciudad Universitaria,
CP:X5000HUA Córdoba, Argentina<EMAIL_ADDRESS>G. Garrigós and T.
Signes, Departamento de Matemáticas, Universidad de Murcia, 30100, Espinardo,
Murcia, Spain<EMAIL_ADDRESS><EMAIL_ADDRESS>Beatriz Viviani
IMAL (UNL-CONICET) y FIQ (Universidad Nacional del Litoral), Colectora Ruta
Nac. N 168, Paraje El Pozo - 3000 Santa Fe - Argentina  viviani@santafe-
conicet.gov.ar Dedicated to the memory of Eleonor _Pola_ Harboure, whose human
qualities
and professional behavior will always be our guidance

###### Abstract.

When $L$ is the Hermite or the Ornstein-Uhlenbeck operator, we find minimal
integrability and smoothness conditions on a function $f$ so that the
fractional power $L^{\sigma}f(x_{0})$ is well-defined at a given point
$x_{0}$. We illustrate the optimality of the conditions with various examples.
Finally, we obtain similar results for the fractional operators
$(-{\Delta}+R)^{\sigma}$, with $R>0$.

###### Key words and phrases:

Fractional Laplacian, Hermite operator, Ornstein-Uhlenbeck semigroup, Poisson
integrals.

###### 2010 Mathematics Subject Classification:

42C10, 35C15, 33C45, 40A10, 35R11.

G.G. was supported in part by grants MTM2017-83262-C2-2-P and
AEI/10.1303/501100011033 from Agencia Estatal de Investigación (Spain), and
grant 20906/PI/18 from Fundación Séneca (Región de Murcia, Spain).

##  1\. Introduction

Let $L$ be a positive self-adjoint differential operator densely defined in a
Hilbert space $L_{2}(\Omega,d\mu)$. Fractional powers $L^{\sigma}$, for
$\sigma>0$, can be defined in various (abstract) equivalent ways, one of the
most standard being via spectral theory:

$\langle
L^{\sigma}f,g\rangle\,=\,\int_{\sigma(L)}{\lambda}^{\sigma}\,dE_{f,g},\quad
f\in\mathop{\rm Dom}(L^{\sigma}),\;\;g\in L_{2}(\Omega,d\mu),$ (1.1)

where $E$ denotes a resolution of the identity associated with $L$; see e.g.
[13, Ch 13]. When the spectrum is discrete
$\sigma(L)=\\{{\lambda}_{n}\\}_{n=0}^{\infty}$, and
$\\{{\varphi}_{n}\\}_{n=0}^{\infty}$ is an orthonormal basis of
eigenfunctions, then (1.1) takes the form

$L^{\sigma}f\,=\,\sum_{n=0}^{\infty}\,{\lambda}_{n}^{\sigma}\,\langle
f,{\varphi}_{n}\rangle\,{\varphi}_{n},$

say for $f\in\mbox{\rm span}\,\\{{\varphi}_{n}\\}$. Alternatively, it is also
possible to express $L^{\sigma}$ in terms of the contraction semigroup
$\\{e^{-tL}\\}_{t>0}$. For instance, when $\sigma\in(0,1)$, via the Bochner
formula

$L^{\sigma}f=\tfrac{1}{{\Gamma}(-\sigma)}\int_{0}^{\infty}(e^{-tL}f-f)\,\frac{dt}{t^{1+\sigma}};$
(1.2)

see e.g. [17, Ch IX.11] and references therein.

When $L$ equals the classical Laplacian $-{\Delta}$ and
$f\in\mathcal{S}({{\mathbb{R}}^{d}})$, both (1.1) and (1.2) lead to the
explicit expression

$(-{\Delta})^{\sigma}f(x)\,=\,c_{d,\sigma}\,\mbox{{\small
PV}}\int_{{{\mathbb{R}}^{d}}}\frac{f(x)-f(y)}{|x-y|^{d+2\sigma}}\,dy,\quad
x\in{{\mathbb{R}}^{d}},\;\;\sigma\in(0,1),$ (1.3)

with a suitable constant $c_{d,\sigma}>0$; see e.g. [9]. This pointwise
formula does actually make sense for a larger class of functions, e.g. when
$f\in L_{1}(dy/(1+|y|)^{d+2\sigma})$ and $f$ is Hölder continuous of order
$2\sigma+{\varepsilon}$ near the point $x$. For general operators $L$,
however, explicit expressions such as (1.3) are not common, and the definition
of $L^{\sigma}f$ as in (1.1)-(1.2) must necessarily be restricted to a
suitable dense class of functions $f$.

Based on work of Caffarelli and Silvestre [2], Stinga and Torrea proposed in
[14] to define $L^{\sigma}f$ as the Neumann boundary value associated with the
elliptic PDE

$\left\\{\begin{array}[]{lll}u_{tt}+\tfrac{1-2\sigma}{t}\,u_{t}\,=\,Lu,&&t>0\\\
u(0,x)=f(x).&&\\\ \end{array}\right.$ (1.4)

More precisely, if $\sigma\in(0,1)$ and
$\mathfrak{c}_{\sigma}=2^{2\sigma-1}{\Gamma}(\sigma)/{\Gamma}(1-\sigma)$, they
set

$L^{\sigma}f(x)\,=\,-\mathfrak{c}_{\sigma}\,\lim_{t\to
0^{+}}\,t^{1-2\sigma}\,\partial_{t}u(t,x),$ (1.5)

where $u(t,x)$ is the solution of (1.4) given by the Poisson-like integral

$u(t,x)=P^{\sigma}_{t}f(x):=\frac{(t/2)^{2\sigma}}{{\Gamma}(\sigma)}\,\int_{0}^{\infty}e^{-\frac{t^{2}}{4s}}\,e^{-sL}f(x)\,\,\frac{ds}{s^{1+\sigma}};$
(1.6)

see [14, Theorem 1.1]. Then the authors specialize to the Hermite operator

$L=-{\Delta}+|x|^{2}\quad\mbox{in $L_{2}({{\mathbb{R}}^{d}})$},$

and prove that the _pointwise limit_ in (1.5) exists at every
$x\in{{\mathbb{R}}^{d}}$, and coincides with (1.2), whenever $f\in
C^{2}({{\mathbb{R}}^{d}})\cap L_{1}(dy/(1+|y|)^{N})$, for some $N>0$; see [14,
Theorem 4.2], and [15, 12] for slightly less restrictive smoothness
assumptions.

The purpose of this paper is to show the validity of (1.5) for a wider class
of functions $f$, with optimal integrability assumptions and very mild
smoothness at a given point $x_{0}\in{{\mathbb{R}}^{d}}$. We also consider the
slightly more general family of operators

$L=-{\Delta}+|x|^{2}+m,\quad\mbox{in $L_{2}({\mathbb{R}}^{d})$},$ (1.7)

with a constant parameter $m\geq-d$ (so that $L$ is positive). For $m=0$ this
is the usual Hermite operator, while for $m=-d$ it can be transformed (after a
change of variables) into the Ornstein-Uhlenbeck operator

$\mathcal{O}=-{\Delta}+2x\cdot\nabla,\quad\mbox{in
}\;L_{2}({{\mathbb{R}}^{d}},e^{-|x|^{2}}\,dx);$

see $\S\ref{S_OU}$ below.

To state our results, we now describe the integrability and smoothness
conditions we shall use below. Given a weight $\Phi(y)\geq 0$, we say that
$f\in L_{1}(\Phi)$ when

$\int_{{\mathbb{R}}^{d}}|f(y)|\,\Phi(y)\,dy<\infty.$ (1.8)

Associated with $L=-{\Delta}+|x|^{2}+m$ in (1.7), and $\sigma\in(0,1)$ we
define the weight

$\Phi_{\sigma}(y)=\left\\{\begin{array}[]{ll}\displaystyle\frac{e^{-|y|^{2}/2}}{(1+|y|)^{\frac{d+m}{2}}[\ln(e+|y|)]^{1+\sigma}}&\mbox{
if }m>-d\\\ &\\\
\displaystyle\frac{e^{-|y|^{2}/2}}{[\ln(e+|y|)]^{\sigma}}&\mbox{ if }m=-d.\\\
\end{array}\right.$ (1.9)

Then, $f\in L_{1}(\Phi_{\sigma})$ ensures that the Poisson-like integral
$P^{\sigma}_{t}f(x)$ in (1.6) is a well-defined (smooth) function when
$(t,x)\in(0,\infty)\times{\mathbb{R}}^{d}$, and this condition is actually
best possible for that property; see [6, Theorem 1.1] (or [8, 5]).

Next, when ${\alpha}\in(0,2)$, we shall say that a (locally integrable)
function $f$ is _${\alpha}$ -smooth at $x_{0}\in{\mathbb{R}}^{d}$_, denoted
$f\in\mathcal{D}^{\alpha}(x_{0})$, if it satisfies the Dini-type condition

$\int_{|h|\leq{\delta}}\frac{|f(x_{0}+h)+f(x_{0}-h)-2f(x_{0})|}{|h|^{d+{\alpha}}}\,dh\,<\,\infty,\quad\mbox{for
some ${\delta}>0$}.$

We say that $f$ is _strictly ${\alpha}$-smooth at $x_{0}$_, denoted
$f\in\mathcal{D}_{\rm st}^{\alpha}(x_{0})$, if

$f\in\mathcal{D}^{\alpha}(x_{0}){\quad\mbox{and}\quad}\int_{|h|\leq{\delta}}\frac{|f(x_{0}+h)-f(x_{0}-h)|}{|h|^{d+{\alpha}-3}}\,dh\,<\,\infty.$

Observe that this last condition is redundant if $d+{\alpha}-3\leq 0$; in
particular, if $d=1$ and ${\alpha}\in(0,2)$, or $d=2$ and ${\alpha}\in(0,1]$.
In other cases however, the classes are different and we shall need this
distinction to obtain our results. We refer to $\S\ref{S_smooth}$ below for
several examples of these notions, and their relation with local Lipschitz
conditions at $x_{0}$.

Our first main result can now be stated as follows.

######  Theorem 1.1.

Let $L$ be the Hermite operator in (1.7), $\sigma\in(0,1)$ and
$\Phi_{\sigma}(y)$ as in (1.9). Suppose that

$f\in L_{1}(\Phi_{\sigma}),{\quad\mbox{and}\quad}f\in\mathcal{D}_{\rm
st}^{2\sigma}(x_{0})\quad\mbox{for some $x_{0}\in{\mathbb{R}}^{d}$}.$

Then, the number $L^{\sigma}f(x_{0})$ exists both, in the limiting sense of
(1.5) and as the absolutely convergent integral in (1.2), and both definitions
agree.

The optimality of the smoothness condition is discussed in §6 below. The
corresponding version for the Ornstein-Uhlenbeck operator takes the following
form.

######  Theorem 1.2.

Let $L=\mathcal{O}=-{\Delta}+2x\cdot\nabla$ and $\sigma\in(0,1)$. Suppose that

$f\in
L_{1}(e^{-|y|^{2}}/[\log(e+y)]^{\sigma}),{\quad\mbox{and}\quad}f\in\mathcal{D}_{\rm
st}^{2\sigma}(x_{0})\quad\mbox{for some $x_{0}\in{\mathbb{R}}^{d}$}.$

Then, the number $\mathcal{O}^{\sigma}f(x_{0})$ exists both, in the limiting
sense of (1.5) and as the absolutely convergent integral in (1.2), and both
definitions agree.

Below, we have attempted to present our results in sufficient generality so
that many of the arguments can be applied to general operators $L$, while only
a few steps require explicit estimates on the kernels. We illustrate this fact
in §5, where with a minimal effort we obtain a version of the above theorems
for the operator ${\mathbb{L}}=-{\Delta}+R$, with $R>0$; see Theorem 5.2
below.

##  2\. Preliminary results for general operators $L$

In this section we shall assume that $L$ is the infinitesimal generator of a
semigroup of operators $\\{e^{-tL}\\}_{t>0}$ in $L_{2}({{\mathbb{R}}^{d}})$,
and that these are described by the integrals

$e^{-tL}f(x)=\int_{{\mathbb{R}}^{d}}h_{t}(x,y)f(y)\,dy,$ (2.1)

for a suitable positive kernel $h_{t}(x,y)$. For each $\sigma\in(0,1)$ we then
consider the family of subordinated operators
$\\{P^{\sigma}_{t}=P^{\sigma,L}_{t}\\}_{t>0}$, formally given by

$P^{\sigma}_{t}f:=\tfrac{(t/2)^{2\sigma}}{{\Gamma}(\sigma)}\,\int_{0}^{\infty}e^{-\frac{t^{2}}{4s}}\,e^{-sL}(f)\,\,\frac{ds}{s^{1+\sigma}}.$

More precisely, we let

$P^{\sigma}_{t}f(x)=\int_{{\mathbb{R}}^{d}}p^{\sigma}_{t}(x,y)f(y)\,dy,$ (2.2)

where the corresponding kernels $p^{\sigma}_{t}(x,y)=p^{\sigma,L}_{t}(x,y)$
are defined by

$p^{\sigma}_{t}(x,y)=\tfrac{(t/2)^{2\sigma}}{\Gamma(\sigma)}\,\int_{0}^{\infty}e^{-\frac{t^{2}}{4s}}\,h_{s}(x,y)\,\frac{ds}{s^{1+\sigma}}.$
(2.3)

Observe that a crude estimate such as $0<h_{s}(x,y)\lesssim s^{-d/2}$ (which
will be satisfied by all the operators $L$ we shall use) guarantees that the
integral in (2.3) is absolutely convergent, and moreover

$\partial_{t}\big{[}p^{\sigma}_{t}(x,y)\big{]}\,=\,\mathfrak{a}_{\sigma}\,t^{2\sigma-1}\int_{0}^{\infty}\Big{(}2\sigma-\frac{t^{2}}{2s}\Big{)}e^{-\frac{t^{2}}{4s}}h_{s}(x,y)\frac{ds}{s^{1+\sigma}},$
(2.4)

with $\mathfrak{a}_{\sigma}=1/(4^{\sigma}{\Gamma}(\sigma))$. It is also not
hard to show that

$t\,\Big{|}\partial_{t}\big{[}p^{\sigma}_{t}(x,y)\big{]}\Big{|}\,\lesssim\,p_{t}^{\sigma}(x,y)+p_{t/\sqrt{2}}^{\sigma}(x,y),$
(2.5)

using the fact that $\sup_{v>0}ve^{-v}<\infty$. However, in order to handle
derivatives of the expression $P^{\sigma}_{t}f(x)$ in (2.2) we shall need more
information on the decay of the kernel $p^{\sigma}_{t}(x,y)$.

In the case that $L$ is a Hermite operator the decay is given by the following
result from [6, Lemma 3.1], which also clarifies the optimal role of the
functions $\Phi_{\sigma}(y)$ in (1.9).

######  Lemma 2.1.

Let $L$ be the Hermite operator in (1.7), $\sigma\in(0,1)$ and
$\Phi_{\sigma}(y)$ as in (1.9). Then, for every $t>0$ and
$x\in{\mathbb{R}}^{d}$, there exist finite numbers $c_{1}(t,x)>0$ and
$c_{2}(t,x)>0$ such that

$c_{1}(t,x)\,\Phi_{\sigma}(y)\,\leq\,p^{\sigma}_{t}(x,y)\,\leq\,c_{2}(t,x)\,\Phi_{\sigma}(y)\;,\quad\forall\;y\in{\mathbb{R}}^{d}.$
(2.6)

In this section we wish to keep the general setting for the operator $L$
described above, but we shall _additionally_ assume that, for each given
$\sigma\in(0,1)$, the kernel $p^{\sigma}_{t}(x,y)$ satisfies

$p^{\sigma}_{t}(x,y)\,\leq\,c_{2}(t,x)\,\Phi(y)\;,\quad\forall\;y\in{\mathbb{R}}^{d}.$
(2.7)

for some function $\Phi(y)$ and some $c_{2}(t,x)>0$. In all the results in
this section we shall not need the explicit expression of $\Phi(y)$.

Our first result will establish a relation between the two _pointwise_
definitions of $L^{\sigma}f(x)$ presented in (1.2) and (1.5) above. We first
consider the following general definition.

######  Definition 2.2.

Let $\sigma\in(0,1)$ and $L$ be an operator such that (2.7) holds for some
$\Phi(y)$. Given $f\in L_{1}(\Phi)$, we say that a point
$x_{0}\in{\mathbb{R}}^{d}$ is $L^{\sigma}$-admissible for $f$, denoted
$x_{0}\in\mathcal{A}_{f}(L^{\sigma})$, if

$\int_{0}^{\infty}\big{|}e^{-sL}f(x_{0})-f(x_{0})\big{|}\frac{ds}{s^{1+\sigma}}<\infty.$
(2.8)

In that case we let

$L^{\sigma}f(x_{0}):=\tfrac{1}{{\Gamma}(-\sigma)}\int_{0}^{\infty}\Big{(}e^{-sL}f(x_{0})-f(x_{0})\Big{)}\,\frac{ds}{s^{1+\sigma}},$
(2.9)

where ${\Gamma}(-\sigma)={\Gamma}(1-\sigma)/(-\sigma)$.

The following result is partially based on the proof of [14, (4.6)].

######  Proposition 2.3.

Let $\sigma\in(0,1)$ and $L$ be an operator such that (2.7) holds for some
$\Phi(y)$. If $f\in L_{1}(\Phi)$ and $x\in\mathcal{A}_{f}(L^{\sigma})$ then

$\lim_{t\to
0^{+}}\,-\mathfrak{c}_{\sigma}\,t^{1-2\sigma}\,\partial_{t}\Big{[}P^{\sigma}_{t}f(x)\Big{]}=L^{\sigma}f(x)\,,$
(2.10)

with $\mathfrak{c}_{\sigma}=2^{2\sigma-1}{\Gamma}(\sigma)/{\Gamma}(1-\sigma)$.

###### Proof.

Using (2.2), (2.4), (2.5) and $f\in L_{1}(\Phi)$ one can justify that

$t^{1-2\sigma}\,\partial_{t}\Big{[}P^{\sigma}_{t}f(x)\Big{]}=\mathfrak{a}_{\sigma}\,\int_{{\mathbb{R}}^{d}}\int_{0}^{\infty}\Big{(}2\sigma-\frac{t^{2}}{2s}\Big{)}e^{-\frac{t^{2}}{4s}}h_{s}(x,y)\,\frac{ds}{s^{1+\sigma}}\,f(y)\,dy,$

with $\mathfrak{a}_{\sigma}=1/(4^{\sigma}{\Gamma}(\sigma))$. We claim that

$I=\int_{0}^{\infty}\Big{(}2\sigma-\frac{t^{2}}{2s}\Big{)}e^{-\frac{t^{2}}{4s}}\frac{ds}{s^{1+\sigma}}=0.$

In fact, using the change $z=t^{2}/4s$ we see that

$I=\frac{2\cdot
4^{\sigma}}{t^{2\sigma}}\int_{0}^{\infty}(\sigma-z)e^{-z}z^{\sigma-1}\,dz=\frac{2\cdot
4^{\sigma}}{t^{2\sigma}}\big{(}\sigma\Gamma(\sigma)-\Gamma(\sigma+1)\big{)}=0.$

Then

$\displaystyle t^{1-2\sigma}\,\partial_{t}\Big{[}P^{\sigma}_{t}f(x)\Big{]}$
$\displaystyle=$
$\displaystyle\mathfrak{a}_{\sigma}\,\int_{0}^{\infty}\Big{(}2\sigma-\frac{t^{2}}{2s}\Big{)}e^{-\frac{t^{2}}{4s}}\,\Big{[}\int_{{\mathbb{R}}^{d}}h_{s}(x,y)f(y)\,dy-f(x)\Big{]}\frac{ds}{s^{1+\sigma}}$
$\displaystyle=$
$\displaystyle\mathfrak{a}_{\sigma}\,\int_{0}^{\infty}\Big{(}2\sigma-\frac{t^{2}}{2s}\Big{)}e^{-\frac{t^{2}}{4s}}\,\Big{[}e^{-sL}f(x)-f(x)\Big{]}\frac{ds}{s^{1+\sigma}}.$

Since we assume that $x\in\mathcal{A}_{f}(L^{\sigma})$, by the Lebesgue
dominated convergence theorem we can take limits as $t\to 0^{+}$, and after
adjusting the constants one easily obtains the result. ∎

In view of Proposition 2.3, we are interested in finding conditions on a
function $f$ which guarantee that a given point
$x\in\mathcal{A}_{f}(L^{\sigma})$. Our next observation shows that only one
part of the integral in (2.8) must be checked.

######  Lemma 2.4.

Let $\sigma\in(0,1)$ and $L$ be an operator such that (2.7) holds for some
$\Phi(y)$. Then, for every $A>0$ and every $x\in{\mathbb{R}}^{d}$ there exists
$c(x,A)>0$ such that

$\int_{A}^{\infty}h_{s}(x,y)\,\frac{ds}{s^{1+\sigma}}\leq
c(x,A)\,\Phi(y),\quad y\in{{\mathbb{R}}^{d}}.$ (2.11)

Moreover, if $f\in L_{1}(\Phi)$ and $|f(x)|<\infty$ then

$\int_{A}^{\infty}|e^{-sL}f(x)-f(x)|\,\frac{ds}{s^{1+\sigma}}<\infty.$

###### Proof.

To prove (2.11), note that

$\int_{A}^{\infty}h_{s}(x,y)\,\frac{ds}{s^{1+\sigma}}\leq
e^{\frac{1}{4A}}\,\int_{0}^{\infty}e^{-\frac{1}{4s}}\,h_{s}(x,y)\,\frac{ds}{s^{1+\sigma}}=c\,p_{1}^{\sigma}(x,y)\leq
c(x,A)\,\Phi(y).$

For the last statement,

$\displaystyle\int_{A}^{\infty}\\!|e^{-sL}f(x)-f(x)|\frac{ds}{s^{1+\sigma}}$
$\displaystyle\leq$
$\displaystyle\int_{{{\mathbb{R}}^{d}}}|f(y)|\int_{A}^{\infty}\\!h_{s}(x,y)\,\frac{ds}{s^{1+\sigma}}\,dy+|f(x)|\,\int_{A}^{\infty}\\!\frac{ds}{s^{1+\sigma}}$
by (2.11) $\displaystyle\lesssim$ $\displaystyle
c(x,A)\int_{{{\mathbb{R}}^{d}}}|f(y)|\,\Phi(y)\,dy+|f(x)|<\infty.$

∎

In order to show that $x_{0}\in\mathcal{A}_{f}(L^{\sigma})$ we expect that
some smoothness of $f$ at the point $x_{0}$ must be required. Actually, the
smoothness of $f$ will only play a _local_ role in the integrals defining the
property $\mathcal{A}_{f}(L^{\sigma})$. This motivates to consider a local
notion of $L^{\sigma}$-admissibility.

######  Definition 2.5.

Let $\sigma\in(0,1)$, and $L$ an operator as above. Given a locally integrable
function $f$, we say that a point $x_{0}\in{\mathbb{R}}^{d}$ is _locally
$L^{\sigma}$-admissible for $f$_, denoted $x_{0}\in\mathcal{A}^{\rm
loc}_{f}(L^{\sigma})$, if there exists ${\delta}>0$ and $A>0$ such that the
integrals

$\mathcal{I}_{\delta}f(x_{0},s)=\int_{|x_{0}-y|<\delta}h_{s}(x_{0},y)\,\big{[}f(y)-f(x_{0})\big{]}dy,\quad
s\in(0,A)$ (2.12)

satisfy the property

$\int_{0}^{A}\big{|}\mathcal{I}_{\delta}f(x_{0},s)\big{|}\,\frac{ds}{s^{1+\sigma}}\,<\,\infty.$
(2.13)

The next lemma gives decay conditions on the kernel and smoothness of $f$ at
$x_{0}$ that guarantee the validity of the previous property.

######  Proposition 2.6.

Let $\sigma\in(0,1)$, and $L$ an operator as above. Let
$x_{0}\in{{\mathbb{R}}^{d}}$ be fixed, and assume that the kernel
$h_{s}(x_{0},\cdot)$ in (2.1) satisfies, for some ${\delta}>0$ and
$A\in(0,\infty]$, the estimates

$\int_{0}^{A}h_{s}(x_{0},x_{0}+y)\frac{ds}{s^{1+\sigma}}\,\leq\frac{c(x_{0})}{|y|^{d+2\sigma}},\quad\mbox{when
}\;\;|y|\leq{\delta}$ (2.14)

and

$\int_{0}^{A}\Big{|}h_{s}(x_{0},x_{0}+y)-h_{s}(x_{0},x_{0}-y)\Big{|}\frac{ds}{s^{1+\sigma}}\,\leq\,\frac{c(x_{0})}{|y|^{(d+2\sigma-3)_{+}}},\quad\mbox{when
}\;\;|y|\leq{\delta}.$ (2.15)

Then, for every locally integrable $f$ it holds

$f\in\mathcal{D}_{\rm st}^{2\sigma}(x_{0})\quad\implies\quad
x_{0}\in\mathcal{A}^{\rm loc}_{f}(L^{\sigma}).$ (2.16)

Moreover, (2.13) holds with the same $A$ and ${\delta}$ as in (2.14) and
(2.15).

Before proving the result, recall the standard notation

$f_{\rm even}(x)=\frac{f(x)+f(-x)}{2}{\quad\mbox{and}\quad}f_{\rm
odd}(x)=\frac{f(x)-f(-x)}{2}.$

We shall use the following elementary lemma.

######  Lemma 2.7.

Let $F$ and $G$ be locally integrable in ${\mathbb{R}}^{d}$, and $B$ a ball
centered at the origin. Then

$\int_{B}F(x)G(x)\,dx=\int_{B}F_{\rm even}(x)G_{\rm
even}(x)\,dx+\int_{B}F_{\rm odd}(x)G_{\rm odd}(x)\,dx.$ (2.17)

We also introduce the notation

$\triangle^{1}_{z}f(x)=f(x+z)-f(x-z),{\quad\mbox{and}\quad}\triangle^{2}_{z}f(x)=f(x+z)+f(x-z)-2f(x).$

Observe that, after dividing by 2, these expressions are respectively the odd
and even parts of the function $z\mapsto f(x+z)-f(x)$.

PROOF of Proposition 2.6:  Changing variables $y=x_{0}+z$ in (2.12), we can
write

$\mathcal{I}_{\delta}f(x_{0},s)=\int_{|z|<\delta}h_{s}(x_{0},x_{0}+z)\,\big{[}f(x_{0}+z)-f(x_{0})\big{]}dz.$

Then, using the identity in (2.17) and simple manipulations, we can rewrite
this expression as

$\displaystyle\mathcal{I}_{\delta}f(x_{0},s)$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\int_{|z|<\delta}h_{s}(x_{0},x_{0}+z)\triangle_{z}^{2}f(x_{0})dz$
(2.18)
$\displaystyle\hskip-28.45274pt+\tfrac{1}{4}\int_{|z|<\delta}\big{(}h_{s}(x_{0},x_{0}+z)-h_{s}(x_{0},x_{0}-z)\big{)}\,\triangle_{z}^{1}f(x_{0})dz$

Thus, using the kernel assumptions in (2.14) and (2.15), we clearly have

$\displaystyle\int_{0}^{A}|\mathcal{I}_{\delta}f(x_{0},s)|\,\frac{ds}{s^{1+\sigma}}\,$
$\displaystyle\leq$
$\displaystyle\tfrac{1}{2}\int_{|z|<\delta}|\triangle_{z}^{2}f(x_{0})|\Big{(}\int_{0}^{A}h_{s}(x_{0},x_{0}+z)\,\frac{ds}{s^{1+\sigma}}\Big{)}dz$
$\displaystyle\hskip-56.9055pt+\tfrac{1}{4}\int_{|z|<\delta}|\triangle_{z}^{1}f(x_{0})|\,\int_{0}^{A}|h_{s}(x_{0},x_{0}+z)-h_{s}(x_{0},x_{0}-z)|\,\frac{ds}{s^{1+\sigma}}dz$
$\displaystyle\lesssim$
$\displaystyle\int_{|z|<\delta}\frac{|\triangle_{z}^{2}f(x_{0})|}{|z|^{d+2\sigma}}\,dz\,+\,\int_{|z|<\delta}\frac{|\triangle_{z}^{1}f(x_{0})|}{|z|^{(d+2\sigma-3)_{+}}}\,dz$

which is a finite quantity when $f\in\mathcal{D}_{\rm st}^{2\sigma}(x_{0})$.

$\Box$

######  Remark 2.8.

When the heat kernel at a given $x_{0}$ satisfies

$h_{s}(x_{0},y)=\rho_{s,x_{0}}(|x_{0}-y|),$

for some function $\rho_{s,x_{0}}$ (for instance, if $h_{s}$ is of convolution
type and radial), then the condition (2.15) is automatically satisfied (since
the integrand is 0). Moreover, in the proof of the proposition the integral in
(2.18) vanishes, so no bound is needed involving $\triangle_{z}^{1}f(x_{0})$.
Thus, in that setting, the conclusion (2.16) of the proposition holds with the
weaker smoothness assumption $f\in\mathcal{D}^{2\sigma}(x_{0})$.

##  3\. The Hermite operator $L=-{\Delta}+|x|^{2}+m$, with $m\geq-d$

In this section we specialize to the case when

$L=-{\Delta}+|x|^{2}+m,\quad\mbox{with $m\geq-d$.}$

We recall the kernel expressions in this setting. For the heat kernel
$h_{t}(x,y)$, associated with $e^{-tL}$, we have the Mehler formula

$h_{t}(x,y)=\,e^{-tm}\,\frac{e^{-\frac{|x-y|^{2}}{2{\operatorname{th}\,}{2t}}-{\operatorname{th}\,}{t}\,x\cdot
y}}{[2\pi{\operatorname{sh}\,}{2t}]^{\frac{d}{2}}}\,,\quad
t>0,\;x,y\in{{\mathbb{R}}^{d}};$

see e.g. [16, (4.3.14)]. Changing variables to
$t=t(s)=\frac{1}{2}\ln(\frac{1+s}{1-s})$ (or equivalently,
$s={\operatorname{th}\,}(t)$), the kernel takes the form

$h_{t(s)}(x,y)=\,\frac{(1-s)^{\frac{m+d}{2}}}{(1+s)^{\frac{m-d}{2}}}\,\frac{e^{-\frac{1}{4}(\frac{|x-y|^{2}}{s}+s|x+y|^{2})}}{(4\pi
s)^{\frac{d}{2}}}\,.$ (3.1)

In the next subsection we shall collect the decay and smoothness estimates of
this kernel that will be needed in the proof of Theorem 1.1.

###  3.1. Kernel estimates

Throughout this section we denote

$\mathcal{K}(x,y):=\int_{0}^{A}h_{t}(x,y)\,\frac{dt}{t^{1+\sigma}},\quad
x,y\in{{\mathbb{R}}^{d}},$ (3.2)

where we select $A>0$ so that ${\operatorname{th}\,}A=1/2$ (any other $A>0$
would also be fine). Performing the change of variables in (3.1) (so that
$dt=\frac{ds}{1-s^{2}}$) we obtain

$\mathcal{K}(x,y)=\int_{0}^{1/2}\,\frac{(1-s)^{\frac{m+d}{2}-1}\,e^{-\frac{1}{4}(\frac{|x-y|^{2}}{s}+s|x+y|^{2})}}{(1+s)^{\frac{m-d}{2}+1}\,(4\pi
s)^{\frac{d}{2}}\,\big{(}\frac{1}{2}\ln\frac{1+s}{1-s}\big{)}^{1+\sigma}}\,ds.$
(3.3)

Observe that in this range of integration we have $\ln\frac{1+s}{1-s}\approx
s$, and $1\pm s\approx 1$, so $\mathcal{K}(x,y)$ becomes comparable to

$\mathcal{K}_{1}(x,y)=\int_{0}^{1/2}\,\frac{e^{-\frac{1}{4}(\frac{|x-y|^{2}}{s}+s|x+y|^{2})}}{s^{1+\sigma+\frac{d}{2}}}\,ds.$
(3.4)

The first lemma shows the decay condition in (2.14). The argument in the proof
is similar to the one used in [8, (4.13)] (where a better estimate is
obtained).

######  Lemma 3.1.

With the notation in (3.3), for every $\sigma>0$ there exists $c=c(\sigma)>0$
such that

$\mathcal{K}(x,y)\leq\frac{c}{|x-y|^{d+2\sigma}},\quad\forall\;x,y\in{\mathbb{R}}^{d}.$

###### Proof.

Changing variables $u=\frac{|x-y|^{2}}{4s}$ in (3.4) we see that

$\displaystyle\mathcal{K}(x,y)$ $\displaystyle\approx$
$\displaystyle\mathcal{K}_{1}(x,y)=\Big{(}\frac{4}{|x-y|^{2}}\Big{)}^{\sigma+\frac{d}{2}}\,\int_{\frac{|x-y|^{2}}{2}}^{\infty}e^{-u}\,e^{-\frac{|x+y|^{2}\,|x-y|^{2}}{16u}}\,u^{\sigma+\frac{d}{2}}\,\frac{du}{u}$
$\displaystyle\leq$
$\displaystyle\frac{4^{\sigma+\frac{d}{2}}}{|x-y|^{d+2\sigma}}\,\int_{0}^{\infty}e^{-u}\,u^{\sigma+\frac{d}{2}}\,\frac{du}{u}=\frac{c}{|x-y|^{d+2\sigma}}.$

∎

######  Remark 3.2.

This lemma may also be proved directly from (3.2) using the property

$h_{t}(x,y)\lesssim t^{-d/2}e^{-c\frac{|x-y|^{2}}{t}},\quad 0<t\lesssim 1.$

This property is known to hold for many other operators $L$.

We now show the smoothness condition in (2.15).

######  Lemma 3.3.

For every $\sigma>0$, there exists $c=c(\sigma)>0$ such that

$\int_{0}^{A}|h_{t}(x,x+y)-h_{t}(x,x-y)|\,\frac{dt}{t^{1+\sigma}}\leq\,\frac{c\,|x|}{|y|^{(d+2\sigma-3)_{+}}},\quad\forall\;x\in{\mathbb{R}}^{d},\;|y|\leq
1.$ (3.5)

In the proof of (3.5) we shall use the following elementary inequality.

######  Lemma 3.4.

If $x,y\in{{\mathbb{R}}^{d}}$ then

$\Big{|}e^{-|x+y|^{2}}-e^{-|x-y|^{2}}\Big{|}\leq 4|x|\,|y|\;e^{-\min|x\pm
y|^{2}}.$

###### Proof.

Using $|x\pm y|^{2}=|x|^{2}+|y|^{2}\pm 2x\cdot y$ we can write

$\Big{|}e^{-|x+y|^{2}}-e^{-|x-y|^{2}}\Big{|}=e^{-|x|^{2}-|y|^{2}}\,\Big{|}e^{2x\cdot
y}-e^{-2x\cdot y}\Big{|}.$

Now, letting $t=2|x\cdot y|$, and using the inequality

$e^{t}-e^{-t}=\int^{t}_{-t}e^{s}\,ds\leq 2te^{t},$

we obtain

$\Big{|}e^{-|x+y|^{2}}-e^{-|x-y|^{2}}\Big{|}\leq 4|x|\,|y|\,e^{2|x\cdot
y|}\,e^{-|x|^{2}-|y|^{2}}=4|x|\,|y|\,e^{-\min|x\pm y|^{2}}.$

∎

PROOF of Lemma 3.3:  Denote by $\mathcal{K}^{\prime}(x,y)$ the left hand side
of (3.5). Then, performing the change of variables in (3.1), and disregarding
the inessential terms (as discussed before (3.4)) we obtain

$\displaystyle\mathcal{K}^{\prime}(x,y)$ $\displaystyle\approx$
$\displaystyle\int_{0}^{1/2}\frac{e^{-\frac{|y|^{2}}{4s}}\,\big{|}e^{-\frac{s}{4}|2x+y|^{2}}-e^{-\frac{s}{4}|2x-y|^{2}}\big{|}}{s^{1+\sigma+\frac{d}{2}}}\,ds$
(by Lemma 3.4) $\displaystyle\lesssim$
$\displaystyle\int_{0}^{1/2}\frac{e^{-\frac{|y|^{2}}{4s}}\,s\,|2x|\,|y|}{s^{1+\sigma+\frac{d}{2}}}\,ds\;\lesssim\;|x|\,|y|\,\int_{0}^{1/2}\frac{e^{-\frac{|y|^{2}}{4s}}}{s^{\sigma+\frac{d}{2}}}\,ds$
($u=|y|^{2}/(4s)$) $\displaystyle=$
$\displaystyle\frac{c\,|x|\,|y|}{|y|^{d+2\sigma-2}}\,\int_{|y|^{2}/2}^{\infty}e^{-u}\,u^{\sigma+\frac{d}{2}-1}\,\frac{du}{u}\;\leq\;\frac{c^{\prime}\,|x|}{|y|^{d+2\sigma-3}},$

the last bound being valid when $\sigma+d/2-1>0$. This is always the case when
$d\geq 2$ and $\sigma>0$, and also if $d=1$ and $\sigma>1/2$.

In the special case that $d=1$ and $\sigma\in(0,1/2]$, one observes that the
integral

$I(y):=\int_{|y|^{2}/2}^{\infty}e^{-u}\,u^{\sigma-\frac{1}{2}}\,\frac{du}{u}\;\approx\left\\{\begin{array}[]{lll}\log(e/|y|),&&\mbox{if
$\sigma=1/2$}\\\ {|y|^{2\sigma-1}},&&\mbox{if
$\sigma\in(0,1/2)$},\end{array}\right.$

when $|y|\leq 1$. Inserting this into the above estimates, it leads to

$\mathcal{K}^{\prime}(x,y)\lesssim\left\\{\begin{array}[]{lll}|x|\,|y|\,\log(e/|y|),&&\mbox{if
$\sigma=1/2$}\\\ |x|\,|y|,&&\mbox{if $\sigma\in(0,1/2)$},\end{array}\right.$

which implies

$\mathcal{K}^{\prime}(x,y)\lesssim|x|,\quad\forall\;|y|\leq 1.$

Note that this matches (3.5) in the special case $d=1$ and $\sigma\leq 1/2$.

$\Box$

Our last result is a strengthening of the decay estimate in Lemma 3.1 when
$|y|\gg|x|$. The proof follows a similar reasoning as in [8, Lemma 4.2].

######  Lemma 3.5.

Let $\sigma>0$. Then there exist $c=c(\sigma)>0$ and ${\gamma}>0$ such that

$\mathcal{K}(x,y)\leq c\,e^{-(\frac{1}{2}+{\gamma})|y|^{2}},\quad\mbox{when
$|y|\geq 10\,\max\\{|x|,1\\}$. }$

###### Proof.

For simplicity denote $a=|x+y|$ and $b=|x-y|$. Note that, the condition
$|y|\geq 10|x|$ implies

$a^{2},b^{2}\geq\big{(}\tfrac{9}{10}\big{)}^{2}\,|y|^{2}.$

Given a small $\eta\in(0,1)$ (to be determined) we have, for all $s\in(0,1/2)$

$\displaystyle e^{-\frac{1}{4}(sa^{2}+\frac{b^{2}}{s})}$ $\displaystyle=$
$\displaystyle e^{-\frac{\eta
b^{2}}{4s}}\,e^{-\frac{1}{4}(sa^{2}+(1-\eta)\frac{b^{2}}{s})}$
$\displaystyle\leq$ $\displaystyle e^{-\frac{\eta
b^{2}}{4s}}\,e^{-\frac{1}{4}\,\big{(}\frac{9}{10}\big{)}^{2}\,|y|^{2}\,\big{(}s+(1-\eta)\frac{1}{s}\big{)}}$
$\displaystyle\leq$ $\displaystyle e^{-\frac{\eta
b^{2}}{4s}}\,e^{-\frac{1}{4}\,\big{(}\frac{9}{10}\big{)}^{2}\,|y|^{2}\,(1-\eta)\,\frac{5}{2}},$

using that $s+\frac{1}{s}\geq 5/2$ when $s\in(0,1/2)$. Note that, if $\eta>0$
is chosen sufficiently small, we can find some ${\gamma}>0$ such that

$\tfrac{1}{4}\,\big{(}\tfrac{9}{10}\big{)}^{2}\,(1-\eta)\,\tfrac{5}{2}\,>\,\tfrac{1}{2}+{\gamma}.$

So we have

$e^{-\frac{1}{4}(sa^{2}+\frac{b^{2}}{s})}\leq e^{-\frac{\eta
b^{2}}{4s}}\,e^{-(\frac{1}{2}+{\gamma})\,|y|^{2}},\quad s\in(0,1/2).$

Thus, inserting these estimates into (3.4), we obtain

$\mathcal{K}(x,y)\approx\mathcal{K}_{1}(x,y)\leq
e^{-(\frac{1}{2}+{\gamma})\,|y|^{2}}\,\int_{0}^{1/2}e^{-\frac{\eta|x-y|^{2}}{4s}}\,\frac{ds}{s^{1+\sigma+\frac{d}{2}}}.$

Finally, in the last integral we perform the change of variables
$u=\frac{\eta|x-y|^{2}}{4s}$ and obtain

$\mathcal{K}(x,y)\lesssim\frac{e^{-(\frac{1}{2}+{\gamma})\,|y|^{2}}}{|x-y|^{2\sigma+d}}\,\int_{0}^{\infty}e^{-u}\,u^{\sigma+\frac{d}{2}}\,\tfrac{du}{u}\lesssim
e^{-(\frac{1}{2}+{\gamma})\,|y|^{2}},$

using in the last step that $|x-y|\approx|y|\geq 1$, under the conditions in
the statement. ∎

###  3.2. Regular positive eigenvectors

######  Definition 3.6.

We say that $\psi(x)\in\mathop{\rm Dom}(L)$ is a _regular positive
eigenvector_ of $L$ if

  1. (a)

$\psi\in C^{\infty}({{\mathbb{R}}^{d}})$

  2. (b)

$\psi(x)>0$, $\forall\,x\in{\mathbb{R}}^{d}$

  3. (c)

$L(\psi)={\lambda}\psi$, for some ${\lambda}\geq 0$.

When $L=-{\Delta}+|x|^{2}+m$, it is elementary to find an explicit regular
positive eigenvector, namely

$\psi(x)=e^{-|x|^{2}/2}.$

Indeed, it is easily verified that $L(\psi)={\lambda}\psi$ with
${\lambda}=m+d\geq 0$.

We have the following simple lemma, which is valid for general operators $L$.

######  Lemma 3.7.

Let $\psi$ be a regular positive eigenvector of $L$. Then, for all
$\sigma\in(0,1)$ and all $x\in{\mathbb{R}}^{d}$ it holds

$\int_{0}^{\infty}\Big{|}e^{-tL}\psi(x)-\psi(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}\,<\,\infty.$

That is, $x\in\mathcal{A}_{\psi}(L^{\sigma})$, for all
$x\in{{\mathbb{R}}^{d}}$.

###### Proof.

Since $e^{-tL}\psi=e^{-t{\lambda}}\psi$, the result is clear if ${\lambda}=0$.
If ${\lambda}>0$, then we have

$\int_{0}^{\infty}\Big{|}e^{-tL}\psi(x)-\psi(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}=\psi(x)\,\int_{0}^{\infty}|e^{-t{\lambda}}-1|\,\frac{dt}{t^{1+\sigma}}.$
(3.6)

Now, from the elementary estimate

$|e^{-t{\lambda}}-1|=\Big{|}\int_{0}^{t}\,{\lambda}\,e^{-s{\lambda}}\,ds\Big{|}\leq\min\\{{\lambda}\,t,2\\}.$

one deduces that (3.6) is a finite expression when $\sigma\in(0,1)$. ∎

######  Remark 3.8.

In this paper we shall not pursue this notion with other operators $L$, but it
is well-known that such eigenvectors exist when $L=-{\Delta}+V(x)$, under very
general conditions on $V(x)$; see e.g. [11, Theorem 11.8].

###  3.3. Proof of Theorem 1.1

In this section we prove Theorem 1.1 for the Hermite operator

$L=-{\Delta}+|x|^{2}+m.$

That is, if $\sigma\in(0,1)$ and $\Phi_{\sigma}$ is given as in (1.9), we must
show that, for $x=x_{0}$,

$\int_{0}^{\infty}\Big{|}e^{-tL}f(x)-f(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}\,<\,\infty,$

under the conditions $f\in L_{1}(\Phi_{\sigma})$ and
$f\in\mathcal{D}^{2\sigma}_{\rm st}(x)$. In that case, the assertions in the
theorem will follow directly from Proposition 2.3. In view of Lemma 2.4, it
suffices to show that

$\int_{0}^{A}\Big{|}e^{-tL}f(x)-f(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}\,<\,\infty,$
(3.7)

where $A>0$ can be chosen as in §3.1.

Let $\psi$ be a regular positive eigenvector for $L$, as described in §3.2.
Since $\psi(x)>0$, a multiplication by this number does not affect the
finiteness of (3.7). Now we have

$\displaystyle J$ $\displaystyle:=$
$\displaystyle\psi(x)\,\int_{0}^{A}\Big{|}e^{-tL}f(x)-f(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}$
$\displaystyle=$
$\displaystyle\int_{0}^{A}\Big{|}\big{(}e^{-tL}f\big{)}(x)\psi(x)-f(x)\psi(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}$
$\displaystyle\leq$
$\displaystyle\int_{0}^{A}\Big{|}\big{(}e^{-tL}f\big{)}(x)\psi(x)-\big{(}e^{-tL}\psi\big{)}(x)f(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}$
$\displaystyle\quad+|f(x)|\,\int_{0}^{A}\Big{|}\big{(}e^{-tL}\psi\big{)}(x)-\psi(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}\,=\,J_{1}+J_{2}.$

Note that $J_{2}<\infty$ by Lemma 3.7, so we must only prove the finiteness of
$J_{1}$. For that term, we have the following inequalities

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{A}\Big{|}\int_{{{\mathbb{R}}^{d}}}h_{t}(x,y)\big{[}f(y)\psi(x)-f(x)\psi(y)\big{]}\,dy\Big{|}\,\frac{dt}{t^{1+\sigma}}$
$\displaystyle\leq$
$\displaystyle\psi(x)\,\int_{0}^{A}\Big{|}\int_{{{\mathbb{R}}^{d}}}h_{t}(x,y)\,\big{[}f(y)-f(x)\big{]}\,dy\Big{|}\,\frac{dt}{t^{1+\sigma}}$
$\displaystyle\quad+|f(x)|\,\int_{0}^{A}\Big{|}\int_{{{\mathbb{R}}^{d}}}h_{t}(x,y)\,\big{[}\psi(x)-\psi(y)\big{]}\,dy\Big{|}\,\frac{dt}{t^{1+\sigma}}\,=\,J_{11}+J_{12}.$

The two summands, $J_{11}$ and $J_{12}$, can be treated similarly, since both
functions $f$ and $\psi$ belong to
$L_{1}(\Phi_{\sigma})\cap\mathcal{D}^{2\sigma}_{\rm st}(x)$, by
assumption***Actually, $\psi$ is much smoother than just
$\mathcal{D}^{2\sigma}_{\rm st}(x)$, so $J_{12}$ is formally easier.. So in
the sequel we will just prove that $J_{11}<\infty$ and this will be enough to
conclude the theorem. In fact, since $\psi(x)>0$, it will suffice to show that

$\mathcal{J}_{11}:=\int_{0}^{A}\Big{|}\int_{{{\mathbb{R}}^{d}}}h_{t}(x,y)\,\big{[}f(y)-f(x)\big{]}\,dy\Big{|}\,\frac{dt}{t^{1+\sigma}}\,<\,\infty.$

At this point we let ${\delta}=11\max\\{|x|,1\\}$, and split the inner
integral into the two regions $\\{|y-x|<{\delta}\\}$ and
$\\{|y-x|\geq{\delta}\\}\subset\big{\\{}|y|\geq 10\max\\{|x|,1\\}\big{\\}}$.
So recalling the notation for $\mathcal{I}_{\delta}f(x,t)$ in (2.12) we have

$\mathcal{J}_{11}\leq\int_{0}^{A}\big{|}\mathcal{I}_{\delta}f(x,t)\big{|}\,\frac{dt}{t^{1+\sigma}}\,+\,\mathcal{J}_{11}^{*},$
(3.9)

where

$\displaystyle\mathcal{J}_{11}^{*}$ $\displaystyle=$
$\displaystyle\int_{0}^{A}\int_{|y|\geq
10\max\\{|x|,1\\}}h_{t}(x,y)\,\big{|}f(y)-f(x)\big{|}\,dy\,\frac{dt}{t^{1+\sigma}}$
$\displaystyle=$ $\displaystyle\int_{|y|\geq
10\max\\{|x|,1\\}}\,\big{|}f(y)-f(x)\big{|}\,\mathcal{K}(x,y)\,dy,$

using this time the notation for $\mathcal{K}(x,y)$ in (3.2). By Lemma 3.5,
this last kernel has a gaussian decay, which leads to

$\mathcal{J}_{11}^{*}\lesssim\int_{{{\mathbb{R}}^{d}}}\big{(}|f(x)|+|f(y)|\big{)}\,e^{-(\frac{1}{2}+{\gamma})|y|^{2}}\,dy\lesssim|f(x)|+\int_{{{\mathbb{R}}^{d}}}|f(y)|\,\Phi_{\sigma}(y)\,dy,$

since, from the definition in (1.9), one has
$e^{-(\frac{1}{2}+{\gamma})|y|^{2}}\lesssim\Phi_{\sigma}(y)$ (actually, for
all $\sigma>0$). Thus, the assumption $f\in L_{1}(\Phi_{\sigma})$ gives
$\mathcal{J}_{11}^{*}<\infty$, and hence, in view of (3.9), we have reduced
matters to verify that

$\int_{0}^{A}\big{|}\mathcal{I}_{\delta}f(x,t)\big{|}\,\frac{dt}{t^{1+\sigma}}<\infty.$

But this is precisely the condition $x\in\mathcal{A}_{f}^{\rm
loc}(L^{\sigma})$ in Definition 2.5. Now, in view of Proposition 2.6, this
property is a consequence of the smoothness assumption
$f\in\mathcal{D}^{2\sigma}_{\rm st}(x)$, since the heat kernel $h_{t}(x,y)$
satisfies the two hypotheses in the proposition, (2.14) and (2.15), due to
Lemmas 3.1 and 3.3. This completes the proof of Theorem 1.1.

$\Box$

######  Remark 3.9.

When $x_{0}=0$, Theorem 1.1 holds with the weaker smoothness condition
$f\in\mathcal{D}^{2\sigma}(x_{0})$. This is because of the observation in
Remark 2.8, since the heat kernel for the Hermite operator, $h_{t}(0,y)$, only
depends on $|y|$; see (3.1).

##  4\. The Ornstein-Uhlenbeck operator
$\mathcal{O}=\,-{\Delta}+2x\cdot\nabla$

###  4.1. Proof of Theorem 1.2

We now turn to the proof of Theorem 1.2 for the operator

$\mathcal{O}=\,-{\Delta}+2x\cdot\nabla,$

which is positive and self-adjoint in $L_{2}(e^{-|y|^{2}}dy)$. In this case,
there is a well-known transference principle, see e.g. [1, Prop 3.3], that
reduces matters to the Hermite operator with $m=-d$, that is

$L=-{\Delta}+|x|^{2}-d,\quad\mbox{in $L_{2}({\mathbb{R}}^{d})$}.$ (4.1)

Indeed, if we set ${\tilde{f}}(x)=e^{-\frac{|x|^{2}}{2}}f(x)$ then it is
easily seen that $\mathcal{O}f(x)=e^{\frac{|x|^{2}}{2}}[L{\tilde{f}}](x)$.
Thus,

$e^{-t\mathcal{O}}f(x)\,=\,e^{\frac{|x|^{2}}{2}}\,e^{-tL}{\tilde{f}}(x){\quad\mbox{and}\quad}P_{t}^{\sigma,\mathcal{O}}f(x)\,=\,e^{\frac{|x|^{2}}{2}}\,P_{t}^{\sigma,L}{\tilde{f}}(x),$

so that the convergence properties of $\mathcal{O}$ and $L$ in (4.1) are
linked by the mapping $f\mapsto{\tilde{f}}$. Indeed, just observe that, if

$\Phi^{\mathcal{O}}_{\sigma}(y)=\frac{e^{-|y|^{2}}}{[\ln(e+|y|)]^{\sigma}},$

then

  1. (i)

$f\in L_{1}(\Phi^{\mathcal{O}}_{\sigma})$ iff ${\tilde{f}}\in
L_{1}(\Phi^{L}_{\sigma})$

  2. (ii)

$\mathcal{O}^{\sigma}(f)(x)=e^{|x|^{2}/2}\,L^{\sigma}({\tilde{f}})(x)$, as
defined in (2.9)

  3. (iii)

$\displaystyle\lim_{t\to
0^{+}}t^{1-2\sigma}\,\partial_{t}\big{[}P^{\sigma,\mathcal{O}}_{t}f(x)\big{]}=e^{\frac{|x|^{2}}{2}}\,\lim_{t\to
0^{+}}t^{1-2\sigma}\,\partial_{t}\big{[}P^{\sigma,L}_{t}{\tilde{f}}(x)\big{]}.$

Since we also have $f\in\mathcal{D}^{2\sigma}_{\rm st}(x)$ iff
${\tilde{f}}\in\mathcal{D}^{2\sigma}_{\rm st}(x)$, then Theorem 1.2 is an
immediate consequence of Theorem 1.1.

$\Box$

##  5\. Results for other operators $L$

One can ask whether Theorem 1.1 continues to hold for other positive self-
adjoint operators $L$. If one aims at _optimal_ integrability conditions on
$f$, the first step would be to find a suitable function
$\Phi^{L}_{\sigma}(y)$ such that

$c_{1}(t,x)\,\Phi^{L}_{\sigma}(y)\,\leq\,p^{\sigma,L}_{t}(x,y)\,\leq\,c_{2}(t,x)\,\Phi^{L}_{\sigma}(y)\;,\quad\forall\;y\in{\mathbb{R}}^{d},$
(5.1)

as stated in Lemma 2.1. Such optimal estimates, for certain families of
operators $L$, have already been investigated by the authors (and their
collaborators) in earlier papers. For instance, besides the already mentioned
reference [8] for Hermite type operators, we have also considered a large
class of _Laguerre type operators_ $L$ in [7], while the Bessel operators (in
the case $\sigma=1/2$) were treated by I. Cardoso in [3].

In this paper we have tried to state our results in sufficient generality, so
that one part of the arguments can be applied to general operators $L$ (such
as in §2), while the other parts concern with specific estimates of the
kernels $h_{t}(x,y)$ associated with $L$ (such as in §3.1). We remark that,
following this line of reasoning, one can derive versions of Theorem 1.1 when
$L$ is any the aforementioned Laguerre or Bessel operators; we expect to take
up these matters in [4].

In this section we illustrate this fact in just one specific but particularly
simple case. For a fixed†††In the sequel we shall not track the dependence of
the constants on $R>0$. parameter $R>0$, we consider the perturbed Laplacian

${\mathbb{L}}=-{\Delta}+R.$

In this case, $e^{-t{\mathbb{L}}}$ has a well-known convolution kernel

$h_{t}(x,y)=e^{-tR}\,W_{t}(x-y),\quad\mbox{where $W_{t}(x)=(4\pi
t)^{-d/2}\,e^{-\frac{|x|^{2}}{4t}}$}.$ (5.2)

It was also proved in [8, §5.2] that (5.1) does hold with

$\Phi^{{\mathbb{L}}}_{\sigma}(y):=\frac{e^{-\sqrt{R(1+|y|^{2})}}}{(1+|y|)^{\frac{d+1}{2}+\sigma}}.$
(5.3)

We now define a similar kernel as in (3.2) (this time letting $A=\infty$)

$\mathcal{K}(x,y):=\int_{0}^{\infty}h_{t}(x,y)\,\frac{dt}{t^{1+\sigma}}.$
(5.4)

######  Lemma 5.1.

With the notation in (5.2), (5.3) and (5.4), if $x\in{{\mathbb{R}}^{d}}$,
there exists $c(x)>0$ such that

$\mathcal{K}(x,y)\leq\,c(x)\,\Phi^{\mathbb{L}}_{\sigma}(y),\quad\mbox{for all
$|y|\geq 2\max\\{|x|,1\\}$.}$

###### Proof.

Note from (5.2) and (5.4) that

$\mathcal{K}(x,y)=(4\pi)^{-d/2}\,\int_{0}^{\infty}e^{-tR}e^{-\frac{|x-y|^{2}}{4t}}\,\frac{dt}{t^{1+\sigma+\frac{d}{2}}}.$
(5.5)

For $\nu>0$, consider the special function

$F_{\nu}(z):=\int_{0}^{\infty}e^{-s}e^{-\frac{z^{2}}{4s}}\,s^{\nu-1}\,ds\,\lesssim(1+z)^{\nu-\frac{1}{2}}\,e^{-z},\quad
z>0,$

where the inequality follows from the asymptotics of the integral; see e.g.
[10, p. 136]. If we change variables $s=|x-y|^{2}/(4t)$ in (5.5) we obtain
that

$\mathcal{K}(x,y)=\,c\,\frac{F_{\sigma+d/2}\big{(}\sqrt{R}\,|x-y|\big{)}}{|x-y|^{2\sigma+d}}\,\lesssim\,\frac{\big{(}1+\sqrt{R}\,|x-y|\big{)}^{\sigma+\frac{d-1}{2}}}{|x-y|^{2\sigma+d}}\,e^{-\sqrt{R}\,|x-y|}.$

If we now assume that $|y|\geq 2\max\\{|x|,1\\}$, the right side is easily
seen to be controlled by $c(x)\,\Phi^{\mathbb{L}}_{\sigma}(y)$; see e.g. [8,
(5.6)] and subsequent lines for a detailed argument. ∎

We can now state the corresponding theorem for the operator
${\mathbb{L}}=-{\Delta}+R$.

######  Theorem 5.2.

Let ${\mathbb{L}}=-{\Delta}+R$ with $R>0$ fixed. Let $\sigma\in(0,1)$ and
$\Phi^{\mathbb{L}}_{\sigma}(y)$ be as in (5.3). Suppose that

$f\in
L_{1}(\Phi^{\mathbb{L}}_{\sigma}){\quad\mbox{and}\quad}f\in\mathcal{D}^{2\sigma}(x_{0})\quad\mbox{at
some $x_{0}\in{{\mathbb{R}}^{d}}$}.$

Then $(-{\Delta}+R)^{\sigma}(x_{0})$ is well defined in the limiting sense of
(2.10), and as the absolutely convergent integral in (2.9), and both
definitions agree.

###### Proof.

By Proposition 2.3 we must show that
$x=x_{0}\in\mathcal{A}_{f}({\mathbb{L}}^{\sigma})$. Observe that
$\psi(y)\equiv 1$ is a regular positive eigenvector for ${\mathbb{L}}$,
according to Definition 3.6. So, by Lemma 3.7 and the inequalities following
(3.3) (applied with $A=\infty$), if suffices to show that

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\Big{|}\big{(}e^{-t{\mathbb{L}}}f\big{)}(x)-f(x)\big{(}e^{-t{\mathbb{L}}}\psi\big{)}(x)\Big{|}\,\frac{dt}{t^{1+\sigma}}$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\Big{|}\int_{{{\mathbb{R}}^{d}}}h_{t}(x,y)\big{[}f(y)-f(x)\big{]}\,dy\Big{|}\,\frac{dt}{t^{1+\sigma}}<\infty.$

We let ${\delta}=3\max\\{|x|,1\\}$, and as before, split the inner integral
into the regions $\\{|y-x|<{\delta}\\}$ and $\\{|y-x|\geq{\delta}\\}$. So,
using the notation for $\mathcal{I}_{\delta}f(x,t)$ in (2.12) we have

$J_{1}\,\leq\,\int_{0}^{\infty}\big{|}\mathcal{I}_{\delta}f(x,t)\big{|}\,\frac{dt}{t^{1+\sigma}}\,+\,J_{1}^{*}$

where

$\displaystyle J_{1}^{*}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\int_{|y-x|\geq{\delta}}h_{t}(x,y)\big{|}f(y)-f(x)\big{|}\,dy\,\tfrac{dt}{t^{1+\sigma}}$
$\displaystyle\leq$
$\displaystyle\int_{|y|\geq\max\\{|x|,1\\}}\,\big{|}f(y)-f(x)\big{|}\,\mathcal{K}(x,y)\,dy,$

with $\mathcal{K}(x,y)$ as in (5.4). By Lemma 5.1,

$J^{*}_{1}\,\leq\,c(x)\,\int_{{{\mathbb{R}}^{d}}}\big{(}|f(x)|+|f(y)|\big{)}\,\Phi^{\mathbb{L}}_{\sigma}(y)\,dy,$

which is a finite expression. So we have reduced matters to show that

$\int_{0}^{\infty}\big{|}\mathcal{I}_{\delta}f(x,t)\big{|}\,\frac{dt}{t^{1+\sigma}}<\infty.$

But under the smoothness assumption that $f\in\mathcal{D}^{2\sigma}(x)$, this
is a consequence of Proposition 2.6 (setting $A=\infty$), since the kernel
$h_{t}(x,y)$ trivially satisfies (2.14) (by Remark 3.2) and (2.15) (whose left
hand side is identically 0; see Remark 2.8). Finally observe, also by Remark
2.8, that only the weaker smoothness condition $f\in\mathcal{D}^{2\sigma}(x)$
is used, due to the convolution structure of the kernel $h_{t}(x,y)$. ∎

##  6\. Smoothness conditions

In this section we give some examples to illustrate the smoothness conditions
from §1. Recall that, for ${\alpha}\in(0,2)$, a locally integrable function
$f\in\mathcal{D}^{{\alpha}}(x_{0})$ if

$\int_{|h|\leq{\delta}}\frac{|f(x_{0}+h)+f(x_{0}-h)-2f(x_{0})|}{|h|^{d+{\alpha}}}\,dh\,<\,\infty,$

for some ${\delta}>0$ (hence for all ${\delta}>0$). Also,
$f\in\mathcal{D}_{\rm st}^{{\alpha}}(x_{0})$ if

$f\in\mathcal{D}^{{\alpha}}(x_{0}){\quad\mbox{and}\quad}\int_{|h|\leq{\delta}}\frac{|f(x_{0}+h)-f(x_{0}-h)|}{|h|^{d+{\alpha}-3}}\,dh\,<\,\infty.$

Observe that if $f$ is bounded near $x_{0}$, this last condition is redundant,
so ${\alpha}$-smooth and strictly ${\alpha}$-smooth agree in this case. The
two classes also coincide if $d+{\alpha}-3\leq 0$, that is

$\mathcal{D}^{{\alpha}}(x_{0})=\mathcal{D}_{\rm
st}^{{\alpha}}(x_{0}),\quad\mbox{if $d=1$, or $d=2$ and ${\alpha}\in(0,1]$.}$
(6.1)

In other cases, the classes are different, as shown by the example in (6.4)
below. Finally, note that strict ${\alpha}$-smoothness can also be
characterized as follows.

######  Lemma 6.1.

Let ${\alpha}\in(0,2)$. Then $f\in\mathcal{D}_{\rm st}^{{\alpha}}(x_{0})$ if
and only if

$f\in\mathcal{D}^{{\alpha}}(x_{0}){\quad\mbox{and}\quad}\int_{|h|\leq{\delta}}\frac{|f(x_{0}+h)-f(x_{0})|}{|h|^{d+{\alpha}-3}}\,dh\,<\,\infty.$
(6.2)

###### Proof.

The implication “$\Leftarrow$” is obvious since

$|f(x_{0}+h)-f(x_{0}-h)|\leq|f(x_{0}+h)-f(x_{0})|+|f(x_{0}-h)-f(x_{0})|.$

For the converse implication “$\Rightarrow$” note that

$2\big{(}f(x_{0}+h)-f(x_{0})\big{)}=\Big{[}f(x_{0}+h)-f(x_{0}-h)\Big{]}+\Big{[}f(x_{0}+h)+f(x_{0}-h)-2f(x_{0})\Big{]}.$

∎

We next collect a few further elementary observations.

  1. (1)

If $f$ is odd about $x_{0}$ and $f(x_{0})=0$, then
$f\in\mathcal{D}^{{\alpha}}(x_{0})$, for all ${\alpha}\in(0,2)$. For instance,
if ${\gamma}\in[0,d)$ then

$f(x)=\mathop{\rm sign\,}(x\cdot\mathbf{e}_{1})/|x|^{\gamma}\quad\mbox{if
$x\not=0$},\quad f(0)=0,$ (6.3)

belongs to $\mathcal{D}^{{\alpha}}(0)$ for all $0<{\alpha}<2$, even though it
is discontinuous there. However, by (6.2), $f\in\mathcal{D}_{\rm
st}^{{\alpha}}(0)$ only if additionally ${\alpha}\in(0,3-{\gamma})$. In
particular, the function

$g(x)=\frac{\mathop{\rm
sign\,}((x-x_{0})\cdot\mathbf{e}_{1})}{|x-x_{0}|^{3-{\alpha}}}\quad\mbox{if
$x\not=x_{0}$},\quad g(x_{0})=0,$ (6.4)

which is locally integrable if $d+{\alpha}-3>0$, shows that $\mathcal{D}_{\rm
st}^{{\alpha}}(x_{0})\subsetneq\mathcal{D}^{{\alpha}}(x_{0})$ in the
complementary range of (6.1).

  2. (2)

There exists a function $f\in\mathcal{D}_{\rm st}^{{\alpha}}(x_{0})$, for all
${\alpha}\in(0,2)$, but which is discontinuous and unbounded at $x_{0}$.
Indeed, if $d\geq 2$ (and $x_{0}=0$), the function $f$ defined in (6.3) with
parameter ${\gamma}=1$ has this property. If $d=1$, one may take any (locally
integrable) odd function unbounded near $x_{0}$.

  3. (3)

If $f\in{\rm Lip}_{\beta}(x_{0})$ for some $\beta\in(0,2]$ then
$f\in\mathcal{D}_{\rm st}^{{\alpha}}(x_{0})$ for all ${\alpha}<\beta$. Here,
$f\in{\rm Lip}_{\beta}(x_{0})$, if $\beta\in(0,1]$, means that

$|f(x_{0}+h)-f(x_{0})|\leq c\,|h|^{\beta},\quad\forall\,|h|\leq{\delta},$

for some $c,{\delta}>0$. If $\beta\in(1,2]$, it means that $f$ is
differentiable at $x_{0}$ and

$\big{|}f(x_{0}+h)-f(x_{0})-\nabla f(x_{0})\cdot h\big{|}\leq
c\,|h|^{\beta},\quad\forall\,|h|\leq{\delta}.$

Indeed, in either case it is clear that $f\in{\rm Lip}_{\beta}(x_{0})$ implies

$|\triangle_{h}^{2}f(x_{0})|=\big{|}f(x_{0}+h)-2f(x_{0})+f(x_{0}-h)\big{|}\,\leq\,2c\,|h|^{\beta},$

and

$|\triangle_{h}^{1}f(x_{0})|=\big{|}f(x_{0}+h)-f(x_{0}-h)\big{|}\,\leq\,c^{\prime}\,|h|^{\min\\{\beta,1\\}},$

which in turn implies $f\in\mathcal{D}_{\rm st}^{{\alpha}}(x_{0})$, for all
${\alpha}<\beta$.

  4. (4)

The following examples relate ${\rm Lip}_{\beta}(x_{0})$ and
$\mathcal{D}^{{\alpha}}(x_{0})$ when $\beta={\alpha}$:

$\displaystyle f(x)=|x-x_{0}|^{\alpha}\in{\rm
Lip}_{\alpha}(x_{0})\setminus\mathcal{D}^{{\alpha}}(x_{0}),\quad\forall\,{\alpha}\in(0,2)$
$\displaystyle g(x)=\mathop{\rm
sign\,}\big{(}(x-x_{0})\cdot\mathbf{e}_{1}\big{)}\,|x-x_{0}|^{\alpha}\in{\rm
Lip}_{\alpha}(x_{0})\cap\mathcal{D}_{\rm
st}^{{\alpha}}(x_{0}),\quad\forall\;{\alpha}\in(0,2).$

  5. (5)

We mention an example relating the above smoothness conditions at a point
$x_{0}$ and the existence of $(-{\Delta})^{\frac{{\alpha}}{2}}f(x_{0})$, as
defined in (1.3). Consider the two functions $f,g$, defined as in point (4)
above but additionally multiplied by a smooth cut-off ${\varphi}(|x-x_{0}|)$,
where ${\varphi}\in C^{\infty}_{c}({\mathbb{R}})$ with ${\varphi}\equiv 1$ in
$[-1,1]$. Then, it is easily seen that

$(-{\Delta})^{\frac{{\alpha}}{2}}f(x_{0})=-\infty\quad\mbox{but}\quad(-{\Delta})^{\frac{{\alpha}}{2}}g(x_{0})=0.$

So in general, $f\in{\rm Lip}_{{\alpha}}(x_{0})$ is not enough to define
pointwise fractional powers, $L^{\frac{{\alpha}}{2}}f(x_{0})$, justifying the
search for a different condition such as $f\in\mathcal{D}^{{\alpha}}(x_{0})$
or $f\in\mathcal{D}_{\rm st}^{{\alpha}}(x_{0})$.

  6. (6)

When $L$ is the Hermite operator in (1.7), one can show that the function
$g(x)$ defined in (6.4), when additionally multiplied by a smooth cut-off
${\varphi}(|x-x_{0}|)$ supported near the point $x_{0}=\mathbf{e}_{1}$, has
the property

$L^{\frac{{\alpha}}{2}}g(x_{0})=\infty.$

Thus, when $x_{0}\not=0$, in Theorem 1.1 one cannot replace the assumption
$f\in\mathcal{D}_{\rm st}^{2\sigma}(x_{0})$ by the weaker condition
$f\in\mathcal{D}^{2\sigma}(x_{0})$ (except of course when the two classes
coincide; see (6.1)). This example also shows that there are compactly
supported $g$ such that

$(-{\Delta})^{\frac{{\alpha}}{2}}g(x_{0})=0\quad\mbox{but}\quad
L^{\frac{{\alpha}}{2}}g(x_{0})=\infty$

(the latter in the sense of Definition 2.2).

### Acknowledgements

We wish to thank the referee for useful comments which have led to an improved
version of this paper.

## References

  * [1] I. Abu-Falahah, J.L. Torrea, Hermite function expansions versus Hermite polynomial expansions. Glasg. Math. J. 48, (2006), 203–215.
  * [2] L. Caffarelli and L. Silvestre, An Extension Problem Related to the Fractional Laplacian. Comm. Partial Differential Equations 32 (8) (2007), 1245–1260.
  * [3] I. Cardoso, On the pointwise convergence to initial data of heat and Poisson problems for the Bessel operator, Jour. Evol. Eq. 17 (2017), 953–977.
  * [4] G. Flores, G. Garrigós, T. Signes, B. Viviani, Pointwise convergence of fractional powers of Laguerre and Bessel operators. Work in progress.
  * [5] G. Flores, B. Viviani, A Calderón theorem for the Poisson semigroups associated with the Ornstein–Uhlenbeck and Hermite operators. Math. Annalen, published online (2022), https://doi.org/10.1007/s00208-022-02399-4.
  * [6] G. Garrigós, A weak 2-weight problem for the Poisson-Hermite semigroup. In _Advanced Courses in Mathematical Analysis VI_ , pp 153–171, eds Martín-Reyes _et al._ World Scientific 2017. Available at webs.um.es/gustavo.garrigos
  * [7] G. Garrigós, S. Hartzstein, T. Signes, B. Viviani, A.e. convergence and 2-weight inequalities for Poisson-Laguerre semigroups. Ann. Matem. Pura Apppl. 196 (2017), 1927–1960.
  * [8] G. Garrigós, S. Hartzstein, T. Signes, J.L. Torrea, B. Viviani, Pointwise convergence to initial data of heat and Laplace equations. Trans. Amer. Math. Soc. 368 (9) (2016), 6575–6600.
  * [9] M. Kwaśnicki, Ten equivalent definitions of the fractional Laplace operator. Fract. Calc. Appl. Anal. 20 (1) (2017), 7–51.
  * [10] N. Lebedev, _Special functions and their applications_. Prentice-Hall Inc., 1965.
  * [11] E. Lieb, M. Loss, _Analysis_ , 2nd ed. Graduate Studies in Math. 14, Amer Math Soc, 2001.
  * [12] L. Roncal and P. Stinga, Fractional Laplacian on the torus. Commun. Contemp. Math. 18 (2016), (3), 1550033, 26 pp.
  * [13] W. Rudin, _Functional Analysis_ , 2nd ed. McGraw-Hill, 1991.
  * [14] P. Stinga, J.L. Torrea, Extension problem and Harnack’s inequality for some fractional operators. Comm. Partial Differential Equations 35 (11) (2010), 2092–2122.
  * [15] P. Stinga, J.L. Torrea, Regularity theory for the fractional harmonic oscillator. Jour. Funct. Anal. 260 (2011), 3097–3131.
  * [16] S. Thangavelu. _Lecture notes on Hermite and Laguerre expansions_. Princeton Univ. Press. Princeton, New Jersey 1993.
  * [17] K. Yosida, _Functional Analysis_ , 6th ed, Springer, Berlin, 1980.


# Reconstructing common latent input from time series with the mapper-coach
network and error backpropagation

Zsigmond Benkő
Dept. of Computational Sciences
Wigner Research Centre for Physics
Budapest, Hungary
<EMAIL_ADDRESS>&Zoltán Somogyvári
Dept. of Computational Sciences
Wigner Research Centre for Physics
Budapest, Hungary
<EMAIL_ADDRESS>
###### Abstract

A two-module, feedforward neural network architecture called mapper-coach
network has been introduced to reconstruct an unobserved, continuous latent
variable input, driving two observed dynamical systems. The method has been
demonstrated on time series generated by two chaotic logistic maps driven by a
hidden third one. The network has been trained to predict one of the observed
time series based on its own past and on the other observed time series by
error-back propagation. It was shown, that after this prediction have been
learned successfully, the activity of the bottleneck neuron, connecting the
mapper and the coach module, correlates strongly with the latent common input
variable. The method has the potential to reveal hidden components of
dynamical systems, where experimental intervention is not possible.

_K_ eywords dynamical systems, artificial neural networks, latent variables,
common cause, intrinsic dimension

##  1 Introduction & Background

Theory of nonlinear dynamical systems has plenty to offer for neural
architecture design. Here we take a dualistic approach, incorporating the
principles of nonlinear time series techniques and the application of simple
feedforward learning algorithms to achieve the reconstruction an unobserved
common input of two observed chaotic dynamical systems.

A dynamical system has a state and an update rule. It can have a discrete time
dynamics or it can be a continuous time flow. In the former, the current state
($s_{t}$) is updated according to the update rule ($f$) thus gives rise to the
future states of the system.

$s_{t+1}=f(s_{t})$ (1)

Here the state is a $p$-dimensional contnuous variable.

If we have an observation ($g$) of the system state producing a time series
($x_{t}$), we can still reconstruct the state up to a continuous
transformation.

$x_{t}=g(s_{t})$ (2)

Takens theorem [1] ensures that this topological equivalence has high
probability, and shows that time delay embedding – shown in (3) – is such a
reconstruction procedure.

$X_{t}=[x_{t},x_{t-1},\quad\ldots\quad,x_{t-(2p-1)}]$ (3)

Where $X_{t}$ is the reconstructed state at time $t$.

In the case of coupled systems, one can reconstruct the cause from the effect
[2, 3, 4], because the directed causal connection leaves the fingerprint of
the former on the latter. So the full coupled system’s state can be
reconstructed or decoded form the observation of the effect.

When two subsystem share a common input, then the dynamics of the latent
variable is encoded in both systems and appears as some kind of redundancy.
This can be leveraged to identify the existence common input between
subsystems [5, 6]. Hirata et al. applied the technique of recurrence maps to
identify the existence of the hidden common cause. Benko et al. measured the
intrinsic dimensionality of reconstructed statespaces, thus identified the
existence of the latent common cause by determining the number of the
redundant degrees of freedom between the two reconstructed dynamics.

There are only a few methods to reconstruct hidden common input from
multivariate time series observations and the use of these is restricted to
specific types of hidden common inputs [7, 8]. Sauer demonstrated, that
reconstruction of common input from time series is possible if the hidden
variable is discrete[7]. Wiskott and Sejnowski introduced slow feature
analysis, which can reconstruct slow latent common input[9].

However, no method existed yet, which can reconstruct a continuous hidden
common input with general dynamics. Here we introduce a new method to
reconstruct continuous hidden common input from time series by a feedforward
neural network.

Figure 1: Directed graph representation of the causal connections in a coupled
logistic map system. The $z$ variable (red) is a hidden common cause of the
two observed variables, $x$ and $y$ (blue).

##  2 Example: Coupled logistic maps

We use logistic maps as simple example systems, to simulate coupled dynamical
systems with hidden common inputs and to generate the observed data series for
the analysis, where the hidden common input is known. The logistic map is a
discrete time dynamical system exhibiting chaotic behaviour if the parameter
of the system is properly selected [10].

Let’s take three logistic maps, which are connected in a way that $z$ is a
common cause of $x$ and $y$ (Fig. 1):

$\begin{split}f:\quad x_{t}&=rx_{t-1}(1-x_{t-1}-\beta_{xz}z_{t-1})\\\ g:\quad
y_{t}&=ry_{t-1}(1-y_{t-1}-\beta_{yz}z_{t-1})\\\ h:\quad
z_{t}&=rz_{t-1}(1-z_{t-1})\end{split}$ (4)

where $x,y$ and $z$ are the state variables, $r$ is the parameter of the
logistic map and $\beta_{ij}$ are the coupling parameters. Furthermore, $x$
and $y$ are observed, however $z$ is hidden.

In the followings, we show an actual mapping that reconstructs the hidden
common cause from one of the observed time series. We know from the theory [2]
that we can represent the $(z_{t},y_{t})$ state of the subsystem solely based
on the observed variable $y$. We can obtain such representation by time delay
embedding:

$Y_{t}=[y_{t},y_{t-1}]$ (5)

where $y_{t}$ is the scalar value of $y$ at time $t$, and $Y_{t}$ is a 2D
vector representing the reconstructed state of the system according to the
Takens’ theorem [1].

To see that $Y_{t}$ truly described the state of the system, we show how one
can express $z$ from $Y$. First, we reorder the equation, which realizes the
$g$ mapping in (4) thus we get the expression for $z_{t-1}$:

$\phi:\quad z_{t-1}=\frac{1-y_{t-1}-\frac{y_{t}}{ry_{t-1}}}{\beta_{yz}}$ (6)

and second, we compose $\phi$ with $h$ to get $z_{t}$:

$z_{t}=(h\circ\phi)(y_{t},y_{t-1})$ (7)

thus we have a mapping from $Y$ to $z$. By this, we demonstrated that there
exists a one-to-one mapping between $Y_{t}$ and $(z_{t},y_{t})$, thus $Y$ is a
state space reconstruction of the subsystem.

On the other side, to predict the value of $x_{t}$, one needs the previous
value of $x_{t-1}$ and $z_{t-1}$ according to the update rule $f$ in (4).
Since $Y$ contains all information about $z$, we can use $Y_{t}$ instead of
$z_{t-1}$, and predict $x_{t}$ given $x_{t-1}$, $Y_{t}$ and the mappings.

When blindly analyzing the observed $x_{t}$ and $y_{t}$ time series from this
coupled system, the reconstruction and update mappings are unknown, and one
has to estimate them from data. For this purpose, it is practical to use
feedforward neural networks as these are general function approximators.

Figure 2: The architecture of the mapper-coach network. The feedforward neural
network predicts present values of $x$. One cannot predict accurately $x(t)$
values based on solely $x(t-1)$, because the lack of information about
$z(t-1)$. This missing information is sipped out from $Y(t)$ through the red
bottleneck At the training phase the full mapper-coach network can be trained
with backpropagation algorithm and – after successful training – the activity
of the red hidden unit has a one-to-one correspondence with values of the
latent $z$ variable. At this reconstruction stage, the coach can be detached
from the mapper, which latter estimates the mapping ($\phi$) from the
reconstructed state space of the effect ($Y$) to the state space of the hidden
variable ($z$).

##  3 Results

###  3.1 Proposed architecture: the mapper-coach network

We reconstruct the hidden common input for a dataset generated from the
logistic map system in (4). To estimate the hidden $z$ input we had to
approximate the $\phi$ mapping, we implemented this function by a feedforward
neural network: the mapper (Fig. 2). The input space is the reconstructed
state space of $Y$ (composed by two successive values of y) and the intended
output is the value of $z$.

But how to find the exact mapping from $Y$ to $z$, when $z$ is unobserved? To
asses this challenge, we attached a coach network (coach) to the output of the
mapper. The coach takes input from the mapper and also directly from $x$, the
other variable influenced by $z$. The output of the coach is a prediction on
the current value of $x$, therefore the subnetwork intends to implement the
$f$ function from (4).

The fusion of the two parts is the mapper-coach network, which predicts $x(t)$
values based on the observation of $x(t-1)$, $y(t-1)$ and $y(t)$.

Figure 3: Learning and prediction performance of the network. A Learning
curves. The learning-process of twenty mapper-coach networks is shown on the
left. 7 instances could not learn, 6 instances are lost in local minima and 7
instances converged to smaller mean squared loss values. B Prediction
performance of the best model on the test set. The correlogram shows the
actual values against the predicted ones. The point-cloud nicely lies around
the diagonal, the coefficient of determination is very high ($r^{2}=0.99$).

###  3.2 Performance on the logistic map system

According to the previous arguments, we trained the mapper-coach network with
the backpropagation algorithm to reconstruct $x$ at the output of the coach
module, and found that the output of the mapper network was equivalent to the
hidden variable.

The network produced very accurate predictions on the present values of $x$
after the learning procedure ($r^{2}=0.99$, Fig. 3). We split the data into
training and test sets ($n=10000$ each), trained the network on the former and
evaluated the results on the latter. We reinitialized the training twenty
times with random weights, and selected the best-performing model for
evaluation. According to the learning characteristics, we identified three
clusters of the models. For seven case out of the twenty, the learning was not
successful (cluster 1), for further six cases the system were stuck in local
minima (cluster 2) and in the third type the learning was successful (cluster
3, 7/20 cases).

We found that the mapper part of the network reconstructed the past values of
the hidden common input quiet accurately: the correlation between the mapper
output and the actual hidden common input was $r^{2}=0.97$, (Fig 4). We
detached the coach and investigated the output of the mapper by providing
$Y(t)$ test inputs ($n=10000$). The output reflected the values of $z(t-1)$,
so the mapper network approximately implemented the $\phi$ mapping in (6).

Figure 4: The output and performance of the mapper in reconstruction-mode on
the test set. A Traces of original unobserved time series (black) and
reconstructed values (red). After reversing and rescaling the amplitudes, the
values are almost identical. B Reconstruction performance on the test set. The
subplot shows the original values against the reconstructed ones. The points
lie around the negative slope diagonal, thus reconstruction has high quality
($r^{2}=0.97$). The negative slope shows, that the reconstructed space is
inverted compared with the original $z$ space.

We showed that the prediction- and the reconstruction performance were
related: the better the model predicted $x$ the better it reconstructed $z$
(Fig. 5). The trained network instances achieved different prediction
performance on the test set, $13$ of the $20$ networks had estimable output.
We plotted the prediction performance against the reconstruction performance
for each model. We saw that better quality predictions implied better quality
reconstructions (Fig. 5).

Figure 5: Prediction performance against the quality of reconstruction. The
figure shows the coefficient of variation metrics for different model-
instances from the learning procedure ($n=13$). Reconstruction quality of $z$
is correlated with prediction performance on $x$.

##  4 Discussion

However we demonstrated, that reconstruction of a continuous latent variable
was possible with the backpropagation algorithm, several challenges remain to
be assessed.

We reconstructed the latent common input from example time series, leveraging
the theoretical properties of the underlying coupled chaotic logistic maps. We
implemented the method by the mapper-coach network, where the mapper part
realized the mapping $z(t-1)=\phi(Y)$ and the coach implemented
$x(t)=f\left(x(t-1),z(t-1)\right)$.

The architecture of the network is asymmetric: the role of the $Y$ and $x$
input is not equivalent. The reconstructed space of $Y$ contains all
information about the hidden variable $z$, but the $x(t-1)$ value lacks this
information. The vital point is that the coach sips out the required
information about $z$ from the $Y$ input through the mapper, and this
information is only available in the $Y$, not in the $x$ input pathway.

To ensure that the mapper outputs information about the hidden variable, the
dimensionality of the coach-input has to be minimal. In more complex systems,
it could be more difficult to obtain this minimality. The number of input
units from the mapper should equal to dimensionality of the common hidden
dynamics. Also, the number of direct input units (from $x$ in the example) has
to equal to the remaining degrees of freedom in $X$ which is independent from
$Y$. Therefore, to determine the proper number of units for mapper output and
$x$-input, one has to investigate the intrinsic dimensionality and the mutual
dimension between the embedded $X$, $Y$ variables[11, 12, 6].

We demonstrated, that the proposed architecture works for a discrete time,
deterministic noise-free dynamical system, but whether this approach is
applicable to continuous flows and how robust is to noise is yet to be
investigated. Theoretically there are no obstacles for the former and – since
state-space reconstruction can be robust up to moderate noise levels[13] – the
latter can be handled in some cases.

Finally, we analyzed the example from a dynamical systems aspect, and clearly,
there are connections e.g. to autoencoders, representation learning and hidden
Markov models which await to be worked out in details.

##  5 Methods

###  5.1 Data handling

We generated $N=20000$ data points according to (4) and split the set: the
first half became training and second half become test data ($r=3.99$,
$\beta=0.2$). We embedded the $y$ variable, thus produced
$Y(t)=[y(t),y(t-1)]$, and aligned the time-scale of $x(t)$ and $z(t)$ time
series according to the time-shift produced by the embedding procedure.

###  5.2 Model implementation

We implemented the mapper-coach network in python[14] with the Keras[15]
framework using the Tensorflow[16] backend. For the sake of simplicity we
chose the number of hidden units to be the same for each hidden layers
($n=20$).

###  5.3 Training and evaluation

We randomly initialized the weights and started training with $2000$ batch
size and for $4000$ epochs. We used the ADAM optimizer and mean squared loss
function. We repeated this procedure twenty times and selected the best model
for evaluation on the test set.

We used the coefficient of determination ($r^{2}$) to evaluate model test
performance. We measured the performance linearly, so the coefficient is the
square of the Pearson correlation coefficient ($r$):

$r_{xy}=\sqrt{\frac{\sum_{i=1}^{N}(x_{i}-\mu_{x})(y_{i}-\mu_{y})}{\sigma_{x}\sigma_{y}}}$
(8)

where $x_{i}$ and $y_{i}$ are the sample points, $\mu_{j}$ are the sample
means and the $\sigma_{j}$-s are the sample standard deviations. This
coefficient describes the linear dependence between the two variables.

We computed $r^{2}$ between $x(t)$ and the predicted values $\hat{x}(t)$, and
also between $z(t-1)$ and the the output of the mapper network
($\hat{z}(t-1)$). We displayed these results on correlograms and on a time
series plot (Fig. 3–4).

We also computed the reconstructed $z(t-1)$ values and the $r^{2}$ values for
the not optimal models, and plotted the prediction performance against the
reconstruction performance of the models (Fig. 5).

## References

  * [1] Floris Takens  “Detecting strange attractors in turbulence”  In _Dynamical Systems and Turbulence, Warwick 1980_ 898, 1981, pp. 366–381  DOI: 10.1007/BFb0091924
  * [2] J Stark  “Delay embeddings for foreced systems. I. Deterministic forcing”  In _Journal of Nonlinear Science_ 9, 1999, pp. 255–332  DOI: 10.1007/s003329900072
  * [3] J. Stark, D. S. Broomhead, M. E. Davies and J. Huke  “Delay embeddings for forced systems. II. stochastic forcing”  In _Journal of Nonlinear Science_ 13.6, 2003, pp. 519–577  DOI: 10.1007/s00332-003-0534-4
  * [4] George Sugihara et al.  “Detecting Causality in Complex Ecosystems”  In _Science_ 338.6106, 2012, pp. 496–500  DOI: 10.1126/science.1227079
  * [5] Yoshito Hirata and Kazuyuki Aihara  “Identifying hidden common causes from bivariate time series: A method using recurrence plots”  In _Physical Review E_ 81.1, 2010, pp. 016203  DOI: 10.1103/PhysRevE.81.016203
  * [6] Zsigmond Benkő et al.  “Complete Inference of Causal Relations between Dynamical Systems”, 2018, pp. 1–44  arXiv: http://arxiv.org/abs/1808.10806
  * [7] Timothy D. Sauer  “Reconstruction of Shared nonlinear dynamics in a network”  In _Physical Review Letters_ 93.19, 2004, pp. 1–4  DOI: 10.1103/PhysRevLett.93.198701
  * [8] Laurenz Wiskott and Terrence J. Sejnowski  “Slow feature analysis: Unsupervised learning of invariances”  In _Neural Computation_ 14.4, 2002, pp. 715–770  DOI: 10.1162/089976602317318938
  * [9] Laurenz Wiskott  “Estimating Driving Forces of Nonstationary Time Series with Slow Feature Analysis”, 2003, pp. 1–8  DOI: 10.1145/947121.947143
  * [10] Robert M. May  “Simple mathematical models with very complicated dynamics”  In _Nature_ 261.5560 CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742: CRC Press, 1976, pp. 459–467  DOI: 10.1038/261459a0
  * [11] Zsigmond Benkő et al.  “Manifold-adaptive dimension estimation revisited”, 2020, pp. 1–21  arXiv: http://arxiv.org/abs/2008.03221
  * [12] Simone Romano et al.  “Measuring dependency via intrinsic dimensionality”  In _2016 23rd International Conference on Pattern Recognition (ICPR)_ IEEE, 2016, pp. 1207–1212  DOI: 10.1109/ICPR.2016.7899801
  * [13] Martin Casdagli, Stephen Eubank, J.Doyne Farmer and John Gibson  “State space reconstruction in the presence of noise”  In _Physica D: Nonlinear Phenomena_ 51.1-3, 1991, pp. 52–98  DOI: 10.1016/0167-2789(91)90222-U
  * [14] Guido Van Rossum and Fred L. Drake  “Python 3 Reference Manual”  Scotts Valley, CA: CreateSpace, 2009
  * [15] Francois Chollet  “Keras”  GitHub, 2015  URL: https://github.com/fchollet/keras
  * [16] Martín Abadi et al.  “Tensorflow: A system for large-scale machine learning”  In _12th $\\{$USENIX$\\}$ Symposium on Operating Systems Design and Implementation ($\\{$OSDI$\\}$ 16)_, 2016, pp. 265–283