# Approximation and Interpolation in Kolmogorov-type groups

Antonello Pesce Dipartimento di Matematica, Università di Bologna, Piazza di
Porta s. Donato 5, 40126 Bologna, Italy. e-mail<EMAIL_ADDRESS>
###### Abstract

We prove a real interpolation characterization for some non Euclidean Hölder
spaces, built on the Lie structure induced by a class of ultra-parabolic
Kolmogorov-type operators satisfying the Hörmander condition. As a by-product
we also obtain an approximation property for intrinsically regular functions
on the whole space.

Keywords: Interpolation, Approximation, Kolmogorov operators, hypoelliptic
operators, Hörmander condition

MSC: 35D99, 35K65, 35H10, 46M35.

##  1 Introduction

###  1.1 Statement of the problem

We consider the problems of approximation and interpolation for some non
Euclidean Hölder spaces built on the intrinsic geometry of the following
Kolmogorov operator

$\mathcal{K}=\frac{1}{2}\sum_{i=1}^{p_{0}}\partial_{x_{i}x_{i}}+Y,\quad
Y=\langle
Bx,\nabla\rangle+\partial_{t},\qquad(t,x)\in\mathbb{R}\times\mathbb{R}^{d},$
(1.1)

where $1\leqslant p_{0}\leqslant d$ and $B$ is a $d\times d$ constant matrix.
If $p_{0}=d$ then $\mathcal{K}$ is a parabolic operator while in general, for
$p_{0}<d$, $\mathcal{K}$ is degenerate, and not uniformly parabolic. However,
$\mathcal{K}$ can be a hypoelliptic operator, provided some structural
assumptions on the matrix $B$ are verified. This kind of PDEs appears in many
linear and non-linear models in physics (see, for instance, [4], [8]) and in
mathematical finance (see, for instance, [2], [7]).

In [15], Lanconelli and Polidoro first studied the non-Euclidean intrinsic Lie
group structure induced by the vector fields
$\partial_{x_{1}},\dots,\partial_{x_{p_{0}}},Y$, and from there on, many
authors carried the study of general Kolmogorov operators in the framework of
the theory of homogeneous groups. We recall, among many, [9], who proved the
existence of a fundamental solution under optimal regularity assumptions,
[10], [12] and [21], who proved Schauder type estimates for the associated
Cauchy and Dirichlet problems, [13], who considered the case of integro-
differential diffusions. One basic tool in the study of regularity properties
of solutions to Kolmogorov operators with variable coefficients is the notion
of Hölder spaces which are intrinsic to the aforementioned geometric
structure. Over the years, different notions of Hölder classes have been
proposed by several authors, based on the Carnot-Carathéodory distance
associated to the group (see, among many, [10], [21], [11]). Eventually
Pagliarani et al. in [23] established an optimal definition of
$C_{B}^{n,\alpha}$ for any $n\in\mathbb{N}_{0}$, which is given in terms of
the regularity in the directions of the vector fields
$\partial_{x_{1}},\dots,\partial_{x_{p_{0}}}$ and $Y$ with their different
formal degrees: for instance, the case $n=0$ consists of functions that are
$C^{\alpha}$ continuous in the directions
$\partial_{x_{1}},\dots,\partial_{x_{p_{0}}}$ and $C^{\frac{\alpha}{2}}$ in
$Y$. With this definition they managed to prove a Taylor-type formula with
related estimates of the remainder, given in terms of the intrinsic distance.

Our main result, Theorem 1.6, shows that these intrinsic spaces interpolate
well with each other through the so called real method, namely that

$\left(C_{B}^{n_{1},\alpha_{1}},C_{B}^{n_{2},\alpha_{2}}\right)_{\theta,\infty}=C_{B}^{n,\alpha},$
(1.2)

for any $n_{1},n_{2}\in\mathbb{N}_{0}$, $\alpha_{1},\alpha_{2}\in[0,1]$, for
some expected parameters $n\in\mathbb{N}_{0}$, $\alpha\in(0,1)$. In the
Euclidean case, this identification is well known (see [20], [27]), and can be
proved by means of the general theory of semigroups. Similar arguments can be
adapted for a different class of Hölder spaces also associated to degenerate
elliptic and parabolic differential equations, the so called anisotropic
Hölder spaces, which, unlike the intrinsic spaces, only refer to the spatial
variables (see Section 1.3 for a more in-depth discussion). Here we choose to
pursue a constructive approach based on explicit approximations of
intrinsically regular functions (Theorem 3.1), which can possibly be of
independent interest. This approach is more or less inspired by the
approximation theory for Sobolev functions (see [1], Chapter 5), it does not
rely on the theory of semigroups, but instead, it heavily exploits the
aforementioned intrinsic Taylor formula.

Classically, characterizations of type (1.2) have remarkable applications in
the study of optimal regularity in Hölder classes for linear elliptic and
parabolic differential operators (see [14], [20], [18], [19]) as well as
approximation theory (see [3]). In the context of Kolmogorov operators, many
works already exploit some kind of interpolation inequalities to obtain
different versions of Schauder estimates in different settings (see, for
instance [12], [10] and [22] among others, see also [17] for the most recent
global estimate for the evolution problem, as well as a complete overview of
the existing literature). Let us also mention that our results, together with
the embeddings of intrinsic Sobolev spaces of integer order in [25] could be
applied to deduce Morrey-type embeddings for suitable notions of Sobolev-
Slobodeckij and Besov intrinsic spaces as well, which, as far as our
understanding, are yet to be explored in the literature in generality. Such
studies will be subjects of future research. For the application of
interpolation results to fractional embeddings in the Euclidean setting, see
for instance, the references [1] or [26].

The paper is organized as follows: in the remaining part of this Section, we
specify the precise functional framework, state the structural hypothesis on
the matrix $B$ and our main interpolation results, Proposition 1.5 and Theorem
1.6, and finally discuss the previous literature on the subject and justify
our approach. In Section 2 we collect some results that are preliminary to the
subsequent proofs; in Section 3 we prove the core result of this work, the
approximation property for intrinsically regular functions, which eventually
leads to the proof of our main results in Section 4.

###  1.2 Hölder spaces and real interpolation

We start by recalling the Lie group structure that is naturally associated to
the differential operator (1.1). In [15] the authors observed that
$\mathcal{K}$ is invariant with respect to the left translations of the group
$(\mathbb{R}^{d+1},\circ)$, where the non-commutative group law is defined by

$(t,x)\circ(s,{\xi})=\ell_{(t,x)}(s,{\xi}):=(s+t,{\xi}+e^{sB}x),\qquad(t,x),\
(s,{\xi})\in\mathbb{R}^{d+1}.$ (1.3)

Precisely, $\ell_{(t,x)}\circ\mathcal{K}=\mathcal{K}\circ\ell_{(t,x)}$ for any
$(t,x)\in\mathbb{R}^{d+1}$. Notice that $(\mathbb{R}^{d+1},\circ)$ is a group
with identity $\text{Id}=(0,0)$ and inverse $(t,x)^{-1}=(-t,e^{-tB}x)$.

The authors also proved that the operator $\mathcal{K}$ is hypoelliptic if and
only if, on a certain basis of $\mathbb{R}^{d+1}$ the matrix $B$ takes the
form

$B=\begin{pmatrix}\ast&\ast&\cdots&\ast&\ast\\\ {\bf
B}_{1}&\ast&\cdots&\ast&\ast\\\ 0_{p_{2}\times p_{0}}&{\bf
B}_{2}&\cdots&\ast&\ast\\\ \vdots&\vdots&\ddots&\vdots&\vdots\\\
0_{p_{r}\times p_{0}}&0_{p_{r}\times p_{1}}&\cdots&{\bf
B}_{r}&\ast\end{pmatrix},$ (1.4)

where $\mathbf{B}_{j}$ is a $p_{j}\times p_{j-1}$ matrix of rank $p_{j}$, with

$p_{0}\geqslant p_{1}\geqslant\dots\geqslant p_{r}\geqslant
1,\quad\sum_{j=1}^{r}p_{j}=d,$

$0_{p_{i}\times p_{j}}$ is a $p_{i}\times p_{j}$ null matrix and the
$\ast$-blocks are arbitrary. Throughout the paper we assume the following

######  Assumption 1.1.

$B$ is a constant $d\times d$ matrix as in (1.4), where each $\ast$-block is
null.

If (and only if) Assumption 1.1 holds, then $\mathcal{K}$ is homogeneus of
degree two with respect to the family of dilations $(D_{\lambda})_{\lambda>0}$
on $\mathbb{R}^{d+1}$ given by

$D_{\lambda}=(\lambda^{2},\lambda
I_{p_{0}},\lambda^{3}I_{p_{1}}\dots,\lambda^{2r+1}I_{p_{r}}),$

where $I_{p_{j}}$ are $p_{j}\times p_{j}$ identity matrices, i.e.
$(\mathcal{K}D_{\lambda}u)(t,x)=\lambda^{2}(\mathcal{K}u)(D_{\lambda}(t,x))$,
for any $(t,x)\in\mathbb{R}^{d+1}$, $\lambda>0$. In this case, the matrix $B$
uniquely identifies the homogeneous Lie group
${\mathcal{G}}_{B}:=(\mathbb{R}^{d+1},\circ,D(\lambda))$. We define the
$D(\lambda)$-homogeneous quasi-norm on ${\mathcal{G}}_{B}$ as follows: let

$\bar{p}_{j}=p_{0}+\dots+p_{j},\qquad j=0,\dots,r,$ (1.5)

and $\bar{p}_{-1}\equiv 0$. Then

$\|(t,x)\|_{B}:=|t|^{\frac{1}{2}}+|x|_{B},\qquad|x|_{B}:=\sum_{j=0}^{r}\sum_{i=\bar{p}_{j-1}+1}^{\bar{p}_{j}}|x_{i}|^{\frac{1}{2j+1}}.$
(1.6)

Next we recall the notions of $B$-intrinsic regularity and $B$-intrinsic
Hölder space introduced in [23]. We start with some useful notations: for any
$(t,x)\in\mathbb{R}^{d+1}$, $i=1,\dots,p_{0}$ we denote

$e^{\delta\partial_{x_{i}}}(t,x)=(t,x+\delta e_{i}),\quad{e^{\delta
Y}(t,x)=(t+\delta,e^{\delta B}x)},\qquad\delta\in\mathbb{R},$ (1.7)

the (unique) integral curves of the vector fields $\partial_{x_{i}}$ and $Y$
respectively.

######  Definition 1.2.

Let $X\in\\{\partial_{x_{1}},\dots,\partial_{x_{p_{0}}},Y\\}$ and
$\alpha\in]0,1]$. We say that $u\in C^{\alpha}_{X}$ if

$[u]_{C^{\alpha}_{X}}:=\sup_{z\in{\mathbb{R}^{d+1}}\atop\delta\in\mathbb{R}\setminus\\{0\\}}\frac{\left|u(e^{\delta
X}z)-u(z)\right|}{|\delta|^{{\alpha}}}<\infty.$ (1.8)

We say that $u$ is $X$-differentiable in $z$ if the function $\delta\mapsto
u(e^{\delta X}z)$ is differentiable in $0$ and refer to the function $z\mapsto
Xu(z):=\frac{d}{d\delta}u(e^{\delta X}z)\mid_{\delta=0}$ as the $X$-Lie
derivative of u.

######  Definition 1.3.

Let $\alpha\in{[0,1]}$, then:

  * i)

If $\alpha=0$, $C^{0,0}_{B}\equiv C(\mathbb{R}^{d+1})$ is the space of bounded
and continuous function on $\mathbb{R}^{d+1}$, endowed with the sup-norm

$\|u\|_{C^{0,0}_{B}}:=|u|_{\infty}.$

If $\alpha\in]0,1]$ we say that $u\in C^{0,\alpha}_{B}$ if $u\in
C^{\frac{\alpha}{2}}_{Y}$ and $u\in C^{\alpha}_{\partial_{x_{i}}}$ for any
$i=1,\dots,p_{0}$, with norm

$\displaystyle\|u\|_{C^{0,\alpha}_{B}}$
$\displaystyle:=|u|_{\infty}+[u]_{C^{\frac{\alpha}{2}}_{Y}}+\sum_{i=1}^{p_{0}}[u]_{C^{\alpha}_{\partial_{x_{i}}}}<\infty,\quad\alpha\in]0,1].$

  * ii)

$u\in C^{1,\alpha}_{B}$ if $u\in C^{\frac{1+\alpha}{2}}_{Y}$ and
$\partial_{x_{i}}u\in C^{\alpha}_{B}$ for any $i=1,\dots,p_{0}$, with norm

$\|u\|_{C^{1,\alpha}_{B}}:=|u|_{\infty}+[u]_{C^{\frac{1+\alpha}{2}}_{Y}}+\sum_{i=1}^{p_{0}}\|\partial_{x_{i}}u\|_{C^{0,\alpha}_{B}}<\infty.$

  * iii)

for $n\in\mathbb{N}$ with $n\geqslant 2$, $u\in C^{n,\alpha}_{B}$ if $Yu\in
C^{n-2,\alpha}_{B}$ and $\partial_{x_{i}}u\in C^{n-1,\alpha}_{B}$ for any
$i=1,\dots,p_{0}$, with norm

$\|u\|_{C^{n}_{B}}:=|u|_{\infty}+\|Yu\|_{C^{n-2,\alpha}_{B}}+\sum_{i=1}^{p_{0}}\|\partial_{x_{i}}u\|_{C^{n-1,\alpha}_{B}}<\infty.$

######  Remark 1.4.

$C^{n,\alpha}_{B}$ is a Banach space. Notice also that
$C^{n,\alpha}_{B}\subseteq C^{n^{\prime},\alpha^{\prime}}_{B}$ for $0\leqslant
n\leqslant n^{\prime}$ and $0\leqslant\alpha^{\prime}\leqslant\alpha\leqslant
1$.

For any two real Banach spaces $Z_{1},Z_{2}$, by $Z_{1}=Z_{2}$ we mean that
$Z_{1}$ and $Z_{2}$ have the same elements with equivalent norms and by
$Z_{1}\subseteq Z_{2}$ we mean that $Z_{1}$ is continuously embedded in
$Z_{2}$. We recall that the couple $\\{Z_{1},Z_{2}\\}$ is called interpolation
couple if both $Z_{1}$ and $Z_{2}$ are continuously embedded in some Hausdorff
topological vector space $\mathcal{Z}$; in this case, the intersection
$Z_{1}\cap Z_{2}$ and the sum $Z_{1}+Z_{2}=\\{u_{1}+u_{2},\ u_{1}\in Z_{1},\
u_{2}\in Z_{2}\\}$, endowed with the norms

$\|u\|_{Z_{1}\cap
Z_{2}}:=\max\\{\|u\|_{Z_{1}},\|u\|_{Z_{1}}\\},\quad\|u\|_{Z_{1}+Z_{2}}:=\inf\\{\|u_{1}\|_{Z_{1}}+\|u_{2}\|_{Z_{1}},u_{1}\in
Z_{1},\ u_{2}\in Z_{2}\\},$

are Banach spaces. Moreover, any Banach space $E$ such that

$Z_{1}\cap Z_{2}\subseteq E\subseteq Z_{1}+Z_{2},$ (1.9)

is called an intermediate space. Among these, we have the real interpolation
space $(Z_{1},Z_{2})_{\theta,\infty}$, which can be defined, for any
$\theta\in(0,1)$ (see, for instance [20]), as

$(Z_{1},Z_{2})_{\theta,\infty}:=\\{u\in Z_{1}+Z_{2},\
\|u\|_{(Z_{1},Z_{2})_{\theta,\infty}}:=\sup_{\lambda\geqslant
0}\lambda^{-\theta}K(\lambda,u;Z_{1},Z_{2})<\infty\\}$ (1.10)

where

$K(\lambda,u;Z_{1},Z_{2}):=\inf\left\\{\|u_{1}\|_{Z_{1}}+\lambda\|u_{2}\|_{Z_{2}},\,u=u_{1}+u_{2},\,u_{1}\in
Z_{1},\,u_{2}\in Z_{2}\right\\}.$

Our first results states that any space $C^{n,0}_{B}$ belongs to a special
class of intermediate spaces between the larger space $C^{n_{1},0}_{B}$,
$n_{1}<n$ and a smaller space $C^{n_{2},0}_{B}$, $n<n_{2}$, with
$n,n_{1},n_{2}\in\mathbb{N}_{0}$.

######  Proposition 1.5.

Let $n,n_{1},n_{2}\in\mathbb{N}_{0}$, $n_{1}<n<n_{2}$.

  * i)

There exists a constant $c_{1}$ such that, for any $u\in C^{n_{2},0}_{B}$, we
have

$\|u\|_{C^{n,0}_{B}}\leqslant
c_{1}\|u\|^{\frac{n_{2}-n}{n_{2}-n_{1}}}_{C^{n_{1},0}_{B}}\|u\|^{\frac{n-n_{1}}{n_{2}-n_{1}}}_{C^{n_{2},0}_{B}};$
(1.11)

  * i)

There exists a constant $c_{2}$ such that, for any $u\in C^{n,0}_{B}$, we have

$K(\lambda,u;C^{n_{1},0}_{B},C^{n_{2},0}_{B})\leqslant
c_{2}\lambda^{\frac{n-n_{1}}{n_{2}-n_{1}}}\|u\|_{C^{n,0}_{B}},\quad\lambda>0.$
(1.12)

Equivalently (see, for instance [20], Proposition 1.20) it holds that

$\left(C^{n_{1},0}_{B},C^{n_{2},0}_{B}\right)_{\frac{n-n_{1}}{n_{2}-n_{1}},1}\subseteq
C^{n,0}_{B}\subseteq\left(C^{n_{1},0}_{B},C^{n_{2},0}_{B}\right)_{\frac{n-n_{1}}{n_{2}-n_{1}},\infty}.$
(1.13)

Next, our main result, gives a characterization of $B$-Hölder spaces of any
order as interpolation spaces.

######  Theorem 1.6.

Let $n_{1},n_{2}\in\mathbb{N}_{0}$ and $\alpha_{1},\alpha_{2}\in[0,1]$, with
$n_{1}+\alpha_{1}\leqslant n_{2}+\alpha_{2}$, and let $\theta\in(0,1)$. Let
$n\in\mathbb{N}_{0}$, $n_{1}\leqslant n\leqslant n_{2}$ such that

$\frac{n-(n_{1}+\alpha_{1})}{(n_{2}+\alpha_{2})-(n_{1}+\alpha_{1})}<\theta<\frac{n+1-(n_{1}+\alpha_{1})}{(n_{2}+\alpha_{2})-(n_{1}+\alpha_{1})}.$
(1.14)

If
$\alpha:=(n_{1}+\alpha_{1})+\theta[n_{2}+\alpha_{2}-(n_{1}+\alpha_{1})]-n\neq
0,1$ then it holds that

$\left(C^{n_{1},\alpha_{1}}_{B},C^{n_{2},\alpha_{2}}_{B}\right)_{\theta,\infty}=C^{n,\alpha}_{B}$
(1.15)

######  Remark 1.7.

If $\Omega$ is an open set of $\mathbb{R}^{d+1}$, one can define the Hölder
spaces $C^{n,\alpha}_{B}(\Omega)$, $n\in\mathbb{N}_{0}$, $\alpha\in(0,1)$ as
in the case $\Omega=\mathbb{R}^{d+1}$, where in particular, for
$X\in\\{\partial_{x_{1}},\dots\partial_{x_{p_{0}}},Y\\}$,

$[u]_{C^{\alpha}_{X}(\Omega)}:=\sup\left\\{\frac{|u(e^{\delta
X}z)-u(z)|}{|\delta|^{\alpha}},\ z\in\Omega,\
\delta\in\mathbb{R}\setminus\\{0\\}\text{ s.t.
}e^{\delta^{\prime}X}z\in\Omega,\ |\delta^{\prime}|<|\delta|\right\\}.$

Then it is likely that our results also hold for the spaces
$C^{n,\alpha}_{B}(\Omega)$, provided that the domain $\Omega$ is smooth enough
(see [27], Theorem 1.2.4 and Section 4.5). The construction of the appropriate
extension operators under optimal regularity assumptions for $\Omega$ is
outside the scope of the present paper and is postponed to future research.

######  Remark 1.8.

Notice that the assumption $n_{1}+\alpha_{1}\leqslant n_{2}+\alpha_{2}$ is not
restrictive. Indeed we can always write

$\left(C^{n_{1},\alpha_{1}}_{B},C^{n_{2},\alpha_{2}}_{B}\right)_{\theta,\infty}=\left(C^{n_{2},\alpha_{2}}_{B},C^{n_{1},\alpha_{1}}_{B}\right)_{1-\theta,\infty}.$

###  1.3 Related works and strategy of the proof

As already mentioned, in the Euclidean case, the interpolation identity (1.2)
can be proved by means of the theory of semigroups ([20], [27]): let $C,C^{1}$
denote the spaces of bounded continuous functions and bounded continuously
differentiable functions with bounded derivatives respectively, and let
$C^{\theta}$ denote the space of bounded and uniformly $\theta$-Hölder
continuous functions, all endowed with the usual norms. Roughly speaking,
denoting with $T_{i}$ the translation semigroup of the $i$-th variable, and
${\mathcal{D}}(\partial_{x_{i}})=\\{f\in C,\ \exists\partial_{x_{i}}f\in
C\\}$, we have

$(C,C^{1})_{\theta,p}=\Big{(}C,\bigcap_{i=1}^{d}{\mathcal{D}}(\partial_{x_{i}})\Big{)}_{\theta,p}=\bigcap_{i=1}^{d}\Big{(}C,{\mathcal{D}}(\partial_{x_{i}})\Big{)}_{\theta,p},$
(1.16)

for any $p\geqslant 1$, where

$\big{(}C,{\mathcal{D}}(\partial_{x_{i}})\big{)}_{\theta,\infty}=\\{f\in C,\
t\mapsto\phi(t):=t^{-\theta}\|T_{i}(t)f-f\|_{\infty}\in
L^{\infty}(0,\infty)\\},$ (1.17)

that is the space of $\theta$-Hölder continuous functions in the $i$-th
variable. Therefore $(C,C^{1})_{\theta,\infty}=C^{\theta}$. In particular
(1.17) holds more generally for $p\geqslant 1$, and substituting $T_{i}$ and
$\partial_{x_{i}}$ with any bounded semigroup $T$ and its infinitesimal
generator on a Banach space, while the second equality in (1.16) precisely
holds because the semigroups $T_{i}$, $i=1,\dots,d$ commute.

In [19], Lunardi adapted these arguments to derive an interpolation result for
anisotropic Hölder spaces associated with operators of the type (1.1), which
are spaces of functions on $\mathbb{R}^{d}$ (only the spatial variables are
considered) defined with respect to the metric induced by $|\cdot|_{B}$ (cfr.
(1.6)). For instance, let us consider the degenerate prototype of (1.1), that
is the Langevin operator

$\mathcal{K}=\frac{1}{2}\partial_{x_{1}x_{1}}+Y,\quad
Y=x_{1}\partial_{x_{2}}+\partial_{t},$

which corresponds to the matrix

$B=\begin{pmatrix}0&0\\\ 1&0\end{pmatrix}.$

In this case the anisotropic Hölder regularity in the space variables reads:

$|u(x)-u({\xi})|\leqslant
C|x-{\xi}|^{\alpha}_{B}\sim|x_{1}-{\xi}_{1}|^{\alpha}+|x_{2}-{\xi}_{2}|^{\frac{\alpha}{3}},\quad
x,{\xi}\in\mathbb{R}^{2}.$ (1.18)

A regularity theory for equations with variable coefficients, built on this
functional setting, is useful when one needs to consider coefficients or data
that are irregular in time (see also [5], [24], [6], [16] for recent
developments). This approach leads to mild or weak/distributional solutions
also because it cannot benefit from any regularizing effect of the semigroup
in time.

On the other hand, for $B$ as above, a function $u\in C^{0,\alpha}_{B}$
satisfies (see Theorem 2.4 in the next Section), for any
$(t,x),(s,{\xi})\in\mathbb{R}^{3}$,

$|u(t,x)-u(s,{\xi})|\leqslant
C\|(s,{\xi})^{-1}\circ(t,x)\|^{\alpha}_{B}\sim|t-s|^{\frac{\alpha}{2}}+|x_{1}-{\xi}_{1}|^{\alpha}+|x_{2}-{\xi}_{2}-(t-s){\xi}_{1}|^{\frac{\alpha}{3}}.$

Notice that for points that are on the same time level, the increments are
controlled as for (1.18); otherwise, as opposed to the standard parabolic
case, the regularity in the euclidean directions is somehow entangled due to
the group law associated to $B$. This functional setting, which exploits some
regularity in the full set of variables, is more suited to obtain classical
solutions (see [10], [21] and the references therein). To the best of our
knowledge this is the first attempt at obtaining a general interpolation
characterization for the spaces $C^{n,\alpha}_{B}$. For this purpose, notice
that while it is true that

$\big{(}C,{\mathcal{D}}(Y)\big{)}_{\theta,\infty}=\\{f\in C,\
t\mapsto\phi(t):=t^{-\theta}\|e^{tY}f-f\|_{\infty}\in
L^{\infty}(0,\infty)\\}=C_{Y}^{\theta},$

since the vector fields $\partial_{x_{1}}$ and $Y$ don’t commute, it’s not at
all clear whether

$\big{(}C,{\mathcal{D}}(\partial^{2}_{x_{1}})\cap{\mathcal{D}}(Y)\big{)}_{\frac{\theta}{2},\infty}=\left(C,{\mathcal{D}}(\partial^{2}_{x_{1}})\right)_{\frac{\theta}{2},\infty}\cap\left(C,{\mathcal{D}}(Y)\right)_{\frac{\theta}{2},\infty}.$
(1.19)

This makes it difficult to pursue an approach based on semigroup theory.
Instead of trying to prove (1.19) we follow a constructive approach based on
approximation: for instance, for any $u\in C^{0,\alpha}_{B}$ we can define

$u_{\varepsilon}(z)=\int_{\mathbb{R}^{d+1}}u({\zeta})\phi\left(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)\right)\frac{d{\zeta}}{\varepsilon^{Q+2}},\quad z\in\mathbb{R}^{d+1},$

where $\phi$ is a test function with unit integral, supported on
$\|y\|_{B}\leqslant 1$. This approximation complies with the Lie structure
induced by $B$, and indeed we prove the following expected controls

$|u-u_{\varepsilon}|_{\infty}\leqslant
c\varepsilon^{\alpha}\|u\|_{C^{0,\alpha}_{B}},\quad\|u_{\varepsilon}\|_{C^{1,0}_{B}}\leqslant
c\varepsilon^{\alpha-1}\|u\|_{C^{0,\alpha}_{B}},$ (1.20)

so that

$\varepsilon^{-\alpha}K(\varepsilon,u;C,C^{1,0}_{B})\leqslant\varepsilon^{-\alpha}(|u-u_{\varepsilon}|_{\infty}+\varepsilon\|u_{\varepsilon}\|_{C^{1,0}_{B}})\leqslant
2c\|u\|_{C^{0,\alpha}_{B}}.$

This way of reasoning is fairly standard (see, for instance, Chapter 1 in
[20]), though the estimates (1.20) are new for the intrinsic spaces and we
present a generalization for any order $n\in\mathbb{N}_{0}$ in Theorem 3.1.

##  2 Preliminaries

Here we collect some remarks and known preliminary results which will play a
central role for the next Sections.

######  Remark 2.1.

Under Assumption 1.1 the matrix $B$ is a nilpotent matrix of degree $r+1$. In
particular

$e^{\delta B}=I_{d}+\sum_{j=1}^{r}\frac{B^{j}}{j!}\delta^{j},$

is a lower triangular matrix with diagonal $(1,\dots,1)$ and therefore it has
determinant equal to 1. Indeed, by Assumption 1.1, a direct computation shows
that, for any $n\leqslant r$ (recalling the notation (1.5) for $\bar{p}_{j}$)
we have

$B^{n}=\begin{pmatrix}0_{\bar{p}_{n-1}\times p_{0}}&0_{\bar{p}_{n-1}\times
p_{1}}&\cdots&0_{\bar{p}_{n-1}\times
p_{r-n}}&0_{\bar{p}_{n-1}\times(\bar{p}_{r}-\bar{p}_{r-n}})\\\
\prod_{j=1}^{n}{\bf B}_{j}&0_{p_{n}\times p_{1}}&\cdots&0_{p_{n}\times
p_{r-n}}&0_{p_{n}\times(\bar{p}_{r}-\bar{p}_{r-n})}\\\ 0_{p_{n+1}\times
p_{0}}&\prod_{j=2}^{n+1}{\bf B}_{j}&\cdots&0_{p_{n+1}\times
p_{r-n}}&0_{p_{n+1}\times(\bar{p}_{r}-\bar{p}_{r-n})}\\\
\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0_{p_{r}\times p_{0}}&0_{p_{r}\times
p_{1}}&\cdots&\prod_{j=r-n+1}^{r}{\bf
B}_{j}&0_{p_{r}\times(\bar{p}_{r}-\bar{p}_{r-n})}\end{pmatrix},\qquad\prod_{j=1}^{n}{\bf
B}_{j}={\bf B}_{n}{\bf B}_{n-1}\cdots{\bf B}_{1},$ (2.1)

and ${B}^{n}=0$ for $n>r$.

######  Remark 2.2.

A simple computation shows that, for any $z,y\in\mathbb{R}^{d+1}$,

$y^{-1}\circ e^{\delta Y}z=e^{\delta Y}(y^{-1}\circ z),\qquad y^{-1}\circ
e^{\delta\partial_{x_{i}}}z=e^{\delta\partial_{x_{i}}}(y^{-1}\circ z),\quad
i=1,\dots,d,$ (2.2)

which is a way to restate the invariance of operator (1.1) w.r.t the group
law.

######  Lemma 2.3.

The following identities hold:

$D_{\lambda}e^{\delta
Y}(z)=e^{\delta\lambda^{2}Y}\left(D_{\lambda}z\right),\qquad
D_{\lambda}e^{\delta\partial_{x_{i}}}(z)=e^{\delta\lambda^{2j+1}\partial_{x_{i}}}D_{\lambda}z,\quad
i=\bar{p}_{j-1}+1,\dots,\bar{p}_{j},$ (2.3)

for any $z\in\mathbb{R}^{d+1}$ and $\delta,\lambda\in\mathbb{R}$.

###### Proof.

Let $z=(t,x)$, and let $D^{0}_{\lambda}$ be the spatial part of the dilation
operator $D_{\lambda}$, namely $D^{0}_{\lambda}=(\lambda
I_{p_{0}},\lambda^{3}I_{p_{1}},\dots,\lambda^{2r+1}I_{p_{r}})$. Denoting with
$x_{\\{i\\}}$, $i=0,\dots,r$, the projection of $x$ on $\mathbb{R}^{p_{i}}$
defined by

$x_{\\{i\\}}:=(x_{\bar{p}_{i-1}+1},\dots,x_{\bar{p}_{i}}),$

by (2.1) we can write

$\left(e^{\delta
B}x\right)_{\\{i\\}}=x_{\\{i\\}}+\sum_{j=0}^{i-1}\delta^{i-j}\left(\prod_{k=j+1}^{i}{\bf
B}_{k}\right)x_{\\{j\\}},\qquad i=0,\dots,r.$ (2.4)

Then we have

$\displaystyle\left(D^{0}_{\lambda}e^{\delta B}x\right)_{\\{i\\}}$
$\displaystyle=\lambda^{2i-1}x_{\\{i\\}}+\lambda^{2i-1}\sum_{j=0}^{i-1}\delta^{i-j}\left(\prod_{k=j+1}^{i}{\bf
B}_{k}\right)x_{\\{j\\}}$
$\displaystyle=\lambda^{2i-1}x_{\\{i\\}}+\sum_{j=0}^{i-1}(\lambda^{2}\delta)^{i-j}\left(\prod_{k=j+1}^{i}{\bf
B}_{k}\right)\lambda^{2j-1}x_{\\{j\\}}=\left(e^{\delta\lambda^{2}B}\left(D^{0}_{\lambda}x\right)\right)_{\\{i\\}}.$

where in the last equality we used that
$\lambda^{2i-1}x_{\\{i\\}}=(D_{\lambda}x)_{\\{i\\}}$ and
$\lambda^{2j-1}x_{\\{j\\}}=(D_{\lambda}x)_{\\{j\\}}$. Thus we get

$D_{\lambda}e^{\delta
Y}z=\left(\lambda^{2}(t+\delta),e^{\delta\lambda^{2}B}\left(D^{0}_{\lambda}x\right)\right)=e^{\delta\lambda^{2}Y}\left(D(\lambda)z\right).$

The second equality is clear from the definitions. ∎

We conclude this Section by recalling the crucial intrinsic Taylor expansion
proved in [23]. Here $\beta=(\beta_{1},\dots,\beta_{d})\in\mathbb{N}^{d}_{0}$
denotes a multi-index, and as usual $\beta!:=\prod_{j=1}^{d}(\beta_{j}!)$.
Moreover, for any $x\in\mathbb{R}^{d}$,
$x^{\beta}=x_{1}^{\beta_{1}}\dots,x_{d}^{\beta_{d}}$ and
$\partial^{\beta}=\partial_{x}^{\beta}=\partial_{x_{1}}^{\beta_{1}}\cdots\partial_{x_{d}}^{\beta_{d}}$.
The $B$-length of $\beta$ is defined as

$|\beta|_{B}:=\sum_{i=0}^{r}(2i+1)\sum_{j=\bar{p}_{i-1}+1}^{\bar{p}_{i}}\beta_{j}.$

######  Theorem 2.4 (Intrinsic Taylor formula).

If $u\in C^{n,\alpha}_{B}$ then we have

$Y^{k}\partial^{\beta}_{x}u\in C_{B}^{n-2k-|\beta|_{B},\alpha},\qquad
0\leqslant 2k+|\beta|_{B}\leqslant n,$ (2.5)

and

$|u(z)-T_{n}u({\zeta},z)|\leqslant
c_{B}\|u\|_{C_{B}^{n,\alpha}}\|{\zeta}^{-1}\circ z\|^{n+\alpha}_{B},\qquad
z,{\zeta}\in\mathbb{R}^{d+1},$ (2.6)

where $c_{B}$ is a positive constant which only depend on $B$ and
$T_{n}u({\zeta},\cdot)$ is the $n$-th order $B$-Taylor polynomial of $u$
around ${\zeta}=(s,{\xi})$ defined as

$T_{n}u({\zeta},z)=\sum_{0\leqslant 2k+|\beta|_{B}\leqslant
n}\frac{1}{k!\beta!}\left(Y^{k}\partial^{\beta}_{\xi}u(s,{\xi})\right)(t-s)^{k}(x-e^{(t-s)B}{\xi})^{\beta},\qquad
z=(t,x)\in\mathbb{R}^{d+1}.$ (2.7)

Notation: Eventually, we will frequently use the notation $\lesssim$. For two
quantities $Q_{1}$ and $Q_{2}$, we mean by $Q_{1}\lesssim Q_{2}$ that there
exists a constant $c_{B}$, only dependent on the matrix $B$, such that
$Q_{1}\leqslant c_{B}Q_{2}$.

##  3 Approximation of $C^{n,\alpha}_{B}$ functions

This Section is mainly devoted to the proof of the following Theorem 3.1,
which establishes an approximation property for our $B$-Hölder continuous
functions.

######  Theorem 3.1 (Approximation property).

Let $n,l\in\mathbb{N}_{0}$ and $\alpha\in[0,1]$. There exists a constant
$c_{B}$, only dependent on $B$ and $d$, such that for any $u\in
C^{n,\alpha}_{B}$ and $0<\varepsilon\leqslant 1$, there exists
$u_{\varepsilon}\in C^{\infty}$ such that

$\displaystyle\|u-u_{\varepsilon}\|_{C^{l,0}_{B}}$ $\displaystyle\leqslant
c_{B}\varepsilon^{n+\alpha-l}\|u\|_{C^{n,\alpha}_{B}},$ $\displaystyle
l\leqslant n,$ (3.1) $\displaystyle\|u_{\varepsilon}\|_{C^{l,0}_{B}}$
$\displaystyle\leqslant
c_{B}\varepsilon^{n+\alpha-l}\|u\|_{C^{n,\alpha}_{B}}\quad$
$\displaystyle{l>n}.$ (3.2)

The proof is based on the intrinsic Hölder expansion of Theorem 2.4 and the
invariance properties of Lemma 2.3 and Remark 2.2. We also need the following
technical Lemma, which remarkably says that we can exchange the
differentiation of the intrinsic Taylor polynomial with the differentiation of
the inner function, taking into account the intrinsic degree of the
derivatives $\partial_{i}$, $i=1,\dots,p_{0}$ and $Y$.

To avoid ambiguity, when necessary, we will write $Y_{z}$, $Y_{\zeta}$ as well
as $\partial_{x_{i}}$, $\partial_{{\xi}_{i}}$ to clearly distinguish the
variables with respect to which the derivation is applied.

######  Lemma 3.2.

For any $u\in C^{n,\alpha}_{B}$ and
$z=(t,x),\,{\zeta}=(s,{\xi})\in\mathbb{R}^{d+1}$ we have

$\displaystyle\partial_{x_{i}}T_{n}u({\zeta},z)$
$\displaystyle=T_{n-1}(\partial_{i}u)({\zeta},z),\quad n\geqslant 1,\quad
i=1,\dots,p_{0},$ (3.3) $\displaystyle Y_{z}T_{n}u({\zeta},z)$
$\displaystyle=T_{n-2}(Yu)({\zeta},z),\quad n\geqslant 2.$ (3.4)

###### Proof.

Equality (3.3) is analogous to the Euclidean case and follows by a direct
computation. Indeed it suffices to notice that, for $i=1,\dots,p_{0}$,
$T_{n}u({\zeta},z)$ depends on $x_{i}$ only in the terms
$(x-e^{(t-s)B}{\zeta})_{i}$ and
$\partial_{x_{j}}(x-e^{(t-s)B}{\zeta})_{i}=\delta_{ij}$ ($\delta_{ij}$ being
the Kronecker Delta) for $i,j=1,\dots,p_{0}$. Equality (3.4) on the other
hand, needs a more careful analysis: the directional derivative $Y_{z}$ is not
null when applied both to the terms $(t-s)^{k}$ and
$(x-e^{(t-s)B}{\zeta})_{i}$ for any $i>p_{0}$ and, most importantly, some
commutators need to be introduced in order to exchange the order of derivation
between $\partial_{{\xi}_{i}}$ and $Y_{\zeta}$, and make $Y_{\zeta}u$ appear
in any term of the sum.

First we introduce some notations: let $v:=x-e^{(t-s)B}{\xi}\in\mathbb{R}^{d}$
and let us denote with $v^{[i]}$, $i=0,\dots,r$, the projection of $v$ on
$\\{0\\}^{\bar{p}_{i-1}}\times\mathbb{R}^{{p}_{i}}\times\\{0\\}^{d-\bar{p}_{i}}$;
then we denote with $X_{i}$ the vector field $X_{i}:=\langle
v^{[i]},\nabla\rangle$, were $\nabla$ represent the usual Nabla operator.
Clearly we have

$\displaystyle Y_{z}X_{0}$ $\displaystyle=0,$ (3.5)

Moreover we have

$\displaystyle Y_{z}X_{i}u({\zeta})$
$\displaystyle=\sum_{j=\bar{p}_{i-1}+1}^{\bar{p}_{i}}(Bv)_{j}\partial_{{\xi}_{j}}u({\zeta})=\langle(Bv)^{[i]},\nabla
u({\zeta})\rangle$ $\displaystyle=\langle Bv^{[i-1]},\nabla
u({\zeta})\rangle=\left[X_{i-1},Y_{{\zeta}}\right]u({\zeta})=:X^{(1)}_{i-1}u({\zeta}).$
(3.6)

To prove (3.4) we proceed by induction. For $n=2$ we directly get

$\displaystyle Y_{z}T_{2}u({\zeta},z)$
$\displaystyle=Y_{z}\left(u({\zeta})+X_{0}u({\zeta})+\frac{1}{2!}X_{0}^{2}u({\zeta})+Y_{\zeta}u({\zeta})(t-s)\right)=Y_{z}Y_{\zeta}u({\zeta})(t-s)=Y_{\zeta}u({\zeta}).$

Then we proceed in two steps, namely, the induction from $2n+1$ to $2n+2$ and
the induction from $2n$ to $2n+1$.
1: Induction from $2n+1$ to $2n+2$
Let $\beta=(\beta_{0},\dots,\beta_{r})\in\mathbb{N}_{0}^{r+1}$ and let us
denote just in the context of this proof

$|\beta|_{B}=\sum_{i=0}^{r}(2i+1)\beta_{i},\quad\beta!=\beta_{1}!\cdots\beta_{r}!.$

Then, with the notation introduced above we can conveniently write

$\displaystyle T_{2n+2}u({\zeta},z)$
$\displaystyle=T_{2n+1}u({\zeta},z)+\sum_{k=0}^{n+1}\sum_{|\beta|_{B}=2(n+1)-2k}\frac{1}{k!\beta!}Y_{\xi}^{k}X_{0}^{\beta_{0}}\cdots
X_{n-k}^{\beta_{n-k}}u({\zeta})(t-s)^{k}$ (3.7)
$\displaystyle=:T_{2n+1}u({\zeta},z)+\tilde{T}_{2n+1}u({\zeta},z).$ (3.8)

By (3.5) and (3.6) we have

$\displaystyle Y_{z}Y_{{\zeta}}^{k}(t-s)^{k}$
$\displaystyle=Y_{{\zeta}}^{k}(t-s)^{k-1},$ (3.9) $\displaystyle
Y_{z}X^{\beta_{i}}_{i}$
$\displaystyle=\beta_{i}X^{\beta_{i}-1}_{i}X^{(1)}_{i-1}$ (3.10)
$\displaystyle Y_{{\zeta}}X^{\beta_{i}}_{i}$
$\displaystyle=X^{\beta_{i}}_{i}Y_{\zeta}-\beta_{i}X^{\beta_{i}-1}_{i}X^{(1)}_{i}.$
(3.11)

Therefore, applying $Y_{z}$ to $\tilde{T}_{2n+1}u({\zeta},z)$ we get
$Y_{z}\tilde{T}_{2n+1}u({\zeta},z)=\sum_{k=0}^{n+1}S_{k}$ with

$\displaystyle S_{0}$
$\displaystyle=\sum_{|\beta|_{B}=2n+2}\frac{1}{\beta!}\sum_{i=0}^{n}{\beta_{i}}X_{0}^{\beta_{0}}\cdots
X^{\beta_{i}-1}_{i}\cdots X^{\beta_{n}}_{n}X^{(1)}_{i-1}u({\zeta})$
$\displaystyle=\sum_{|\beta|_{B}=2n+2}\sum_{i=0}^{n-1}\frac{1}{\beta_{0}!\cdots(b_{i+1}-1)!\cdots\beta_{n}!}X_{0}^{\beta_{0}}\cdots
X^{\beta_{i+1}-1}_{i+1}\cdots X^{\beta_{n}}_{n}X^{(1)}_{i}u({\zeta})$
$\displaystyle=\sum_{i=0}^{n-1}\sum_{|\beta|_{B}=2(n-i)-1}\frac{1}{\beta!}X_{0}^{\beta_{0}}\cdots
X^{\beta_{n-i-1}}_{n-i-1}X^{(1)}_{i}u({\zeta}),$

and, for $k=1,\dots,n+1$,

$\displaystyle S_{k}=$
$\displaystyle\sum_{|\beta|_{B}=2(n+1)-2k}\frac{(t-s)^{k-1}}{(k-1)!\beta!}Y_{{\zeta}}^{k-1}\left(X_{0}^{\beta_{0}}\cdots
X_{n-k}^{\beta_{n-k}}Y_{\zeta}-\sum_{i=0}^{n-k}\beta_{i}X_{0}^{\beta_{0}}\cdots
X_{i}^{\beta_{i}-1}\cdots X_{n-k}^{\beta_{n-k}}X^{(1)}_{i}\right)u({\zeta})$
$\displaystyle\quad+\sum_{|\beta|_{B}=2(n+1)-2k}\frac{(t-s)^{k}}{k!\beta!}Y_{\zeta}^{k}\sum_{i=0}^{n-k}\beta_{i}X_{0}^{\beta_{0}}\cdots
X^{\beta_{i}-1}_{i}\cdots X^{\beta_{n-k}}_{n-k}X^{(1)}_{i-1}u({\zeta})$
$\displaystyle=$ $\displaystyle
S_{k}^{(1)}-\sum_{i=0}^{n-k}\sum_{|\beta|_{B}=2(n-k-i)+1}\frac{1}{(k-1)!\beta!}Y_{{\zeta}}^{k-1}X_{0}^{\beta_{0}}\cdots
X^{\beta_{n-i-k}}_{n-i-k}X^{(1)}_{i}u({\zeta})(t-s)^{k-1}$
$\displaystyle\quad+\sum_{i=0}^{n-k-1}\sum_{|\beta|_{B}=2(n-k-i)-1}\frac{1}{k!\beta!}Y_{{\zeta}}^{k}X_{0}^{\beta_{0}}\cdots
X^{\beta_{n-i-k-1}}_{n-i-k-1}X^{(1)}_{i}u({\zeta})(t-s)^{k}$ $\displaystyle=$
$\displaystyle S_{k}^{(1)}+S_{k}^{(2)}+S_{k}^{(3)}.$

Now notice that

$S_{0}+S_{1}^{(2)}=0\qquad S_{k}^{(3)}+S_{k+1}^{(2)}=0,\quad k=1,\dots,n.$
(3.12)

Moreover $S_{n}^{(3)}=S_{n+1}^{(2)}=S_{n+1}^{(3)}=0$ by (3.9). Thus we finally
get

$\displaystyle Y_{z}\tilde{T}_{2n+2}u({\zeta},z)$
$\displaystyle=\sum_{k=0}^{n}S^{(1)}_{k+1}=\sum_{k=0}^{n}\sum_{|\beta|_{B}=2n-2k}\frac{1}{k!\beta!}Y_{{\zeta}}^{k}X_{0}^{\beta_{0}}\cdots
X_{n-k-1}^{\beta_{n-k-1}}Y_{\zeta}u({\zeta})(t-s)^{k}=\tilde{T}_{2n}Yu({\zeta},z),$

and therefore, using the induction hypotheses

$Y_{z}T_{2n+2}u({\zeta},z)=T_{2n-1}Yu({\zeta},z)+\tilde{T}_{2n}Yu({\zeta},z)=T_{2n}Yu({\zeta},z).$

2: Induction from $2n$ to $2n+1$
This Step only differs from the previous one by the tracking of the indexes
throughout the computations and the fact that the polynomial
$\tilde{T}_{2n+1}$ always contains at least one field of type $X_{i}$. We
write $Y_{z}\tilde{T}_{2n+1}u({\zeta},z)=\sum_{k=0}^{n}S_{k}$ and we derive,
with similar computations than in the previous step

$S_{0}=\sum_{i=0}^{n-1}\sum_{|\beta|_{B}=2(n-i)}\frac{1}{\beta!}X_{0}^{\beta_{0}}\cdots
X_{n-i-1}^{\beta_{n-i-1}}X^{(1)}_{i}u({\zeta}),$

and, for any $k=1,\dots,n-1$,

$\displaystyle S_{k}=$
$\displaystyle\sum_{|\beta|_{B}=2(n-k)+1}\frac{1}{(k-1)!\beta!}Y_{{\zeta}}^{k-1}X_{0}^{\beta_{0}}\cdots
X_{n-k}^{\beta_{n-k}}Y_{\zeta}u({\zeta})(t-s)^{k-1}$
$\displaystyle\quad-\sum_{i=0}^{n-k}\sum_{|\beta|_{B}=2(n-k-i)}\frac{1}{(k-1)!\beta!}Y_{{\zeta}}^{k-1}X_{0}^{\beta_{0}}\cdots
X^{\beta_{n-i-k}}_{n-i-k}X^{(1)}_{i}u({\zeta})(t-s)^{k-1}$
$\displaystyle\quad+\sum_{i=0}^{n-k-1}\sum_{|\beta|_{B}=2(n-k-i-1)}\frac{1}{k!\beta!}Y_{{\zeta}}^{k}X_{0}^{\beta_{0}}\cdots
X^{\beta_{n-i-k-1}}_{n-i-k-1}X^{(1)}_{i}u({\zeta})(t-s)^{k}$ $\displaystyle=$
$\displaystyle S_{k}^{(1)}+S_{k}^{(2)}+S_{k}^{(3)}.$

In particular, $S_{0}+S_{1}^{(2)}=0$, $S_{k}^{(3)}+S_{k+1}^{(2)}=0$ for any
$k=1,\dots,n-1$ and $S_{n}^{(3)}=0$ by (3.5). Therefore we can eventually
derive (3.4). ∎

We are now ready to prove Theorem 3.1.

###### Proof of Theorem 3.1.

Let $u\in C^{n,\alpha}_{B}$ and let $\phi$ be a test function supported on
$\|z\|_{B}\leqslant 1$ with unit integral. We define our candidate
approximation by

$\displaystyle
u^{(n)}_{\varepsilon}(z)=T_{n}u(\cdot,z)\star_{B}\phi_{\varepsilon}(z):=\int_{\mathbb{R}^{d+1}}T_{n}u({\zeta},z)\phi\left(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)\right)\frac{d{\zeta}}{\varepsilon^{Q+2}},$

where $T_{n}u({\zeta},z)$ is the Taylor polynomial in (2.7) and $Q$ is the
spatial homogeneous dimension of $\mathbb{R}^{d+1}$ w.r.t
$(D_{\lambda})_{\lambda>0}$, that is the positive integer

$Q=p_{0}+3p_{1}+\cdots(2r+1)p_{r}.$

Notice that, by the change of variables
$\bar{{\zeta}}=D_{\varepsilon^{-1}}({\zeta}^{-1}\circ z)$, recalling Remark
2.1, it is easy to verify that

$\int_{\mathbb{R}^{d+1}}\phi\left(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)\right)\frac{d{\zeta}}{\varepsilon^{Q+2}}=\int_{\|{\zeta}\|_{B}\leqslant
1}\phi({\zeta})d{\zeta}=1.$ (3.13)

In particular we have

$\displaystyle u(z)-u^{(n)}_{\varepsilon}(z)=$
$\displaystyle\int_{\mathbb{R}^{d+1}}\left(u(z)-T_{n}u({\zeta},z)\right)\phi\left(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)\right)\frac{d{\zeta}}{\varepsilon^{Q+2}}.$

Observe now that the differentiation of $u^{(n)}_{\varepsilon}$ falls back
both on the test function and the polynomial $T_{n}u$, the latter being
handled by Lemma 3.2; for the former case, by Lemma 2.3 and Remark 2.2 we
deduce

$\displaystyle Y_{z}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ z))$
$\displaystyle=\varepsilon^{-2}Y\phi(\bar{\zeta})\mid_{\bar{\zeta}=D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)},$
$\displaystyle\partial_{z_{j}}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ z))$
$\displaystyle=\varepsilon^{-(2i+1)}\partial_{j}\phi(\bar{\zeta})\mid_{\bar{\zeta}=D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)},\quad\bar{p}_{i-1}<j\leqslant\bar{p}_{i},$

and, more generally, for any $\beta\in\mathbb{N}_{0}^{d}$,

$Y^{k}_{z}\partial_{z}^{\beta}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z))=\varepsilon^{-2k-|\beta|_{B}}Y^{k}\partial^{\beta}\phi(\bar{z})\mid_{\bar{z}=D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)}.$ (3.14)

1: Preliminary controls
Let us denote a general intrinsic derivative of order $m$,
${\mathfrak{D}}^{m}=\sum_{2k+|\beta|_{B}=m}Y^{k}\partial^{\beta}$, and let

$I^{(n,m)}_{\varepsilon}u(z):=\int_{\mathbb{R}^{d+1}}\left(u(z)-T_{n}u({\zeta},z)\right){\mathfrak{D}}_{z}^{\beta}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z))\frac{d{\zeta}}{\varepsilon^{Q+2}}.$

We want to prove that

$\displaystyle|I^{(n,m)}_{\varepsilon}u|_{\infty}$
$\displaystyle\lesssim\varepsilon^{n+\alpha-m}\|u\|_{C^{n,\alpha}_{B}},$
(3.15) $\displaystyle[I^{(n,m)}_{\varepsilon}u]_{C^{\frac{1}{2}}_{Y}}$
$\displaystyle\lesssim\varepsilon^{n+\alpha-(m+1)}\|u\|_{C^{n,\alpha}_{B}},\quad
n\geqslant 1.$ (3.16)

By (3.14), the change of variable
$\bar{\zeta}=D_{\varepsilon^{-1}}({\zeta}^{-1}\circ z)$, and Theorem 2.4, we
have

$\displaystyle|I^{(n,m)}_{\varepsilon}u|_{\infty}$
$\displaystyle\leqslant\int_{\mathbb{R}^{d+1}}\left|\left(u(z)-T_{n}u({\zeta},z)\right){\mathfrak{D}}^{m}_{z}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z))\right|\frac{d{\zeta}}{\varepsilon^{Q+2}}$
$\displaystyle\lesssim\|u\|_{C^{n,\alpha}_{B}}\int_{\mathbb{R}^{d+1}}\|{\zeta}^{-1}\circ
z\|_{B}^{n+\alpha}\varepsilon^{-m}{\mathfrak{D}}^{m}\phi(\bar{\zeta})\mid_{\bar{\zeta}=D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z)}\frac{d{\zeta}}{\varepsilon^{Q+2}}$
$\displaystyle\lesssim\|u\|_{C^{n,\alpha}_{B}}\int_{\|\bar{\zeta}\|_{B}\leqslant
1}\|D_{\varepsilon}\bar{\zeta}\|_{B}^{n+\alpha}\varepsilon^{-m}{\mathfrak{D}}^{m}\phi(\bar{\zeta})d\bar{\zeta}\lesssim\varepsilon^{n+\alpha-m}\|u\|_{C^{n,\alpha}_{B}},$

were we used that
$\|D_{\varepsilon}\bar{\zeta}\|_{B}=\varepsilon\|\bar{\zeta}\|_{B}$ in the
last inequality, and this proves (3.15). On the other hand we have

$\displaystyle|I^{(n,m)}_{\varepsilon}u(z)-I^{(n,m)}_{\varepsilon}u(e^{\delta
Y}z)|$
$\displaystyle\qquad\leqslant\int_{\mathbb{R}^{d+1}}\left|u(z)-T_{n}u({\zeta},z)\right|\left|{\mathfrak{D}}_{z^{\prime}}^{m}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z^{\prime}))-{\mathfrak{D}}^{m}_{z}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z))\right|_{z^{\prime}=e^{\delta Yz}}\frac{d{\zeta}}{\varepsilon^{Q+2}}$
$\displaystyle\qquad\quad+\int_{\mathbb{R}^{d+1}}\left|u(e^{\delta
Y}z)-T_{n}u({\zeta},e^{\delta
Y}z)-\left(u(z)-T_{n}u({\zeta},z)\right)\right|\left|{\mathfrak{D}}^{m}_{z}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z))\right|\frac{d{\zeta}}{\varepsilon^{Q+2}}$ $\displaystyle\qquad=:\Delta
I^{(n,m)}_{\varepsilon,1}+\Delta I^{(n,m)}_{\varepsilon,2}.$

By (3.14), Theorem 3.3, Lemma 2.3, and the usual change of variables
$\bar{\zeta}=D_{\varepsilon^{-1}}({\zeta}^{-1}\circ z)$, we have

$\displaystyle\Delta I^{(n,m)}_{\varepsilon,1}$
$\displaystyle\lesssim\|u\|_{C^{n,\alpha}_{B}}\int_{\mathbb{R}^{d+1}}\|{\zeta}^{-1}\circ
z\|^{n+\alpha}_{B}\varepsilon^{-m}\left|{\mathfrak{D}}^{m}\phi(\bar{{\zeta}}^{\prime})-{\mathfrak{D}}^{m}\phi(\bar{{\zeta}})\right|_{\bar{{\zeta}}^{\prime}=e^{\frac{\delta}{\varepsilon^{2}}Y}\bar{{\zeta}}}\frac{d{\zeta}}{\varepsilon^{Q+2}}$
$\displaystyle\lesssim\|u\|_{C^{n,\alpha}_{B}}\int_{\|\bar{\zeta}\|_{B}\leqslant
1}\varepsilon^{n+\alpha-m}\left|\frac{\delta}{\varepsilon^{2}}\right|^{\frac{1}{2}}\|\phi\|_{C^{m+1,0}_{B}}d\bar{\zeta}\lesssim|\delta|^{\frac{1}{2}}\varepsilon^{n+\alpha-(m+1)}\|u\|_{C^{n,\alpha}_{B}}.$

To control $\Delta I^{(n,m)}_{\varepsilon,2}$ we write

$\Delta I^{(n,m)}_{\varepsilon,2}=\Delta
I^{(n,m)}_{\varepsilon,2}\mathbb{I}_{\delta>\varepsilon^{2}}+\Delta
I^{(n,m)}_{\varepsilon,2}\mathbb{I}_{\delta\leqslant\varepsilon^{2}}.$

We have

$\displaystyle\Delta
I^{(n,m)}_{\varepsilon,2}\mathbb{I}_{\delta>\varepsilon^{2}}$
$\displaystyle\leqslant\int_{\mathbb{R}^{d+1}}\left(|u(e^{\delta
Y}z)-T_{n}u({\zeta},e^{\delta
Y}z)|+|u(z)-T_{n}u({\zeta},z)|\right)\varepsilon^{-m}\left|{\mathfrak{D}}^{m}\phi(\bar{\zeta})\right|\frac{d{\zeta}}{\varepsilon^{Q+2}}\mathbb{I}_{\delta>\varepsilon^{2}}$
(using that ${\zeta}^{-1}\circ e^{\delta
Y}z=D_{\varepsilon}e^{\delta\varepsilon^{2}Y}\bar{\zeta}$ by Lemma 2.3 and
Remark 2.2)
$\displaystyle\lesssim\|u\|_{C^{n,\alpha}_{B}}\int_{\|\bar{\zeta}\|_{B}\leqslant
1}\varepsilon^{n+\alpha-m}(\|e^{\delta\varepsilon^{2}Y}\bar{\zeta}\|^{n+\alpha}_{B}+\|\bar{\zeta}\|^{n+\alpha}_{B})\left|{\mathfrak{D}}^{m}\phi(\bar{\zeta})\right|d\bar{\zeta}\mathbb{I}_{\delta>\varepsilon^{2}}$
$\displaystyle\lesssim\varepsilon^{n+\alpha-m}\|u\|_{C^{n,\alpha}_{B}}\mathbb{I}_{\delta>\varepsilon^{2}}\lesssim|\delta|^{\frac{1}{2}}\varepsilon^{n+\alpha-(m+1)}\|u\|_{C^{n,\alpha}_{B}}.$

Finally, assume for a moment $n\geqslant 2$. By the standard mean value
theorem and Lemma 3.2, there exists $\lambda\in[0,1]$ such that

$\displaystyle\Delta
I^{(n,m)}_{\varepsilon,2}\mathbb{I}_{\delta\leqslant\varepsilon^{2}}$
$\displaystyle\leqslant|\delta|\int_{\mathbb{R}^{d+1}}\left|Yu(z^{\prime})-T_{n-2}Yu({\zeta},z^{\prime})\right|_{{\zeta}^{\prime}=e^{\lambda\delta
Y}z}\varepsilon^{-m}\left|{\mathfrak{D}}^{m}\phi(\bar{\zeta})\right|\frac{d{\zeta}}{\varepsilon^{Q+2}}\mathbb{I}_{\delta\leqslant\varepsilon^{2}}$
$\displaystyle\lesssim|\delta|\|Yu\|_{C^{n-2,\alpha}_{B}}\int_{\mathbb{R}^{d+1}}\|{\zeta}^{-1}\circ
e^{\lambda\delta
Y}z\|^{n-2+\alpha}_{B}\varepsilon^{-m}\|\phi\|_{C^{m,0}_{B}}\frac{d{\zeta}}{\varepsilon^{Q+2}}\mathbb{I}_{\delta\leqslant\varepsilon^{2}}$
$\displaystyle\lesssim|\delta|\|u\|_{C^{n,\alpha}_{B}}\int_{\|{\zeta}\|_{B}\leqslant
1}\varepsilon^{n-2+\alpha-m}\|e^{\lambda\delta\varepsilon^{2}Y}\bar{\zeta}\|^{n-2+\alpha}_{B}{d\bar{\zeta}}\mathbb{I}_{\delta\leqslant\varepsilon^{2}}$
$\displaystyle\lesssim|\delta|\varepsilon^{n-2+\alpha-m}\|u\|_{C^{n,\alpha}_{B}}\mathbb{I}_{\delta\leqslant\varepsilon^{2}}\lesssim|\delta|^{\frac{1}{2}}\varepsilon^{n+\alpha-(m+1)}\|u\|_{C^{n,\alpha}_{B}}.$

If $n=0$ or $n=1$ we have $T_{1}u({\zeta},e^{\delta Y}z)=T_{1}u({\zeta},z)$
and $T_{0}u({\zeta},z)=T_{0}u({\zeta},e^{\delta Y}z)=u({\zeta})$, so that the
expression simplifies and it suffices to exploit the regularity of $u$ w.r.t
the field $Y$: this proves (3.16).
2: Conclusions
Let

$J^{(n,m)}_{\varepsilon}u(z):=\int_{\mathbb{R}^{d+1}}T_{n}u({\zeta},z){\mathfrak{D}}^{m}_{z}\phi(D_{\varepsilon^{-1}}({\zeta}^{-1}\circ
z))\frac{d{\zeta}}{\varepsilon^{Q+2}}.$

By iterative applications of the generalized Leibniz formula and Lemma 3.2,
for $l\leqslant n$:

$\displaystyle\|u-u^{(n)}_{\varepsilon}\|_{C^{l,0}_{\alpha}}$
$\displaystyle\lesssim\sum_{i=0}^{l}\left|I^{(n-i,l-i)}_{\varepsilon}{\mathfrak{D}}^{i}u\right|_{\infty}+\sum_{i=0}^{l-1}\left[I^{(n-i,l-i-1)}_{\varepsilon}{\mathfrak{D}}^{i}u\right]_{C^{\frac{1}{2}}_{Y}}$
(using that ${\mathfrak{D}}^{i}u\in C^{n-i,\alpha}_{B}$ by Theorem 2.4, and
(3.15)-(3.16) with $n-i$ instead of $n$ and $m=l-i$)
$\displaystyle\lesssim\sum_{i=0}^{l}\varepsilon^{n+\alpha-l}\|{\mathfrak{D}}^{i}u\|_{C^{n-i,\alpha}_{B}}+\sum_{i=0}^{l-1}\
\varepsilon^{n+\alpha-l}\|{\mathfrak{D}}^{i}u\|_{C^{n-i,\alpha}_{B}}\lesssim\varepsilon^{n+\alpha-l}\|u\|_{C^{n,\alpha}_{B}}.$
(3.17)

Similarly, for $l>n$, we have

$\displaystyle\|u^{(n)}_{\varepsilon}\|_{C^{l,0}_{\alpha}}\lesssim\sum_{i=0}^{n}\left(\left|J^{(n-i,l-i)}_{\varepsilon}{\mathfrak{D}}^{i}u\right|_{\infty}+\left[J^{(n-i,l-i-1)}_{\varepsilon}{\mathfrak{D}}^{i}u\right]_{C^{\frac{1}{2}}_{Y}}\right)$

Since $l>n$ any term of the first sum sports, at least, one derivative applied
on $\phi$; since the integral of the mollifier (3.13) is constant in $z$, the
cancellation property
$J^{(n-i,l-i)}_{\varepsilon}{\mathfrak{D}}^{i}u=I^{(n-i,l-i)}_{\varepsilon}{\mathfrak{D}}^{i}u$
holds for any $i=0,\dots,n$. Similarly for the second sum. Therefore we
finally get

$\displaystyle\|u^{(n)}_{\varepsilon}\|_{C^{l,0}_{\alpha}}$
$\displaystyle\lesssim\sum_{i=0}^{n}\left(\left|I^{(n-i,l-i)}_{\varepsilon}{\mathfrak{D}}^{i}u\right|_{\infty}+\left[I^{(n-i,l-i-1)}_{\varepsilon}{\mathfrak{D}}^{i}u\right]_{C^{\frac{1}{2}}_{Y}}\right)$
$\displaystyle\lesssim\sum_{i=0}^{n}\varepsilon^{n+\alpha-l}\|{\mathfrak{D}}^{i}u\|_{C^{n-i,\alpha}_{B}}\lesssim\varepsilon^{n+\alpha-l}\|u\|_{C^{n,\alpha}_{B}}.$
(3.18)

The proof is completed. ∎

##  4 Proofs of Proposition 1.5 and Theorem 1.6

###### Proof of Proposition 1.5.

Let $u\in C^{n,0}_{B}$. By Proposition 3.1 we have

$K(\lambda,u;C^{n_{1},0}_{B},C^{n_{2},0}_{B})\lesssim\|u-u_{\varepsilon}\|_{C^{n_{1},0}_{B}}+\lambda\|u_{\varepsilon}\|_{C^{n_{2},0}_{B}}\lesssim(\varepsilon^{n-n_{1}}+\lambda\varepsilon^{n-n_{2}})\|u\|_{C^{n,0}_{B}}.$

Then, taking $\varepsilon=\lambda^{\frac{1}{n_{2}-n_{1}}}$ we directly get
$K(\lambda,u;C^{n_{1},0}_{B},C^{n_{2},0}_{B})\lesssim\lambda^{\frac{n-n_{1}}{n_{2}-n_{1}}}\|u\|_{C^{n,0}_{B}}$,
which gives the set inclusion on the right. It remains to prove that, for any
$u\in C^{n_{2},0}_{B}$

$\|u\|_{C^{n,0}_{B}}\lesssim\|u\|_{C^{n_{1},0}_{B}}^{1-\frac{n-n_{1}}{n_{2}-n_{1}}}\|u\|_{C^{n_{2},0}_{B}}^{\frac{n-n_{1}}{n_{2}-n_{1}}}.$
(4.1)

We check that (4.1) holds for $(n_{1},n_{2})=(n-1,n+1)$, $n\in\mathbb{N}$, in
which case, the inequality reads

$\|u\|_{C^{n,0}_{B}}\lesssim\|u\|_{C^{n-1,0}_{B}}^{\frac{1}{2}}\|u\|_{C^{n+1,0}_{B}}^{\frac{1}{2}}.$
(4.2)

Then it is straightforward to see that (4.1) follows by repeated applications
of (4.2) with itself. We proceed by induction on $n$. By the standard Mean
Value Theorem we have

$\displaystyle|u(e^{\delta\partial_{x_{i}}}z)-u(z)-\partial_{x_{i}}u(z)\delta|$
$\displaystyle\leqslant\frac{1}{2}|\partial^{2}_{x_{i}}u|_{\infty}\delta^{2},\quad
i=1,\dots,p_{0}$ $\displaystyle|u(e^{\delta^{2}Y}z)-u(z)|$
$\displaystyle\leqslant|Yu|_{\infty}\delta^{2},$

and therefore

$\sum_{i=1}^{p_{0}}|\partial_{x_{i}}u|_{\infty}+[u]_{C^{1}_{Y}}\lesssim\delta^{-1}|u|_{\infty}+\delta\left(|\partial^{2}_{x_{i}}u|_{\infty}+|Yu|_{\infty}\right).$

Taking the optimal $\delta>0$ we get

$\|u\|_{C^{1,0}_{B}}\lesssim|u|_{\infty}+|u|^{\frac{1}{2}}_{\infty}\left(|\partial^{2}_{x_{i}}u|_{\infty}+|Yu|_{\infty}\right)^{\frac{1}{2}}\lesssim|u|^{\frac{1}{2}}_{\infty}\|u\|^{\frac{1}{2}}_{C^{2,0}_{B}}.$
(4.3)

This proves (4.2) for $n=1$. Next, by the mean value Theorem along the vector
field $Y$ we have that, for every $z\in R^{d+1}$ and $\delta\neq 0$, there
exists $\bar{\delta}$, $|\bar{\delta}|\leqslant|\delta|$ such that

$u(e^{\delta Y}z)-u(z)-\delta
Yu(z)=\delta\left(Yu(e^{\bar{\delta}Y}z)-Yu(z)\right).$

Then, dividing by $|\delta|^{\frac{1}{2}}$ we easily derive

$|\delta|^{\frac{1}{2}}|Yu|_{\infty}\lesssim[u]_{C^{\frac{1}{2}}_{Y}}+|\delta|[Yu]_{C^{\frac{1}{2}}_{Y}}$
(4.4)

By (4.3) and (4.4), taking the optimal $\delta$, we eventually get

$\displaystyle\|u\|_{C^{2,0}_{B}}$
$\displaystyle=|u|_{\infty}+|Yu|_{\infty}+\sum_{i=0}^{p_{0}}\|\partial_{x_{i}}u\|_{C^{1,0}_{B}}$
$\displaystyle\lesssim|u|_{\infty}+[u]^{\frac{1}{2}}_{C^{\frac{1}{2}}_{Y}}[Yu]^{\frac{1}{2}}_{C^{\frac{1}{2}}_{Y}}+\sum_{i=0}^{p_{0}}|\partial_{x_{i}}u|^{\frac{1}{2}}_{\infty}\|\partial_{x_{i}}u\|^{\frac{1}{2}}_{C^{2,0}_{B}}\lesssim\|u\|^{\frac{1}{2}}_{C^{1,0}}\|u\|^{\frac{1}{2}}_{C^{3,0}}.$

This proves (4.2) for $n=2$. The general case simply follows by the iterative
definition of the spaces and the induction hypothesis. ∎

Before we proceed with the proof of Theorem 1.6, we recall another important
tool in the theory of interpolation, that is the well known Reiteration
Theorem (see [20], Theorem 1.23 or [27], Section 1.10.2).

######  Theorem 4.1 (Reiteration Theorem).

Let $\\{Z_{1},Z_{2}\\}$ be an interpolation couple, and let $E_{1},E_{2}$ be
some intermediate spaces between $Z_{1}$ and $Z_{2}$. If

$\big{(}Z_{1},Z_{2}\big{)}_{\theta_{i},1}\subseteq
E_{i}\subseteq\big{(}Z_{1},Z_{2}\big{)}_{\theta_{i},\infty},\quad i=1,2$

for some $\theta_{i}$ such that
$0\leqslant\theta_{1}<\theta_{2}\leqslant\infty$, then

$\big{(}E_{1},E_{2}\big{)}_{\alpha,p}=\big{(}Z_{1},Z_{2}\big{)}_{(1-\alpha)\theta_{1}+\alpha\theta_{2},p},\quad\alpha\in(0,1),\
p\geqslant 1.$

We are finally ready to prove Theorem 1.6.

###### Proof of Theorem 1.6.

The proof is a fairly standard application of Theorem 3.1, Proposition 1.5 and
Theorem 4.1. We proceed in two steps:

Step 1: We prove (1.15) for $\alpha_{1}=\alpha_{2}=0$ and $n_{1}=n$,
$n_{2}=n+1$. Precisely, for any $n\in\mathbb{N}_{0}$, and $0<\alpha<1$,

$\left(C^{n,0}_{B},C^{n+1,0}_{B}\right)_{\alpha,\infty}=\left(C^{n,0}_{B},C^{n,1}_{B}\right)_{\alpha,\infty}=C^{n,\alpha}_{B}.$

$(\supseteq)$: This inclusion is a direct consequence of the approximation
result of Theorem 3.1. Indeed, for any $u\in C^{n,\alpha}_{B}$ and
$\lambda\in[0,1)$ we have

$K(\lambda,u;C^{n,0}_{B},C^{n+1,0}_{B})\lesssim\|u-u_{\varepsilon}\|_{C^{n,0}_{B}}+\lambda\|u_{\varepsilon}\|_{C^{n+1,0}_{B}}\lesssim(\varepsilon^{\alpha}+\lambda\varepsilon^{\alpha-1})\|u\|_{C^{n,\alpha}_{B}}.$

Therefore, taking $\epsilon=\lambda$ we get

$K(\lambda,u;C^{n,0}_{B},C^{n+1,0}_{B})\lesssim\lambda^{\alpha}\|u\|_{C^{n,\alpha}_{B}}.$

On the other hand, for $\lambda\geqslant 1$ it suffices to take
$u_{\varepsilon}\equiv 0$.
$(\subseteq)$: We proceed by induction on $n$. Let $u\in C^{0,\alpha}_{B}$,
then, for any choice of $a\in C$, $b\in C^{1,0}_{B}$ such that $u=a+b$ we have
$|u|_{\infty}\leqslant|a|_{\infty}+|b|_{\infty}$ and therefore
$|u|_{\infty}\leqslant
K(1,u;C,C^{1,0}_{B})\leqslant\|u\|_{(C,C^{1,0}_{B})_{\alpha,\infty}}$.
Similarly, for any $i=1,\dots,p_{0}$

$|u(e^{\delta\partial_{x_{i}}}z)-u(z)|\leqslant
2|a|_{\infty}+[b]_{C^{1}_{\partial_{x_{i}}}}|\delta|,\quad|u(e^{\delta
Y}z)-u(z)|\leqslant
2|a|_{\infty}+[b]_{C^{\frac{1}{2}}_{Y}}|\delta|^{\frac{1}{2}},\quad
z\in\mathbb{R}^{d+1},$

and thus, for any $i=1,\dots,p_{0}$

$\displaystyle|u(e^{\delta\partial_{x_{i}}}z)-u(z)|$ $\displaystyle\leqslant
2K(|\delta|,u;C,C^{1,0}_{B})\leqslant
2|\delta|^{\alpha}\|u\|_{(C,C^{1,0}_{B})_{\alpha,\infty}},$ (4.5)
$\displaystyle|u(e^{\delta Y}z)-u(z)|$ $\displaystyle\leqslant
2K(|\delta|^{\frac{1}{2}},u;C,C^{1,0}_{B})\leqslant
2|\delta|^{\frac{\alpha}{2}}\|u\|_{(C,C^{1,0}_{B})_{\alpha,\infty}}.$ (4.6)

Gathering together (4.5)-(4.6), we get the inclusion in the case $n=0$. Next,
let $u\in C^{1,\alpha}_{B}$. Clearly $|u|_{\infty}\leqslant
K(1,u;C^{1,0}_{B},C^{2,0}_{B})$. Moreover,for any choice of $a\in
C^{1,0}_{B}$, $b\in C^{2,0}_{B}$ such that $u=a+b$ we have, for any
$i=1,\dots,p_{0}$

$\displaystyle|u(e^{\delta\partial_{x_{i}}}z)-u(z)-\delta\partial_{x_{i}}u(z)|$
$\displaystyle\leqslant
2|\partial_{x_{i}}a|_{\infty}|\delta|+\delta^{2}[\partial_{x_{i}}b]_{C^{1}_{\partial_{x_{i}}}}|\delta^{2}|,$
$\displaystyle|u(e^{\delta Y}z)-u(z)|$
$\displaystyle\leqslant[a]_{C^{\frac{1}{2}}_{Y}}|\delta|^{\frac{1}{2}}+|Yb|_{\infty}|\delta|,$
$\displaystyle|\partial_{x_{i}}u(z^{\prime})-\partial_{x_{i}}u(z)|_{z^{\prime}=e^{\delta
Y}z}$ $\displaystyle\leqslant
2|\partial_{x_{i}}a|_{\infty}+[\partial_{x_{i}}b]_{C^{\frac{1}{2}}_{Y}}|\delta|^{\frac{1}{2}},$

and therefore, for any $i=1,\dots,p_{0}$

$\displaystyle|u(e^{\delta\partial_{x_{i}}}z)-u(z)-\delta\partial_{x_{i}}u(z)|$
$\displaystyle\leqslant
2|\delta|K(|\delta|,u;C^{1,0}_{B},C^{2,0}_{B})\leqslant
2|\delta|^{1+\alpha}\|u\|_{(C^{1,0}_{B},C^{2,0}_{B})_{\alpha,\infty}},$ (4.7)
$\displaystyle|u(e^{\delta Y}z)-u(z)|$
$\displaystyle\leqslant|\delta|^{\frac{1}{2}}K(|\delta|^{\frac{1}{2}},u;C^{1,0}_{B},C^{2,0}_{B})\leqslant|\delta|^{\frac{1+\alpha}{2}}\|u\|_{(C^{1,0}_{B},C^{2,0}_{B})_{\alpha,\infty}},$
(4.8)
$\displaystyle|\partial_{x_{i}}u(z^{\prime})-\partial_{x_{i}}u(z)|_{z^{\prime}=e^{\delta
Y}z}$ $\displaystyle\leqslant
2K(|\delta|^{\frac{1}{2}},u;C^{1,0}_{B},C^{2,0}_{B})\leqslant
2|\delta|^{\frac{\alpha}{2}}\|u\|_{(C^{1,0}_{B},C^{2,0}_{B})_{\alpha,\infty}}.$
(4.9)

Gathering together (4.7)-(4.9), we get the inclusion in the case $n=1$.
Finally, for $u\in C^{n,\alpha}_{B}$, $n\geqslant 2$, we have

$\displaystyle\|u\|_{C^{n,\alpha}_{B}}$
$\displaystyle=|u|_{\infty}+\|Yu\|_{C^{n-2,\alpha}_{B}}+\sum_{i=1}^{p_{0}}\|\partial_{x_{i}}u\|_{C^{n-1,\alpha}_{B}}$
$\displaystyle\lesssim|u|_{\infty}+\|Yu\|_{(C^{n-2,0}_{B},C^{n-1,0}_{B})_{\alpha,\infty}}+\sum_{i=1}^{p_{0}}\|\partial_{x_{i}}u\|_{(C^{n-1,0}_{B},C^{n,0}_{B})_{\alpha,\infty}}\lesssim\|u\|_{(C^{n,0}_{B},C^{n+1,0}_{B})_{\alpha,\infty}},$

where we used that
$K(\lambda,\partial_{x_{i}}u;C^{n-1,0}_{B},C^{n,0}_{B})\leqslant
K(\lambda,u;C^{n,0}_{B},C^{n+1,0}_{B})$ as well as
$K(\lambda,Yu;C^{n-2,0}_{B},C^{n-1,0}_{B})\leqslant
K(\lambda,u;C^{n,0}_{B},C^{n+1,0}_{B})$, for any $\lambda\geqslant 0$,
$i=1,\dots,p_{0}$.

Step 2: Let now $n_{1},\,n_{2}\in\mathbb{N}_{0}$, $n_{1}<n_{2}$, and let
$\alpha_{1},\alpha_{2}\in[0,1]$. By the interpolation result of Step 1, and
the Reiteration Theorem paired with Proposition 1.5, we get

$\displaystyle\left(C^{n_{1},\alpha_{1}}_{B},C^{n_{2},\alpha_{2}}_{B}\right)_{\theta,\infty}$
$\displaystyle=\left(\left(C^{n_{1},0}_{B},C^{n_{1}+1,0}_{B}\right)_{\alpha_{1},\infty},\left(C^{n_{2},0}_{B},C^{n_{2}+1,0}_{B}\right)_{\alpha_{2},\infty}\right)_{\theta,\infty}$
$\displaystyle=\left(\left(C^{n_{1},0}_{B},C^{n_{2}+1,0}_{B}\right)_{\frac{\alpha_{1}}{n_{2}+1-n_{1}},\infty},\left(C^{n_{1},0}_{B},C^{n_{2}+1,0}_{B}\right)_{\frac{n_{2}+\alpha_{2}-n_{1}}{n_{2}+1-n_{1}},\infty}\right)_{\theta,\infty}$
$\displaystyle=\left(C^{n_{1},0}_{B},C^{n_{2}+1,0}_{B}\right)_{(1-\theta)\frac{\alpha_{1}}{n_{2}+1-n_{1}}+\theta\frac{n_{2}+\alpha_{2}-n_{1}}{n_{2}+1-n_{1}},\infty}$
$\displaystyle=:\left(C^{n_{1},0}_{B},C^{n_{2}+1,0}_{B}\right)_{\theta^{\prime},\infty}$

Taking $n$ as in (1.14) we finally obtain, again by the Reiteration Theorem,

$\displaystyle\left(C^{n_{1},\alpha_{1}}_{B},C^{n_{2},\alpha_{2}}_{B}\right)_{\theta,\infty}$
$\displaystyle=\left(C^{n,0}_{B},C^{n+1,0}_{B}\right)_{\theta^{\prime}(n_{2}+1-n_{1})-(n-n_{1}),\infty}=C^{n,\alpha_{1}+\theta[(n_{2}+\alpha_{2})-(n_{1}+\alpha_{1})]-(n-n_{1})}_{B}.$

The proof is completed. ∎

## Acknowledgments

This research was supported by the Gruppo Nazionale per l’Analisi Matematica,
la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di
Alta Matematica (INdAM), and the University of Bologna. We would like to thank
Professor A. Pascucci for his interest in our work and for several helpful
conversations.

## References

  * [1] Adams, R. A. Sobolev spaces.  Academic Press, New York-London, 1975.  Pure and Applied Mathematics, Vol. 65.
  * [2] Barucci, E., Polidoro, S., and Vespri, V. Some results on partial differential equations and Asian options.  Math. Models Methods Appl. Sci. 11, 3 (2001), 475–497.
  * [3] Butzer, P. L., and Berens, H. Semi-groups of operators and approximation.  Die Grundlehren der mathematischen Wissenschaften, Band 145. Springer-Verlag New York, Inc., New York, 1967.
  * [4] Cercignani, C. The Boltzmann equation and its applications.  Springer-Verlag, New York, 1988.
  * [5] Chaudru de Raynal, P.-E., and Menozzi, S. Regularization effects of a noise propagating through a chain of differential equations: an almost sharp result.  Trans. Amer. Math. Soc. 375, 1 (2022), 1–45.
  * [6] Chaudru de Raynal, P.-E., Menozzi, S., Pesce, A., and Xicheng, Z. Heat kernel and gradient estimates for kinetic SDEs with low regularity coefficients.  arXiv:2203.11515 (2022).
  * [7] Cibelli, G., Polidoro, S., and Rossi, F. Sharp estimates for Geman-Yor processes and applications to Arithmetic Average Asian options.  Journal de Mathématiques Pures et Appliquées 129 (2019), 87–130.
  * [8] Desvillettes, L., and Villani, C. On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: the linear Fokker-Planck equation.  Comm. Pure Appl. Math. 54, 1 (2001), 1–42.
  * [9] Di Francesco, M., and Pascucci, A. On a class of degenerate parabolic equations of Kolmogorov type.  AMRX Appl. Math. Res. Express 3 (2005), 77–116.
  * [10] Di Francesco, M., and Polidoro, S. Schauder estimates, Harnack inequality and Gaussian lower bound for Kolmogorov-type operators in non-divergence form.  Adv. Differential Equations 11, 11 (2006), 1261–1320.
  * [11] Frentz, M., Nyström, K., Pascucci, A., and Polidoro, S. Optimal regularity in the obstacle problem for Kolmogorov operators related to American Asian options.  Math. Ann. 347, 4 (2010), 805–838.
  * [12] Imbert, C., and Mouhot, C. The Schauder estimate in kinetic theory with application to a toy non-linear model.  Annales H. Lebesgue 4 (2021), 369–405.
  * [13] Imbert, C., and Silvestre, L. The Schauder estimate for kinetic integral equations.  Analysis and PDE 14 (2021), 171–204.
  * [14] Krylov, N. V. Lectures on elliptic and parabolic equations in Hölder spaces, vol. 12 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, 1996.
  * [15] Lanconelli, E., and Polidoro, S. On a class of hypoelliptic evolution operators.  Rend. Sem. Mat. Univ. Politec. Torino 52, 1 (1994), 29–63.
  * [16] Lucertini, G., Pagliarani, S., and Pascucci, A. Optimal regularity for degenerate Kolmogorov equations with rough coefficients.  To appear in J. Evol. Equ. (2023).
  * [17] Lucertini, G., Pagliarani, S., and Pascucci, A. Optimal Schauder estimates for kinetic Kolmogorov equations with time measurable coefficients.  arXiv:2304.13392 (2023).
  * [18] Lunardi, A. Analytic Semigroups and Optimal Regularity in Parabolic Problems.  Modern Birkhäuser Classics. Birkhäuser Basel, 1995.
  * [19] Lunardi, A. Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in ${\mathbb{R}}^{N}$.  Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24, 1 (1997), 133–164.
  * [20] Lunardi, A. Interpolation theory, vol. 16 of Lecture Notes. Scuola Normale Superiore di Pisa.  Edizioni della Normale, Pisa, 2018.
  * [21] Manfredini, M. The Dirichlet problem for a class of ultraparabolic equations.  Adv. Differential Equations 2, 5 (1997), 831–866.
  * [22] Manfredini, M. The Dirichlet problem for a class of ultraparabolic equations.  Adv. Differential Equations 2, 5 (1997), 831–866.
  * [23] Pagliarani, S., Pascucci, A., and Pignotti, M. Intrinsic Taylor formula for Kolmogorov-type homogeneous groups.  J. Math. Anal. Appl. 435, 2 (2016), 1054–1087.
  * [24] Pascucci, A., and Pesce, A. On stochastic Langevin and Fokker-Planck equations: the two-dimensional case.  J. Differential Equations 310 (2022), 443–483.
  * [25] Pascucci, A., and Pesce, A. Sobolev embeddings for kinetic Fokker-Planck equations.  arXiv:2209.05124 (2022).
  * [26] Tartar, L. An introduction to Sobolev spaces and interpolation spaces, vol. 3 of Lecture Notes of the Unione Matematica Italiana.  Springer, Berlin; UMI, Bologna, 2007.
  * [27] Triebel, H. Interpolation Theory, Function Spaces, Differential Operators, vol. 18 of North-Holland Mathematical Library.  1978\.