Entire Monogenic Functions of Given Proximate Order and Continuous Homomorphisms

Abstract

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution as initial data for the Schrödinger equation. Inspired by the infinite order differential operators arising in quantum mechanics, in this paper we investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions. Precisely, we consider homomorphisms acting on functions in the kernel of the Dirac operator. For this class of functions, often called monogenic functions, we introduce the proximate order and prove some fundamental properties. As an important application, we are able to characterize infinite order differential operators that act continuously on spaces of monogenic entire functions.

1 Introduction

Infinite-order differential operators have been studied already since a long time. In recent years they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial data for the Schrödinger equation. Superoscillatory functions arise in several areas of science and technology, for example in quantum mechanics they are the outcome of Aharonov’s weak values, see [1, 8]. To study their time evolution as initial data of quantum field equations represents an important problem in quantum mechanics.

The study of the evolution of superoscillatory functions under the Schrödinger equation is highly non-trivial. A natural functional analytic setting is the space of entire functions with certain growth conditions. In fact, the Cauchy problem for the Schrödinger equation with superoscillatory initial data leads to infinite order differential operators of the type

$$\begin{aligned} \mathcal {U}(t,z;\partial _z)=\sum _{m=1}^\infty u_m(t,z)\partial _z^m, \end{aligned}$$

where the coefficients $u_m(t,z)$ depend on the Green’s function of the time dependent Schrödinger equation for a given potential V, t is the time variable and z is the complexification of the space variable. For more details, see for example the monograph [6] and [2,3,4, 7, 14, 16, 17, 30].

Another application of infinite order differential operators within the scope of the theory of superoscillatory functions is the extension of this theory to several super oscillating variables, see [5].

For $p\ge 1$ the natural spaces on which operators such as $\mathcal {U}(t,z;\partial _z)$ act are the spaces of entire functions with growth order either order lower than p or equal to p and finite type. In other words, they consist of entire functions f for which there exist constants $B, C >0$ such that $|f(z)|\le C e^{B|z|^p}.$

The problem to extend infinite order differential operators to the hypercomplex setting is treated in the recent paper [9] where the authors investigate the continuity of a class of infinite order differential operators acting on spaces of entire hyperholomorphic functions that include monogenic functions. In [9] we find the hypercomplex version of some results obtained in [10, 13]; the class of monogenic functions is the most delicate case to investigate. Even though the classical exponential function is not in the kernel of the Dirac operator, monogenic functions with exponential bounds play a crucial role in the study of the continuity of a class of infinite order differential operators in the hypercomplex settings. The hypercomplex setting is non-trivial and requires some efforts because of the structure of monogenic functions and of the fact that they admit series expansions in terms of the so-called Fueter polynomials. Precisely, the Fueter’s polynomials $V_k(x)$, see for instance formula (4) of [21] page 794 or elsewhere, are defined by

$$\begin{aligned} V_k(x):= \frac{k!}{|k|!}\sum _{\sigma \in perm(|k|)} z_{j_{\sigma (1)}}z_{j_{\sigma (2)}} \ldots z_{j_{\sigma (|k|)}}, \end{aligned}$$

where k represents a multi-index. They play the role of the sequence of the complex polynomials $(z^n)_n$ when z is a complex variable. To preserve the monogenicity we need a special product, called Cauchy–Kowalewski product, for short CK-product, that does not coincide with the ordinary pointwise product. The CK-product of two left entire monogenic f and g is denoted by $f \odot _L g$ is given in Definition 2.4 in terms of the Fueter polynomials. A similar definition is given for the right monogenic functions.

In the paper [9] we obtained the following result regarding monogenic functions. Let $p\ge 1$ and set $\mathbb {N}_0=\mathbb {N} \cup \{0\}$. Let $(u_m)_{m\in (\mathbb {N}_0)^n }:\mathbb {R}^{n+1}\rightarrow \mathbb {R}_n$ be left entire monogenic functions such that for every $\varepsilon >0$ there exist $B_\varepsilon >0$, $C_\varepsilon >0$ for which

$$\begin{aligned} |u_m(x)|\le C_\varepsilon \frac{\varepsilon ^{|m|}}{(|m|!)^{1/q}}\exp (B_\varepsilon |x|^p), \ \ \ \mathrm{for \ all} \ \ \ m\in (\mathbb {N}_0)^n, \end{aligned}$$

(1.1)

where $1/p+1/q=1$ and where we set $1/q:=0$ when $p=1$, and m is a multi-index. We considered the formal infinite order differential operator

$$\begin{aligned} U_L(x,\partial _{x})f(x):=\sum _{|m|=0}^\infty u_m(x)\odot _L \partial _{x}^m f(x), \end{aligned}$$

(1.2)

for left entire monogenic functions f where $\partial _{x}^m:= \partial _{x_1}^{m_1}\dots \partial _{x_n}^{m_n}$ and $\odot _L$ denotes the CK-product. Then for $p\ge 1$, we proved that the operator $U_L(x,\partial _{x})$ acts continuously on the space of left monogenic functions with the condition $|f(x)|\le C e^{B|x|^p}.$

In this paper, we characterize the continuous homomorphisms of type (1.2) acting on monogenic functions. To do this we introduce proximate orders for monogenic functions and we study some fundamental properties. After that, we investigate the monogenic counterpart of the differential operator representation of continuous homomorphisms between the spaces of entire functions of given proximate order proved by T. Aoki and co-authors in [11, 12].

The plan of the paper. Section 1 provides an introduction. In Sect. 2 we state some preliminary results on monogenic functions. In Sect. 3 we study entire monogenic functions where the growth is determined by a proximate order. We establish some important properties on the proximate order of monogenic functions. In Sect. 4 we then apply these results to characterize the continuous homomorphisms. We observe that Sects. 3 and 4 closely follow Section 3 and Section 4 of [11]. The points where the adaptation to the case of monogenic functions are not straightforward are Theorem 3.4, Theorem 4.4 and Theorem 4.5. In particular, in the last two theorems we have used the new Lemmas 3.12 and 3.13.

2 Preliminary Results on Monogenic Functions

In this section, we recall some results on monogenic functions, whose proofs can be found in [15]. We recall that $\mathbb {R}_n$ is the real Clifford algebra over n imaginary units $e_1,\ldots ,e_n$. The element $(x_0,x_1,\ldots ,x_n)\in \mathbb {R}^{n+1}$ will be identified with the paravector $\textbf{x}=x_0+\underline{x}=x_0+ \sum _{\ell =1}^nx_\ell e_\ell$ and the real part $x_0$ of $\textbf{x}$ will also be denoted by $\textrm{Re}(\textbf{x})$. An element y in $\mathbb {R}_{n}$, is called a Clifford number. If $A=\{i_1,\ldots ,i_r\}$ is an element in the power set $P(1,\ldots ,n)$, then the element $e_{i_1}\ldots e_{i_r}$ can be written as $e_{i_1...i_r}$ or, in short, $e_A$. Thus, we can write a Clifford number as $y=\sum _Ay_A e_A.$ Possibly using the defining relations $e_i^2=-1$, $e_ie_j+e_je_i=0$, $i,j\in \{1,\ldots , n\}$, $i\not =j$, we will order the indices in A as $i_1< \ldots <i_r$. When $A=\emptyset$ we set $e_\emptyset =1$. The Euclidean norm of an element $y\in \mathbb {R}_n$ is given by $|y|^2=\sum _{A} |y_A|^2$, in particular the norm of the paravector $\textbf{x}\in \mathbb {R}^{n+1}$ is $|\textbf{x}|^2=x_0^2+x_1^2+\ldots +x_n^2$.

Definition 2.1

We call $f:{\mathbb {R}}^{n+1}\rightarrow {\mathbb {R}}_n$ an entire left (right) monogenic function if $f\in \mathcal {C}^{1}({\mathbb {R}}^{n+1},{\mathbb {R}}_n)$ and $\mathcal {D}f\equiv 0$ (resp. $f\mathcal {D}\equiv 0$) where

$$\begin{aligned} \mathcal {D}:=\partial _{x_0}+\sum _{i=1}^n e_i \partial _{x_i}. \end{aligned}$$

We will restrict ourselves to consider only the set of the left monogenic functions since for right monogenic functions analogous computations hold.

We will denote the set of left entire monogenic function by ${\mathcal {M}}_L({\mathbb {R}}^{n+1})$. The Cauchy formula for monogenic functions and their derivatives is computed on the boundary of $U \subset \mathbb {R}^{n+1}$ where $\overline{U}$ is contained in the set of monogenicity of f.

Theorem 2.2

If f is a left monogenic function in a neighborhood of ${\overline{U}}$ then for any $\textbf{x}\in U$

$$\begin{aligned} f(\textbf{x})= \frac{1}{A_{n+1}}\int _{\partial U} q_0(\textbf{x}-\xi ) \textrm{d}\tau (\xi ) f(\xi ) \end{aligned}$$

where $A_{n+1}$ is the n-dimensional surface area of the $n+1$-dimensional unit ball,

$$\begin{aligned} \textrm{d}\tau (\xi )=\sum _{j=0}^n(-1)^j e_j \widehat{\textrm{d}\xi _j} \quad \text {and}\quad q_0(\textbf{x})=\frac{{\bar{\textbf{x}}}}{\Vert \textbf{x}\Vert ^{n+1}} \end{aligned}$$

with $\overline{\textbf{x}} = x_0 - \underline{x}$ and

$$\begin{aligned} \widehat{\textrm{d}\xi _j}=\textrm{d}\xi _0\wedge \dots \wedge \textrm{d}\xi _{j-1}\wedge \textrm{d}\xi _{j+1}\wedge \dots \wedge \textrm{d}\xi _n. \end{aligned}$$

Theorem 2.3

Let $\textbf{m}:=(m_1,\dots , m_n)\in {\mathbb {N}}_0^n$ be a multi-index. We denote by $|\textbf{m}|=m_1+\dots +m_n$. If f is a left monogenic function in a ball $\Vert \textbf{x}\Vert < R$, then for all $\Vert \textbf{x}\Vert <r$ with $0<r<R$, its Taylor series expansion is given by

$$\begin{aligned} f(\textbf{x})=\sum _{|\textbf{m}|=0}^\infty V_{\textbf{m}}(\textbf{x}) a_{\textbf{m}} \end{aligned}$$

where $V_{\textbf{m}}$ are the Fueter polynomials and

$$\begin{aligned} a_{\textbf{m}}=\frac{1}{\textbf{m}! A_{n+1}}\int _{\Vert \xi \Vert \le r}q_{\textbf{m}}(\xi ) \textrm{d}\tau (\xi ) f(\xi ) \end{aligned}$$

where $q_{\textbf{m}}(\textbf{x})=\partial _{x_1}^{m_1}\dots \partial _{x_n}^{m_n}q_0(\textbf{x})$. We have the following Cauchy inequality

$$\begin{aligned} \Vert a_{\textbf{m}}\Vert \le c(n,\textbf{m}) \frac{M(r,f)}{r^{|\textbf{m}|}}, \end{aligned}$$

where

$$\begin{aligned} c(n,\textbf{m}):=\frac{n(n+1)\cdot \dots \cdot (n+|\textbf{m}|-1)}{\textbf{m}!} \end{aligned}$$

and $M(r,f)=\sup _{\Vert \textbf{x}\Vert =r} \Vert f(\textbf{x})\Vert$. In particular, setting $\textbf{m}:=(m_0,\dots , m_n)\in {\mathbb {N}}^{n+1}_0$, we have (cf. [20])

$$\begin{aligned} \Vert \partial _{x_0}^{m_0}\dots \partial _{x_n}^{m_n}q_0(\textbf{x})\Vert \le \frac{n(n+1)\cdots (n+|\textbf{m}|-1)}{\Vert \textbf{x}\Vert ^{n+|\textbf{m}|}}. \end{aligned}$$

(2.1)

Definition 2.4

Let $f,\, g\in {\mathcal {M}}_L({\mathbb {R}}^{n+1})$ (resp. $f,\, g\in {\mathcal {M}}_R({\mathbb {R}}^{n+1})$). Using their Taylor series representation

$$\begin{aligned} f(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } V_{\textbf{m}}(\textbf{x}) a_{\textbf{m}},\quad \left( \text {resp. } f(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } a_{\textbf{m}} V_{\textbf{m}}(\textbf{x})\right) \end{aligned}$$

and

$$\begin{aligned} g(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } V_{\textbf{m}}(\textbf{x}) b_{\textbf{m}},\quad \left( \text {resp. } g(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } b_{\textbf{m}} V_{\textbf{m}}(\textbf{x})\right) \end{aligned}$$

we define

$$\begin{aligned} f\odot _L g:= & {} \sum _{|\textbf{m}|=0}^{\infty }\sum _{|\textbf{k}|=0}^{\infty } V_{\textbf{m}+\textbf{k}}(\textbf{x}) a_\textbf{m}b_\textbf{k}\quad \\{} & {} \times {\left( \text {resp. } f\odot _R g:= \sum _{|\textbf{m}|=0}^{\infty }\sum _{|\textbf{k}|=0}^{\infty } a_\textbf{m}b_\textbf{k}V_{\textbf{m}+\textbf{k}}(\textbf{x}) \right) }. \end{aligned}$$

We mention that a different product, that is defined just for the subclass of axially monogenic function (see [23] and [24]), can also be used to define infinite order differential operators in the monogenic setting.

Next, following [21], we recall the definition of the standard growth order of an entire monogenic function:

Definition 2.5

Let $f: \mathbb {R}^{n+1} \rightarrow \mathbb {R}_n$ be an entire left monogenic function. Then its growth order is said to be

$$\begin{aligned} \rho = \rho (f) = \limsup \limits _{r \rightarrow +\infty } \frac{\log ^+ \log ^+ (M(r,f))}{\log r}, \end{aligned}$$

where $\log ^+(r) = \max \{0,\log (r)\}$.

It may occur that $0 \le \rho \le +\infty$. For the case where $0< \rho < +\infty$ Constales et al. defined in [19] the growth type of an entire monogenic function of growth order $\rho$ by

$$\begin{aligned} \sigma = \sigma (f) = \limsup \limits _{r \rightarrow +\infty } \frac{\log ^+ M(r,f)}{r^{\rho }}. \end{aligned}$$

As shown in [18] the growth order of a monogenic function can be directly computed by its Taylor coefficients, namely by

$$\begin{aligned} \rho (f) = \limsup \limits _{|\textbf{m}| \rightarrow + \infty } \frac{|\textbf{m}| \log |\textbf{m}|}{-\log \Big |\frac{1}{c(n,\textbf{m}) }a_\textbf{m} \Big |}. \end{aligned}$$

Similarly, as shown in [19] the growth type can be expressed by

$$\begin{aligned} \sigma (f) = \frac{1}{e\rho } \limsup \limits _{r \rightarrow +\infty }|\textbf{m}|\Big (|a_\textbf{m}| \Big )^{\frac{\rho }{|\textbf{m}|}}. \end{aligned}$$

Remark 2.6

To also get a finer classification of functions with slow growth $\rho =0$ and fast growth $\rho =\infty$ Seremeta [31] and Shah [32] introduced the notion of generalized growth in the complex analysis setting which has been generalized to the monogenic setting in [22, 26, 27, 33]. More precisely the authors considered functions $\alpha (\cdot )$, $\beta (\cdot )$ and $\gamma (\cdot )$ satisfying particular conditions mentioned concisely for example in Definition 1 of [22] and introduced the notions of the generalized growth and type in the way

$$\begin{aligned} \rho _{\alpha ,\beta }(f) = \limsup \limits _{r \rightarrow +\infty } \frac{\alpha (\log ^+ M(r,f))}{\beta (\log (r))} \end{aligned}$$

and

$$\begin{aligned} \sigma _{\alpha ,\beta ,\gamma }(f) = \limsup \limits _{r \rightarrow +\infty } \frac{\alpha (\log ^+ M(r,f))}{\beta ((\gamma (r))^{\rho _{\alpha ,\beta }})}, \end{aligned}$$

(see for instance Definition 2 of [22] or [27]). In the particular case where $\alpha (r)=\log (r)$ and where $\beta$ is the identity function one re-obtains the classical definition of the growth order and growth type. Another generalization of the classical growth order that includes the classical growth order as a special case, and even the generalized growth orders under particular conditions, is the definition of the proximate growth order.

We introduce:

Definition 2.7

A differentiable function $\rho (r)\ge 0$ defined for $r\ge 0$ is said to be a proximate order for the order $\rho \ge 0$ if it satisfies

• (1) $\lim _{r\rightarrow +\infty } \rho (r)=\rho$,

• (2) $\lim _{r\rightarrow +\infty } \rho '(r)r\ln (r)=0.$

We observe that for any proximate order function $\rho (r)$ there exists a positive constant $r_0>0$ such that for any $r>r_0$ the function $r^{\rho (r)}$ is strictly increasing and tending to $+\infty$.

Definition 2.8

For any proximate order function $\rho (r)$ we can always take another proximate order function, called the normalization of the proximate order function $\rho (r)$, $\hat{\rho }(r)$ such that there exists a constant $r_1>0$ for which $\hat{\rho }(r)=\rho (r)$ for any $r\ge r_1$ and $r^{\hat{\rho }(r)}$ is strictly increasing on $r>0$ and maps the interval $(0,+\infty )$ to $(0,+\infty )$.

We denote by $\varphi :(0,+\infty )\rightarrow (0+\infty )$ the inverse function of the function $t=r^{\hat{\rho }(r)}$. Moreover, we set

$$\begin{aligned} G_q=G_{\hat{\rho },q}:=\frac{\varphi (q)^q}{(e\rho )^{q/\rho }},\quad \text {for } q\in {\mathbb {N}}. \end{aligned}$$

(2.2)

Let $\rho (r)$ be a proximate order for a positive order $\rho >0$. For any $\sigma >0$, we consider the Banach space

$$\begin{aligned} A_{\rho ,\sigma }:=\{f\in {\mathcal {M}}_L({\mathbb {R}}^{n+1}):\, \Vert f\Vert _{\rho ,\sigma }:=\sup _{\textbf{x}\in {\mathbb {R}}^{n+1}} \Vert f(\textbf{x})\Vert \exp (-\sigma \Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )})<+\infty \} \end{aligned}$$

with the norm $\Vert \cdot \Vert _{\rho ,\sigma }$. Moreover, we use the following notation: for any $\textbf{m}=(m_1,\dots , m_n)\in {\mathbb {N}}^n$ we write

$$\begin{aligned} \partial ^\textbf{m}_\textbf{x}f(\textbf{x}):=\partial _{x_1}^{m_1}\dots \partial _{x_n}^{m_n} f(\textbf{x}). \end{aligned}$$

We recall some basic properties of proximate orders that will be used in the following. The proofs of these results can be found in Section 2 of [11] and in the references therein.

Lemma 2.9

There exist constants $k>0$ and $B>0$ depending only on $\hat{\rho }$ such that

$$\begin{aligned} \forall r>0,\, \forall s>0,\quad (r+s)^{\hat{\rho }(r+s)}\le k(r^{\hat{\rho }(r)}+s^{\hat{\rho }(s)})+B. \end{aligned}$$

Precisely speaking, we can choose k depending only on the order $\rho =\lim _{r\rightarrow +\infty } \hat{\rho }(r)$

Lemma 2.10

The sequence $\{G_p\}_p$ is super multiplicative, that is,

$$\begin{aligned} G_pG_q\le G_{p+q},\quad \text {for any } p,\, q\in {\mathbb {N}}. \end{aligned}$$

Lemma 2.11

For every $\delta >0$ with $\delta <\frac{1}{\rho }$, there exists $T_0>0$ such that if $t\ge T_0$, we have

$$\begin{aligned} \left( \frac{1}{\rho }-\delta \right) \frac{\textrm{d}}{\textrm{d}t} \ln (t)<\frac{\textrm{d}}{\textrm{d}t}\ln \varphi (t)<\left( \frac{1}{\rho }+\delta \right) \frac{d}{\textrm{d}t} \ln (t). \end{aligned}$$

Lemma 2.12

For $u,\, t,\, \sigma >0$ we define

$$\begin{aligned} y_\sigma (u,t):=\ln \frac{\varphi (t)}{\varphi (u)}-\sigma \frac{t}{u}. \end{aligned}$$

Then for any $\sigma '$ with $0<\sigma '<\sigma$, there exists $T_1$ such that

$$\begin{aligned} y_\sigma (u,t)+\frac{1}{\rho }\ln (e\rho )\le -\frac{1}{\rho }\ln (\sigma '),\quad \text {for any } u,\, t\ge T_1. \end{aligned}$$

Keeping in mind the above results we can now introduce the notion of monogenic functions of proximate order and we can study some properties.

3 Some Properties of Monogenic Functions of Proximate Order

In the following we will use some results on monogenic entire functions contained in [18,19,20]. We prove some new properties of entire slice monogenic functions that appear here for the first time to the best of the knowledged of the authors. Some of the difficulties in proving our results rely on the series expansion of these functions in terms of the Fueter polynomials.

Lemma 3.1

If $\sigma _2>\sigma _1>0$, then the inclusion map is compact.

Proof

We will show that $B:=\{f\in A_{\rho ,\sigma _1}:\, \Vert f\Vert _{\rho ,\sigma _1}\le 1\}$ is relatively compact in $A_{\rho ,\sigma _2}$, i.e., any sequence $\{f_j\}_{j\in {\mathbb {N}}}\subset B$ has an accumulation point with respect to the norm of $A_{\rho ,\sigma _2}$.

First, we will prove that any sequence $\{f_j\}_{j\in {\mathbb {N}}}\subset B$ admits a convergent subsequence in the uniform convergence topology on compact subsets to an entire monogenic function. By the Arzelá–Ascoli Theorem it is sufficient to prove that $\{f_j\}_{j\in {\mathbb {N}}}$ is equicontinuous and uniformly bounded in any convex compact set $K\subseteq {\mathbb {R}}^{n+1}$. We fix a compact subset K of ${\mathbb {R}}^{n+1}$. Since $\{f_j\}_{j\in {\mathbb {N}}}\subset B$, the sequence is uniformly bounded in K. Moreover, we have

$$\begin{aligned} \Vert f_j(\textbf{x})-f_j(\textbf{y})\Vert \le C_j\Vert \textbf{x}-\textbf{y}\Vert , \end{aligned}$$

where $\nabla$ is the usual gradient, $C_j=\sup _{\textbf{x}\in K}\Vert \nabla f_j(\textbf{x})\Vert$ and $\textbf{x},\, \textbf{y}\in K$. We choose r large enough in a such way that $K\subset B(0,r)$. Thus, there exists a positive constant $C_K$ which only depends on K such that for any $\textbf{x}\in K$ and for any $j\in {\mathbb {N}}$, we have

$$\begin{aligned} \Vert \nabla f_j(\textbf{x})\Vert= & {} \left\| \frac{1}{A_{n+1}}\int _{\partial B(0,r)}\nabla q_0(\textbf{x}-\xi )\textrm{d}\tau (\xi ) f_j(\xi )\right\| \\\le & {} C_1 \int _{\partial B(0,r)} \frac{1}{\Vert \textbf{x}-\xi \Vert ^{1+n}} |\textrm{d}\tau (\xi )| M(r,f_j)\\\le & {} C_2 \frac{M(r,f_j)}{{\text {dist}}(\textbf{x},\partial B(0,r))^{1+n}}\le C_K. \end{aligned}$$

where $C_1$ and $C_2$ are suitable positive constants while in the first inequality we have used (2.1). In particular, for any $\textbf{x},\, \textbf{y}\in K$ and for any $j\in {\mathbb {N}}$ we have

$$\begin{aligned} \Vert f_j(\textbf{x})-f_j(\textbf{y})\Vert \le C_K\Vert \textbf{x}-\textbf{y}\Vert , \end{aligned}$$

i.e., $\{f_j\}_{j\in {\mathbb {N}}}$ is equicontinuous. After taking a subsequence if necessary, we can suppose that the sequence $\{f_j\}_{j\in {\mathbb {N}}}$ converges to $f\in {\mathcal {M}}_L({\mathbb {R}}^{n+1})$ in the topology of the uniform convergence on compact subsets. Now we will prove that the sequence $\{f_j\}_{j\in {\mathbb {N}}}$ is a Cauchy sequence in $A_{\rho ,\sigma _2}$. We fix $\delta >0$ and we observe that for any $R>0$ we have

$$\begin{aligned}{} & {} \sup _{\textbf{x}\in {\mathbb {R}}^{n+1}}\Vert f_j(\textbf{x})-f_\ell (\textbf{x})\Vert \exp (-\sigma _2\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )})\\{} & {} \quad =\max \left\{ \sup _{\Vert \textbf{x}\Vert \le R}\Vert f_j(\textbf{x})-f_\ell (\textbf{x})\Vert \exp (-\sigma _2\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )}),\right. \\{} & {} \qquad \left. \sup _{\Vert \textbf{x}\Vert \ge R}\Vert f_j(\textbf{x})-f_\ell (\textbf{x})\Vert \exp (-\sigma _2\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )})\right\} . \end{aligned}$$

With respect to the second supremum, we can choose $R>0$ large enough such that:

$$\begin{aligned} \exp ((\sigma _1-\sigma _2)\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )})\le \frac{\delta }{2}, \ \ \ \mathrm{for\ any}\ \Vert \textbf{x}\Vert \ge R. \end{aligned}$$

Thus, since $f_j,\, f_\ell \in B$, we have that

$$\begin{aligned}{} & {} \sup _{\Vert \textbf{x}\Vert \ge R}\Vert f_j(\textbf{x})-f_\ell (\textbf{x})\Vert \exp (-\sigma _2\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )})\\{} & {} \quad =\sup _{\Vert \textbf{x}\Vert \ge R}\Vert f_j(\textbf{x})-f_\ell (\textbf{x})\Vert \exp (-\sigma _1\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )}) \exp ((\sigma _1-\sigma _2)\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )})\\{} & {} \quad \le 2\cdot \frac{\delta }{2}=\delta . \end{aligned}$$

Moreover, with respect to the first supremum, by the uniform convergence of the sequence $\{f_j\}_{j\in {\mathbb {N}}}$ on the compact subset of ${\mathbb {R}}^{n+1}$, there exists a positive integer N such that for any $j,\, \ell \ge N$ we have

$$\begin{aligned} \sup _{\Vert \textbf{x}\Vert \le R}\Vert f_j(\textbf{x})-f_\ell (\textbf{x})\Vert \exp (-\sigma _2\Vert \textbf{x}\Vert ^{\rho (\Vert \textbf{x}\Vert )})\le \sup _{\Vert \textbf{x}\Vert \le R}\Vert f_j(\textbf{x})-f_\ell (\textbf{x})\Vert \le \delta . \end{aligned}$$

Thus we have proved that the sequence $\{f_j\}_{j\in {\mathbb {N}}}$ is a Cauchy sequence. $\square$

Definition 3.2

(The spaces $A_\rho$ and $A_{\rho ,\sigma +0}$) We define the space

$$\begin{aligned} A_\rho :=\lim _{\underset{\sigma >0}{\rightarrow }} A_{\rho ,\sigma } \end{aligned}$$

i.e. $A_{\rho }=\cup _{\sigma >0}A_{\rho ,\sigma }$ and we say that a sequence $\{f_j\}_{j\in {\mathbb {N}}}\subseteq A_{\rho }$ converges to $f\in A_\rho$ if there exists $\sigma >0$ such that $\{f_j\}_{j\in {\mathbb {N}}}\subseteq A_{\rho ,\sigma }$, $f\in A_{\rho ,\sigma }$ and $\lim _{j\rightarrow +\infty } \Vert f_j-f\Vert _{\rho ,\sigma }=0$.

We also define the space

$$\begin{aligned} A_{\rho ,\sigma +0}:=\lim _{\underset{\epsilon >0}{\leftarrow }} A_{\rho , \sigma +\epsilon } \end{aligned}$$

i.e., $A_{\rho ,\sigma +0}:=\cap _{\epsilon >0} A_{\rho ,\sigma +\epsilon }$ and we say that a sequence $\{f_j\}_{j\in {\mathbb {N}}}\subseteq A_{\rho ,\sigma +0}$ converges to $f\in A_{\rho ,\sigma +0}$ if for any $\epsilon >0$ we have $\{f_j\}_{j\in {\mathbb {N}}}\subseteq A_{\rho ,\sigma +\epsilon }$, $f\in A_{\rho ,\sigma +\epsilon }$ and $\lim _{j\rightarrow +\infty } \Vert f_j-f\Vert _{\rho ,\sigma +\epsilon }=0$.

Remark 3.3

The spaces $A_\rho$ and $A_{\hat{\rho }}$ coincide with each other and they share the same locally convex topologies as well, and the same holds for $A_{\rho ,\sigma +0}$ and $A_{\hat{\rho },\sigma +0}$.

For any fixed $f(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } V_\textbf{m}(\textbf{x}) a_\textbf{m}\in {\mathcal {M}}_L({\mathbb {R}}^{n+1})$ we define

$$\begin{aligned} K_q:=\sup _{\Vert \textbf{x}\Vert \le 1}\left\| \sum _{|\textbf{m}|=q} V_\textbf{m}(\textbf{x})a_\textbf{m}\right\| \end{aligned}$$

and

$$\begin{aligned} P_q({\textbf{x}}):=\sum _{|{\textbf{m}}|=q} V_{\textbf{m}}({\textbf{x}})a_{\textbf{m}}. \end{aligned}$$

Theorem 3.4

Let $f(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } V_\textbf{m}(\textbf{x}) a_\textbf{m}\in {\mathcal {M}}_L({\mathbb {R}}^{n+1})$ be a left monogenic entire function of finite order $\rho >0$ and of proximate order $\rho (r)$. Then its type $\sigma$ with respect to $\rho (r)$ is given by

$$\begin{aligned} \frac{1}{\rho }\ln (\sigma )=\limsup _{q\rightarrow \infty }\left( \frac{1}{q}\ln K_q+\ln \varphi (q)\right) -\frac{1}{\rho }-\frac{\ln (\rho )}{\rho }. \end{aligned}$$

(3.1)

Proof

First we prove that

$$\begin{aligned} \frac{1}{\rho }\ln (\sigma )\ge \limsup _{q\rightarrow \infty }\left( \frac{1}{q}\ln K_q+\ln \varphi (q)\right) -\frac{1}{\rho }-\frac{\ln (\rho )}{\rho }. \end{aligned}$$

As we have already defined, we put

$$\begin{aligned} P_q(\textbf{x})=\sum _{|\textbf{m}|=q} V_{\textbf{m}}(\textbf{x})a_{\textbf{m}} \end{aligned}$$

and suppose that $\textbf{w}_q\in {\mathbb {R}}^{n+1}$ be such that $\Vert P_q(\textbf{w}_q)\Vert =K_q$ and $\Vert \textbf{w}_q\Vert =1$. There exists $\textbf{b}\in {\mathbb {R}}_{n}$ (see [25, Theorem 3.20]) such that

• $\textbf{b}=\textbf{z}_1\cdots \textbf{z}_r$ where $\textbf{z}_i\in {\mathbb {R}}^{n+1}$ and $\Vert \textbf{z}_i\Vert =1$ for any $i=1,\dots r$,

• $\textbf{w}_q:= \textbf{b}(x\textbf{e}_1) {\bar{\textbf{b}}}$ for some $x\in {\mathbb {R}}$ with $|x|\le 1$ where ${\bar{\textbf{b}}}:= {\bar{\textbf{z}}}_r\cdots {\bar{\textbf{z}}}_1$.

The function $g(\textbf{x}):={\bar{\textbf{b}}} f(\textbf{b}\textbf{x}{\bar{\textbf{b}}})$ is a left monogenic entire function (see [34] for the case $\textbf{x}\in {\mathbb {R}}^{n+1}$ or [29] for the more general case $\textbf{x}\in {\mathbb {C}}^{n+1}$) whose Taylor series is

$$\begin{aligned} g(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } V_\textbf{m}(\textbf{x}) a_\textbf{m}'. \end{aligned}$$

We define

$$\begin{aligned} {\tilde{P}}_q(\textbf{x}):=\sum _{|\textbf{m}|=q} V_{\textbf{m}}(\textbf{x})a_\textbf{m}'. \end{aligned}$$

Thus we have

$$\begin{aligned} K_q= & {} \Vert P_q(\textbf{w}_q)\Vert =\Vert {\tilde{P}}_q(x\textbf{e}_1)\Vert =\Vert x^q a_{\textbf{m}_q}'\Vert \le \Vert a_{\textbf{m}_q}'\Vert \le \frac{c(n,\textbf{m}_q) M(r,g)}{r^q}\\= & {} \frac{c(n,\textbf{m}_q) M(r,f)}{r^q}, \end{aligned}$$

where $\textbf{m}_q=(q,0,\dots ,0)$. If ${\tilde{\sigma }}>\sigma$ then for r large we have:

$$\begin{aligned} M(r,f)\le \exp \left( {\tilde{\sigma }} r^{\rho (r)}\right) , \end{aligned}$$

and

$$\begin{aligned} \ln (K_q)\le \ln (c(n,\textbf{m}_q))+{\tilde{\sigma }} r^{\rho (r)}-q\ln (r). \end{aligned}$$

(3.2)

If q is large enough, then we define $r_q$ to be the real number such that $q={\tilde{\sigma }} \cdot \rho \cdot (r_q)^{\rho (r_q)}$ and we have $\varphi \left( \frac{q}{{\tilde{\sigma }} \rho }\right) =r_q$. Thus, for q large enough, dividing by q and summing $\log (\varphi (q))$ to both side of inequality (3.2), we have

$$\begin{aligned} \ln \left( \varphi (q)K_q^{\frac{1}{q}}\right) <\ln \left( c(n,\textbf{m}_q)^{\frac{1}{q}}\right) +\frac{1}{\rho }+\ln \left( \frac{\varphi (q)}{\varphi \left( \frac{q}{{\tilde{\sigma }} \rho }\right) }\right) . \end{aligned}$$

(3.3)

By [28, (1) Theorem 1.23] we have

$$\begin{aligned} \lim _{q\rightarrow +\infty }\frac{\varphi (q)}{\varphi \left( \frac{q}{{\tilde{\sigma }}\rho }\right) }=({\tilde{\sigma }} \rho )^{\frac{1}{\rho }}. \end{aligned}$$

Moreover, since $\lim _{q\rightarrow +\infty } \left( c(n,\textbf{m}_q)\right) ^{\frac{1}{q}}=1$, taking the $\limsup$ to both side of (3.3), we have

$$\begin{aligned} \limsup _{q\rightarrow +\infty } \ln \left( \varphi (q) K_q^{\frac{1}{q}} \right) \le \ln \left( (e{\tilde{\sigma }} \rho )^{\frac{1}{\rho }}\right) \end{aligned}$$

for any ${\tilde{\sigma }}>\sigma$. Thus we have proved that

$$\begin{aligned} \limsup _{q\rightarrow +\infty } \ln \left( \varphi (q) K_q^{\frac{1}{q}} \right) \le \ln \left( (e\sigma \rho )^{\frac{1}{\rho }}\right) . \end{aligned}$$

The other side of the inequality follows as in [28, (3) Theorem 1.23] by the properties of $\varphi$. $\square$

Lemma 3.5

A left monogenic entire function $f(\textbf{x})=\sum _{|\textbf{m}|=0}^{+\infty }V_\textbf{m}(\textbf{x}) a_\textbf{m}\in {\mathcal {M}}_L({\mathbb {R}}^{n+1})$ belongs to $A_{\rho ,\sigma +0}$ if and only if we have

$$\begin{aligned} \limsup _{q\rightarrow \infty }(K_qG_{\hat{\rho },q})^{\frac{\rho }{q}}\le \sigma . \end{aligned}$$

Proof

We have that $f\in A_{\rho ,\sigma +0}$ if and only if for any $\epsilon >0$ there exists a $D_\epsilon >0$ such that

$$\begin{aligned} \Vert f({\textbf{x}})\Vert \le D_\epsilon \exp ((\sigma +\epsilon ) \Vert {\textbf{x}}\Vert ^{\rho ({\textbf{x}})}) \quad \text {for all } {\textbf{x}}\in {\mathbb {R}}^{n+1}. \end{aligned}$$

This is equivalent to

$$\begin{aligned} \limsup _{r\rightarrow +\infty } \frac{\sup _{\Vert {\textbf{x}}\Vert \le r} \ln (\Vert f({\textbf{x}})\Vert )}{r^{\rho (r)}} \le \sigma . \end{aligned}$$

(3.4)

Since

$$\begin{aligned} \limsup _{r\rightarrow +\infty } \frac{\sup _{\Vert {\textbf{x}}\Vert \le r} \ln (\Vert f({\textbf{x}})\Vert )}{r^{\rho (r)}} \end{aligned}$$

is the type of f with respect to $\rho (r)$, by Theorem 3.4 and inequality (3.4) we have

$$\begin{aligned} \limsup _{q\rightarrow +\infty } \left( \frac{\rho }{q} \ln (K_q)+\rho \ln (\varphi (q)) \right) -\ln (e\rho )\le \ln (\sigma ). \end{aligned}$$

Finally, we can conclude that

$$\begin{aligned} \limsup _{q\rightarrow +\infty }\, \ln \left( K_q^{\rho /q}\frac{\varphi (q)^\rho }{e\rho } \right) \le \ln (\sigma ) \end{aligned}$$

which gives

$$\begin{aligned} \limsup _{q\rightarrow +\infty }\left( K_q \frac{\varphi (q)^q}{(e\rho )^{q/\rho }} \right) ^{\rho /q}=\limsup _{q\rightarrow +\infty }\left( K_q G_{\hat{\rho },q} \right) ^{\rho /q}\le \sigma . \end{aligned}$$

$\square$

Corollary 3.6

If $f(\textbf{x})=\sum _{|\textbf{m}|=q}V_\textbf{m}(\textbf{x}) a_\textbf{m}\in {\mathcal {M}}_L({\mathbb {R}}^{n+1})$ belongs to $f\in A_{\rho , \sigma +0}$, then we have

$$\begin{aligned} \limsup _{q\rightarrow \infty }\left( \max _{|\textbf{m}|=q } \Vert a_\textbf{m}\Vert G_{\hat{\rho },q}\right) ^{\rho /q}\le n^\rho \sigma . \end{aligned}$$

(3.5)

Conversely, if $\{a_\textbf{m}\}$ satisfies (3.5), then we have $f(\textbf{x})\in A_{\rho , n^\rho \sigma +0}$.

Proof

We have

$$\begin{aligned} \Vert a_{\textbf{m}}\Vert= & {} \frac{1}{{\textbf{m}}!}\Vert \partial _{\textbf{x}}^{\textbf{m}}P_q(0)\Vert \le \frac{c(n,{\textbf{m}}) \sup _{\Vert x\Vert \le r} \Vert P_q({\textbf{x}})\Vert }{r^q}\\= & {} c(n,{\textbf{m}}) \sup _{\Vert x\Vert \le 1} \Vert P_q({\textbf{x}})\Vert =c(n,{\textbf{m}}) K_q. \end{aligned}$$

Thus we get that for any $\epsilon >0$ there exists $q_0>0$ such that for any $q>q_0$ we have

$$\begin{aligned} \left( \max _{|{\textbf{m}}|=q} \Vert a_{\textbf{m}}\Vert G_{\rho ,q} \right) ^{\rho /q}\le & {} \left( \max _{|{\textbf{m}}|=q} c(n,{\textbf{m}}) K_q G_{\rho , q} \right) ^{\rho /q}\\\le & {} \left( \left( \sum _{|{\textbf{m}}|=q} c(n,{\textbf{m}})\right) ^{1/q}\right) ^\rho (K_q G_{\rho , q})^{\rho /q} \\\le & {} (n+\epsilon )^\rho (\sigma +\epsilon ), \end{aligned}$$

where the last inequality is a consequence of the last inequality in the proof of Lemma 3.5 and the fact that: $\limsup _{q\rightarrow +\infty }\left( \sum _{|{\textbf{m}}|=q} c(n,{\textbf{m}})\right) ^{1/q} =n$, see Lemma 1 in [19]. In conclusion, we have

$$\begin{aligned} \limsup _{q\rightarrow +\infty } \, \left( \max _{|{\textbf{m}}|=q} \Vert a_{\textbf{m}}\Vert G_{\rho , q} \right) ^{\rho /q}\le n^\rho \sigma . \end{aligned}$$

Conversely, if we have (3.5), then for any $\epsilon >0$ there exists $C>0$ and $q_0>0$ such that for any $q>q_0$ we have

$$\begin{aligned} K_q= & {} \sup _{\Vert \textbf{x}\Vert \le 1} \Vert P_q({\textbf{x}})\Vert \le \sup _{\Vert \textbf{x}\Vert \le 1} \sum _{\Vert {\textbf{m}}\Vert =q} \Vert a_{\textbf{m}}\Vert \Vert {\textbf{x}}\Vert ^q \le \max _{|{\textbf{m}}|=q} \Vert a_{\textbf{m}}\Vert G_{\rho ,q} \frac{(q+1)^{n-1}}{G_{\rho ,q}} \\\le & {} (n^\rho \sigma +\epsilon )^{q/\rho } \frac{(q+1)^{n-1}}{G_{\rho ,q}}. \end{aligned}$$

Thus, we have

$$\begin{aligned} \limsup _{q\rightarrow +\infty }(K_q G_{\rho ,q})^{\rho /q}\le n^\rho \sigma \end{aligned}$$

and $f\in A_{\rho ,n^\rho \sigma +0}$. $\square$

The following proposition is a direct consequence of the previous corollary.

Proposition 3.7

A left monogenic entire function $f(\textbf{x})=\sum _{|\textbf{m}|=q}V_\textbf{m}(\textbf{x}) a_\textbf{m}\in {\mathcal {M}}_L({\mathbb {R}}^{n+1})$ belongs to $A_\rho$ if and only if we have

$$\begin{aligned} \limsup _{q\rightarrow \infty }\left( \max _{|\textbf{m}|=q } \Vert a_\textbf{m}\Vert G_{\hat{\rho },q}\right) ^{\rho /q}< \infty . \end{aligned}$$

We now need some estimates of norms of Fueter polynomials.

Lemma 3.8

Suppose $\rho >0$. For any $\sigma$ and $\sigma '$ with $0<\sigma '<\sigma$, there exists $C>0$ such that for any $\textbf{m}\in {\mathbb {N}}^n_0$, we have

$$\begin{aligned} \Vert V_\textbf{m}(\textbf{x})\Vert _{\rho ,\sigma }\le C\sigma '^{-|\textbf{m}|/\rho }G_{\hat{\rho },|\textbf{m}|}. \end{aligned}$$

Proof

Since the norms $\Vert \cdot \Vert _{\rho ,\sigma }$ and $\Vert \cdot \Vert _{{\hat{\rho }},\sigma }$ are equivalent, we may assume from the beginning that ${\hat{\rho }}(r)$ is as in Definition 2.8 and ${\hat{\rho }}(r)=\rho (r)$.

First we prove, for sufficiently large $r=\Vert {\textbf{x}}\Vert$ and $q=|{\textbf{m}}|$, that

$$\begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}} \Vert \le (\sigma ')^{-|{\textbf{m}}|/\rho } G_{\hat{\rho },|{\textbf{m}}|}. \end{aligned}$$

Let $\varphi$ be the inverse function of $t=r^{\rho (r)}$. We define $r=\varphi (t)=\Vert {\textbf{x}}\Vert$. Thus we have

$$\begin{aligned} \begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}}\Vert G^{-1}_{{\hat{\rho }}, q}&\le r^qe^{-\sigma r^{\rho (r)}} G^{-1}_{{\hat{\rho }}, q}\\&=\exp \left( q\left( \ln (\varphi (t))-\sigma \frac{t}{q}-\frac{1}{q}\ln (G_{{\hat{\rho }},q})\right) \right) \\&=\exp \left( q\left( \ln (\varphi (t))-\sigma \frac{t}{q}-\ln (\varphi (q))+\frac{1}{\rho }\ln (e\rho )\right) \right) \\&=\exp \left( q\left( \ln \left( \frac{\varphi (t)}{\varphi (q)} \right) -\sigma \frac{t}{q}+\frac{1}{\rho }\ln (e\rho )\right) \right) \\ \end{aligned} \end{aligned}$$

By Lemma 2.12, for any $0<\sigma '<\sigma$ there exists $T_1\ge 0$ such that

$$\begin{aligned} \ln \left( \frac{\varphi (t)}{\varphi (q)} \right) -\sigma \frac{t}{q}+\frac{1}{\rho }\ln (e\rho )\le -\frac{1}{\rho }\ln (\sigma ')\quad \text {for any } q,t\ge T_1. \end{aligned}$$

This implies

$$\begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}}\Vert G^{-1}_{{\hat{\rho }},q} \le \exp \left( -\frac{q}{\rho }\ln \sigma ' \right) =(\sigma ')^{-q/\rho } \end{aligned}$$

(3.6)

for $|{\textbf{m}}|=q\ge T_1$ and $t\ge T_1$ (i.e. $\Vert {\textbf{x}}\Vert \ge \varphi (T_1)$). Next we consider the case $\Vert {\textbf{x}}\Vert \le \varphi (T_1)$ (i.e. $t\le T_1$) and $q=|{\textbf{m}}|>>1$. We have

$$\begin{aligned} \begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}}\Vert G^{-1}_{{\hat{\rho }}, q}&\le r^qe^{-\sigma r^{\rho (r)}} G^{-1}_{{\hat{\rho }}, q}\\&=\exp \left( q\left( \ln \left( \frac{\varphi (t)}{\varphi (q)} \right) -\sigma \frac{t}{q}+\frac{1}{\rho }\ln (e\rho )\right) \right) \\&\le \exp \left( q\left( \ln \left( \frac{\varphi (T_1)}{\varphi (q)} \right) +\frac{1}{\rho } \ln (e\rho )\right) \right) . \end{aligned} \end{aligned}$$

Note that for any given $\sigma '>0$, we can take $T_2\ge T_1$:

$$\begin{aligned} \ln \left( \frac{\varphi (T_1)}{\varphi (T_2)} \right) +\frac{1}{\rho }\ln (e\rho )\le -\frac{1}{\rho }\ln (\sigma '). \end{aligned}$$

Thus we have

$$\begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}}\Vert G^{-1}_{{\hat{\rho }},q} \le (\sigma ')^{-q/\rho } \end{aligned}$$

(3.7)

for $|{\textbf{m}}|=q\ge T_2$ and $\Vert {\textbf{x}}\Vert \le \varphi (T_1)$. By (3.6) and (3.7) for any ${\textbf{x}}\in {\mathbb {R}}^{n+1}$ and for any $|{\textbf{m}}|=q\ge T_2$, we have

$$\begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}}\Vert G^{-1}_{{\hat{\rho }},q} \le (\sigma ')^{-q/\rho }. \end{aligned}$$

We know that there exists $C>0$ such that $\Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}}\Vert \le C$ for any $|{\textbf{m}}|<T_2$ and for any ${\textbf{x}}\in {\mathbb {R}}^{n+1}$. Thus

$$\begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}}) e^{-\sigma r^{\rho (r)}}\Vert G^{-1}_{{\hat{\rho }},q} \le C' (\sigma ')^{-q/\rho } \end{aligned}$$

where

$$\begin{aligned} C'=\sup _{|{\textbf{m}}|\le T_2}\left( 1, \frac{C}{(\sigma ')^{-q/\rho } G_{{\hat{\rho }}, q}}\right) . \end{aligned}$$

$\square$

Lemma 3.9

There exists a constant k depending only on $\rho$ for which the following statement holds: For any $\sigma >0$, we can take $C(\sigma )$ such that for any $f\in A_{\hat{\rho },\sigma }$, and any $\textbf{m}\in {\mathbb {N}}_0^n$, the inequality

$$\begin{aligned} \frac{1}{\textbf{m}!} \Vert \partial ^\textbf{m}_{\textbf{x}} f(\textbf{x})\Vert _{\hat{\rho }, k\sigma }\le C(\sigma )\Vert f\Vert _{\hat{\rho },\sigma }\frac{(2k(c(n,{\textbf{m}}))^{\rho /q}\sigma )^{q/\rho }}{G_{\hat{\rho },q}} \end{aligned}$$

holds. Here we write $q=|\textbf{m}|$.

Proof

There exists a constant $k>0$ which depends on $\rho$ and there exists a constant $B>0$ which depends on ${\hat{\rho }}(r)$ such that

$$\begin{aligned} (r+s)^{{\hat{\rho }}(r+s)}\le k(r^{{\hat{\rho }}(r)}+s^{{\hat{\rho }}(s)})+B\quad \text {for all } r,s>0. \end{aligned}$$

The Cauchy estimates give us, for $\Vert {\textbf{x}}\Vert \le r$ and $|{\textbf{m}}|=q$ the chain of inequalities

$$\begin{aligned} \begin{aligned} \frac{\Vert \partial ^{\textbf{m}}_{\textbf{x}}f({\textbf{x}})\Vert }{{\textbf{m}}!}&\le \inf _{s>0} \frac{c(n,{\textbf{m}}) \max _{|\xi |=s} \Vert f({\textbf{x}}+\xi )\Vert }{s^{|{\textbf{m}}|}} \\&\le \Vert f\Vert _{{\hat{\rho }},\sigma } \inf _{s>0} \frac{c(n,{\textbf{m}})}{s^q} \exp \left( \sigma (r+s)^{{\hat{\rho }}(r+s)} \right) \\&\le \Vert f\Vert _{{\hat{\rho }},\sigma } \inf _{s>0} \frac{c(n,{\textbf{m}})}{s^q} \exp \left( k\sigma (r^{{\hat{\rho }}(r)}+s^{{\hat{\rho }}(s)}) + B\sigma \right) \\&= e^{B\sigma }\Vert f\Vert _{{\hat{\rho }},\sigma } c(n,{\textbf{m}}) \exp \left( k\sigma r^{{\hat{\rho }}(r)}\right) \inf _{s>0} \frac{ \exp \left( k\sigma s^{{\hat{\rho }}(s)}\right) }{s^q} \\&\le e^{B\sigma }\Vert f\Vert _{{\hat{\rho }},\sigma } \exp \left( k\sigma r^{{\hat{\rho }}(r)}\right) \frac{\left( c(n,{\textbf{m}})^{\rho /q} e\right) ^{q/\rho }}{\varphi (q)^q} \left( \frac{\varphi (q)}{\varphi (q/(k\sigma \rho ))} \right) ^q \\&\le C(\sigma )\Vert f\Vert _{{\hat{\rho }},\sigma } \exp \left( k\sigma r^{{\hat{\rho }}(r)}\right) \frac{\left( 2k c(n,{\textbf{m}})^{\rho /q} \sigma \right) ^{q/\rho }}{G_{{\hat{\rho }},q}}, \end{aligned} \end{aligned}$$

where the last two inequalities are obtained as in [11, Lemma 3.8]. In particular, we have

$$\begin{aligned} \frac{\Vert \partial _\textbf{x}^\textbf{m}f(\textbf{x})\Vert }{\textbf{m}!} \exp \left( k\sigma \Vert \textbf{x}\Vert ^{\hat{\rho }(\Vert \textbf{x}\Vert )} \right) \le C(\sigma )\Vert f\Vert _{{\hat{\rho }},\sigma } \frac{\left( 2k c(n,{\textbf{m}})^{\rho /q} \sigma \right) ^{q/\rho }}{G_{{\hat{\rho }},q}}, \end{aligned}$$

which implies the inequality in the statement. $\square$

Remark 3.10

Observe that since

$$\begin{aligned} \limsup _{q\rightarrow +\infty } \left( \sum _{|{\textbf{m}}|=q} c(n,{\textbf{m}}) \right) ^{1/q}=n \end{aligned}$$

there exists a constant C(n) which depends only on n such that for any ${\textbf{m}}\in {\mathbb {N}}^n$ we have

$$\begin{aligned} (c(n,{\textbf{m}}))^{1/|{\textbf{m}}|}\le C(n). \end{aligned}$$

Thus, in Lemma 3.9 the estimate can be rewritten in the following way

$$\begin{aligned} \frac{1}{\textbf{m}!} \Vert \partial ^\textbf{m}_{\textbf{x}} f(\textbf{x})\Vert _{\hat{\rho }, k\sigma }\le C(\sigma )\Vert f\Vert _{\hat{\rho },\sigma }\frac{(2k(C(n))^{\rho }\sigma )^{q/\rho }}{G_{\hat{\rho },q}}. \end{aligned}$$

In view of the above-stated properties, we can now prove the following crucial results.

Proposition 3.11

For an entire left monogenic function $f(\textbf{x})$ belonging to $A_{\rho ,\sigma +0}$, its Taylor expansion $\sum _{\textbf{m}\in {\mathbb {N}}^n}V_\textbf{m}(\textbf{x}) a_\textbf{m}$ converges to $f(\textbf{x})$ in the space $A_{\rho ,n^\rho \sigma +0}$. In particular, the set of Fueter polynomials is dense in $A_{\rho ,+0}$ and also dense in $A_\rho$.

Proof

For the former statement, it suffices to show that

$$\begin{aligned} \sum _{{\textbf{m}}\in {\mathbb {N}}^n} \Vert V_{\textbf{m}}({\textbf{x}}) a_{\textbf{m}}\Vert _{\rho ,n^\rho (\sigma +\epsilon )} \end{aligned}$$

is finite for any $\epsilon >0$. By Lemma 3.8 we have

$$\begin{aligned} \Vert V_{\textbf{m}}({\textbf{x}})\Vert _{\rho ,n^\rho (\sigma +\epsilon )}\le C_0(n^\rho (\sigma +\epsilon /2))^{-|{\textbf{m}}|/\rho } G_{{\hat{\rho }},|{\textbf{m}}|}. \end{aligned}$$

On the other hand, by Corollary 3.6, we have

$$\begin{aligned} \max _{|{\textbf{m}}|=q} \Vert a_{\textbf{m}}\Vert G_{{\hat{\rho }},q}\le C_1(n^\rho (\sigma +\epsilon /4))^{q/\rho }. \end{aligned}$$

Therefore, we have

$$\begin{aligned} \begin{aligned} \sum _{{\textbf{m}}\in {\mathbb {N}}^n} \Vert V_{\textbf{m}}({\textbf{x}})a_{\textbf{m}}\Vert _{\rho , n^\rho (\sigma +\epsilon )}&=\sum _{q\in {\mathbb {N}}}\sum _{|{\textbf{m}}|=q} \Vert a_{\textbf{m}}\Vert G_{{\hat{\rho }},q} \Vert V_{\textbf{m}}({\textbf{x}})\Vert _{\rho ,n^{\rho }(\sigma +\epsilon )} G^{-1}_{{\hat{\rho }},q}\\&\le \sum _{q\in {\mathbb {N}}}\sum _{|{\textbf{m}}|=q} C_0 (n^\rho (\sigma +\epsilon /2))^{-q/\rho } C_1 (n^\rho (\sigma +\epsilon /4))^{q/\rho }\\&\le C_0C_1 \sum _{q\in {\mathbb {N}}} (q+1)^{n-1} \left( \frac{\sigma +\epsilon /4}{\sigma +\epsilon /2} \right) ^{q/\rho }, \end{aligned} \end{aligned}$$

where the last series is convergent. For the latter statement in the case $f\in A_{\rho ,+0}$, it follows from the former one with $\sigma =0$ that

$$\begin{aligned} \lim _{q\rightarrow +\infty } \sum _{|{\textbf{m}}|\le q} V_{\textbf{m}}({\textbf{x}}) a_{\textbf{m}}=f({\textbf{x}}) \end{aligned}$$

in the space $A_{\rho ,+0}$. In the case $f\in A_\rho$, there exists $\sigma >0$ such that $f\in A_{\rho ,\sigma +0}$. Then the same convergence holds in the space $A_{\rho ,n^\rho \sigma +0}$ and therefore also in the space $A_\rho$. $\square$

Lemma 3.12

Let $f\in A_{\rho ,\sigma }$ and $f=\sum _{|\textbf{m}|=0}^\infty V_{\textbf{m}}(\textbf{x}) f_{\textbf{m}}$. Let s be a real positive number. Then, for any $\eta >0$ there exists $C_\eta >0$ such that for any $\textbf{m}\in {\mathbb {N}}^n_0$, we have

$$\begin{aligned} \Vert f_{\textbf{m}}\Vert \exp (-\sigma (1+\eta )(s+1)^\rho r^{\rho (r)})\le \exp (\sigma C_\eta ) \frac{\Vert f\Vert _{\rho ,\sigma } c(n,\textbf{m})}{(sr)^{|\textbf{m}|}}. \end{aligned}$$

Proof

We have that

$$\begin{aligned} \begin{aligned} \Vert f_{\textbf{m}}\Vert&\!=\!\frac{\Vert \partial ^\textbf{m}f (0)\Vert }{\textbf{m}!}\!\le \! \sup _{\Vert \textbf{x}\Vert =r}\frac{\Vert \partial ^\textbf{m}f(\textbf{x})\Vert }{\textbf{m}!}\!\le \! \sup _{\Vert \textbf{x}\Vert =r}\frac{ (\sup _{\Vert \zeta -\textbf{x}\Vert =s\Vert \textbf{x}\Vert } \Vert f(\zeta )\Vert ) c(n,\textbf{m})}{(sr)^{|\textbf{m}|}}\\&\le \exp (\sigma ((s+1)r)^{\rho ((s+1)r)})\\&\quad \times \frac{ \left( \sup _{\Vert \textbf{x}\Vert =(s+1)r} \Vert f(\textbf{x})\Vert \exp (-\sigma ((s+1)r)^{\rho ((s+1)r)})\right) c(n,\textbf{m})}{(sr)^{|\textbf{m}|}}\\&\le \exp (\sigma (1+\eta )(s+1)^\rho r^{\rho (r)}+\sigma C_\eta ) \frac{\Vert f\Vert _{\rho ,\sigma } c(n,\textbf{m})}{(sr)^{|\textbf{m}|}}, \end{aligned} \end{aligned}$$

where in the first inequality we have used the maximum modulus principle, in the second inequality we have used the Cauchy inequalities in the ball centered at $\textbf{x}$ with radius $s\Vert \textbf{x}\Vert$, in the third inequality we have used the fact that all the balls centered at $\textbf{x}$ with $\Vert \textbf{x}\Vert =r$ of radius $s\Vert \textbf{x}\Vert$ are contained in the ball centered at 0 with radius $(s+1)r$ and in the last inequality we have used the fact that for any $\eta >0$ there exists a positive constant $C_\eta$ such that

$$\begin{aligned} (kr)^{\rho (kr)} \le (1+\eta ) k^\rho r^{\rho (x)} + C_\eta , \end{aligned}$$

see [28, p. 16, Proposition 1.20]. $\square$

Lemma 3.13

Let $g_1(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } V_\textbf{m}(\textbf{x})a_{\textbf{m}}\in A_{\rho , \tau _1}$ and $g_2(\textbf{x})=\sum _{|\textbf{m}|=0}^{\infty } V_\textbf{m}(\textbf{x})b_{\textbf{m}}\in A_{\rho , \tau _2}$. Let $\delta$ be a positive constant then for any $\eta >0$ there exists $C(n, \eta ,\tau _1,\tau _2)>0$ such that

$$\begin{aligned} \Vert g_1\odot _{CK} g_2(\textbf{x})\Vert _{A_{\rho , (1+\eta )(n+\delta +1)^\rho (\tau _1+\tau _2)}}\le C(n, \eta ,\tau _1,\tau _2) \Vert g_1\Vert _{\rho ,\tau _1} \Vert g_2\Vert _{\rho ,\tau _2}. \end{aligned}$$

Proof

Choosing $s=(n+\delta )$ in Lemma 3.12, we have

$$\begin{aligned} \begin{aligned}&\Vert g_1\odot _{CK} g_2(\textbf{x})\Vert _{\rho , (1+\eta )(n+\delta +1)^\rho (\tau _1+\tau _2)} \le 2^n \sup _{\textbf{x}\in {\mathbb {R}}^{n+1}}\\&\qquad \times \left( \sum _{|\textbf{m}|=0}^\infty \sum _{|\mathbf{\ell }|=0}^{\infty } \Vert V_{\textbf{m}+\mathbf{\ell }} (\textbf{x})\Vert \Vert a_m\Vert \Vert b_\ell \Vert \exp (-(1+\eta )(n+\delta +1)^\rho (\tau _1+\tau _2) \Vert \textbf{x}\Vert ^{\rho (\textbf{x})})\right) \\&\quad \le C'(n,\eta ,\tau _1,\tau _2)\Vert g_1\Vert _{\rho ,\tau _1}\Vert g_2\Vert _{\rho ,\tau _2} \sum _{|\textbf{m}|=0}^\infty \sum _{|\ell |=0}^{\infty } \Vert x\Vert ^{|\textbf{m}|+|\ell |} \frac{c(n,\textbf{m})}{(n+\delta )^{|\textbf{m}|}\Vert \textbf{x}\Vert ^{|\textbf{m}|}}\\&\qquad \times \frac{c(n,\mathbf{\ell })}{ (n+\delta )^{|\mathbf{\ell }|} \Vert \textbf{x}\Vert ^{|\ell |}} \le C(n,\eta ,\tau _1,\tau _2)\Vert g_1\Vert _{\rho ,\tau _1}\Vert g_2\Vert _{\rho ,\tau _2}, \end{aligned} \end{aligned}$$

where in the last inequality we used the fact that

$$\begin{aligned} \sum _{|\textbf{m}|=0}^{\infty } \frac{c(n,\textbf{m})}{(n+\delta )^{|\textbf{m}|}}< \infty \end{aligned}$$

since

$$\begin{aligned} \limsup _{r\rightarrow \infty }\left( \frac{1}{(n+\delta )^r}\sum _{|m|=r} c(n,\textbf{m}) \right) ^{\frac{1}{r}}=\frac{n}{(n+\delta )}<1. \end{aligned}$$

$\square$

The results of this section will be used to characterize continuous homomorphisms in terms of differential operators in the sense that will be specified in the next session.

4 Differential Operators, Representations of Continuous Homomorphisms

Before studying continuous homomorphisms between $A_{\rho _i}$ $(i=1,\, 2)$ and those between $A_{\rho _i,+0}$ $(i=1,\, 2)$, we define a differential operator representation of homomorphisms from

$$\begin{aligned} {\mathbb {R}}_n [{\textbf{x}}]:=\left\{ \sum _{|{\textbf{m}}|=0}^q V_{{\textbf{m}}}({\textbf{x}}) a_{{\textbf{m}}}:\, a_{{\textbf{m}}}\in {\mathbb {R}}_n \quad \text {and}\quad q\in {\mathbb {N}}_0 \right\} \end{aligned}$$

to

$$\begin{aligned} {\mathbb {R}}_n [[{\textbf{x}}]]:=\left\{ \sum _{|{\textbf{m}}|=0}^\infty V_{{\textbf{m}}}({\textbf{x}}) a_{{\textbf{m}}}:\, a_{{\textbf{m}}}\in {\mathbb {R}}_n\right\} . \end{aligned}$$

In what follows, we abbreviate a right (left) ${\mathbb {R}}_n$ homomorphism under the CK-product as a right (left) linear mapping.

We define the space of formal right linear differential operators of infinite order with coefficients in ${\mathbb {R}}_n [[{\textbf{x}}]]$ by

$$\begin{aligned} \hat{D}:=\left\{ P=\sum _{{\textbf{m}}\in {\mathbb {N}}^n_0} u_{{\textbf{m}}}({\textbf{x}})\odot _L\partial ^{\textbf{m}}_{\textbf{x}}:\, u_{\textbf{m}}({\textbf{x}})\in {\mathbb {R}}_n [[{\textbf{x}}]] \right\} . \end{aligned}$$

Note that $\hat{D}$ is linearly isomorphic to $\prod _{{\textbf{m}}\in {\mathbb {N}}^n} {\mathbb {R}}_n [[{\textbf{x}}]]$, by the correspondence

$$\begin{aligned} \sum _{{\textbf{m}}\in {\mathbb {N}}^n_0} u_{{\textbf{m}}}({\textbf{x}}) \odot _L\partial ^{\textbf{m}}_{\textbf{x}}\mapsto (u_{{\textbf{m}}}({\textbf{x}}))_{{\textbf{m}}\in {\mathbb {N}}^n}. \end{aligned}$$

Proposition 4.1

There are two left linear isomorphisms:

$$\begin{aligned} \hat{D}\leftrightarrow {\text {Hom}}_{{\mathbb {R}}_n}({\mathbb {R}}_n [{\textbf{x}}],\, {\mathbb {R}}_n [[{\textbf{x}}]])\leftrightarrow \prod _{{\textbf{m}}\in {\mathbb {N}}^n} {\mathbb {R}}_n [[{\textbf{x}}]], \end{aligned}$$

where the first and second mappings are given by

$$\begin{aligned} \begin{aligned}&\sum _{{\textbf{m}}\in {\mathbb {N}}^n_0}\! u_{{\textbf{m}}}({\textbf{x}})\odot _L\partial ^{\textbf{m}}_{\textbf{x}}\mapsto \! \left( \! {\mathbb {R}}_n[{\textbf{x}}]\ni \!\sum _{{\textbf{m}}} V_{{\textbf{p}}}({\textbf{x}}) f_{{\textbf{p}}}\mapsto \! \sum _{{\textbf{m}}\le {\textbf{p}}} \frac{{\textbf{p}}!}{({\textbf{p}}\!-\!{\textbf{m}})!} u_{{\textbf{m}}} ({\textbf{x}})\odot _L V_{{\textbf{p}}-{\textbf{m}}} ({\textbf{x}}) f_{{\textbf{p}}} \!\right) ,\\&F\mapsto (F(V_{{\textbf{m}}} ({\textbf{x}}))/{\textbf{m}}!)_{{\textbf{m}}\in {\mathbb {N}}^n_0}, \end{aligned} \end{aligned}$$

respectively.

Proof

We can easily see that both mappings are injective left linear mappings between vector spaces and that their composition is given by

$$\begin{aligned} \sum _{{\textbf{m}}} u_{{\textbf{m}}}({\textbf{x}}) \odot _L \partial ^{\textbf{m}}_{\textbf{x}}\mapsto \left( \sum _{{\textbf{m}}\le {\textbf{p}}} u_{{\textbf{m}}}({\textbf{x}}) \odot _L \frac{V_{{\textbf{p}}-{\textbf{m}}}({\textbf{x}})}{({\textbf{p}}-{\textbf{m}})!}\right) _{{\textbf{p}}\in {\mathbb {N}}^n_0}. \end{aligned}$$

Therefore, it suffices to show that the relations

$$\begin{aligned} b_{\textbf{p}}({\textbf{x}})=\sum _{{\textbf{m}}\le {\textbf{p}}} u_{{\textbf{m}}}({\textbf{x}}) \odot _L \frac{V_{{\textbf{p}}-{\textbf{m}}}({\textbf{x}})}{({\textbf{p}}-{\textbf{m}})!},\quad {\textbf{p}}\in {\mathbb {N}}^n_0, \end{aligned}$$

(4.1)

induce a bijection

$$\begin{aligned} \prod _{{\textbf{m}}\in {\mathbb {N}}^n_0} {\mathbb {R}}_n [[{\textbf{x}}]] \ni (u_{{\textbf{m}}}({\textbf{x}}))_{{\textbf{m}}\in {\mathbb {N}}^n}\mapsto (b_{\textbf{p}}({\textbf{x}}))_{{\textbf{p}}\in {\mathbb {N}}^n} \in \prod _{{\textbf{p}}\in {\mathbb {N}}^n_0} {\mathbb {R}}_n[[{\textbf{x}}]]. \end{aligned}$$

This follows from the fact that the relation (4.1) can be inverted as

$$\begin{aligned} u_{{\textbf{m}}}({\textbf{x}})=\sum _{{\textbf{p}}\le {\textbf{m}}} b_{{\textbf{p}}}({\textbf{x}}) \odot _L \frac{V_{{\textbf{m}}-{\textbf{p}}}(-{\textbf{x}})}{({\textbf{m}}-{\textbf{p}})!}. \end{aligned}$$

(4.2)

In fact, we can calculate the ${\textbf{p}}$-element of the image of $(u_{{\textbf{m}}}({\textbf{x}}))_{{\textbf{m}}\in {\mathbb {N}}_0^n}$ by composition of (4.2) and (4.1) as

$$\begin{aligned} \begin{aligned}&\sum _{{\textbf{s}}\le {\textbf{p}}} \sum _{{\textbf{m}}\le {\textbf{s}}} \left( u_{{\textbf{m}}}({\textbf{x}})\odot _L \frac{V_{{\textbf{s}}-{\textbf{m}}}({\textbf{x}})}{({\textbf{s}}-{\textbf{m}})!} \right) \odot _L \frac{V_{{\textbf{p}}-{\textbf{s}}}(-{\textbf{x}})}{({\textbf{p}}-{\textbf{s}})!} \\&\quad =\sum _{{\textbf{s}}\le {\textbf{p}}} \sum _{{\textbf{m}}\le {\textbf{s}}} u_{{\textbf{m}}}({\textbf{x}})\odot _L \left( \frac{V_{{\textbf{s}}-{\textbf{m}}}({\textbf{x}})}{({\textbf{s}}-{\textbf{m}})!} \odot _L \frac{V_{{\textbf{p}}-{\textbf{s}}}(-{\textbf{x}})}{({\textbf{p}}-{\textbf{s}})!} \right) \\&\quad =\sum _{{\textbf{m}}\le {\textbf{p}}} u_{{\textbf{m}}}({\textbf{x}}) \odot _L \left( \sum _{{\textbf{m}}\le {\textbf{s}}\le {\textbf{p}}} \frac{V_{{\textbf{s}}-{\textbf{m}}}({\textbf{x}})\odot _L V_{{\textbf{p}}-{\textbf{s}}}(-{\textbf{x}})}{({\textbf{s}}-{\textbf{m}})!({\textbf{p}}-{\textbf{s}})!} \right) \\&\quad =\sum _{{\textbf{m}}\le {\textbf{p}}} u_{{\textbf{m}}}({\textbf{x}}) \odot _L \left( \sum _{{\textbf{m}}\le {\textbf{s}}\le {\textbf{p}}} \frac{({\textbf{p}}-{\textbf{m}})! (-1)^{|{\textbf{p}}-{\textbf{s}}|} V_{{\textbf{p}}-{\textbf{m}}}({\textbf{x}}) }{({\textbf{s}}-{\textbf{m}})!({\textbf{p}}-{\textbf{s}})!({\textbf{p}}-{\textbf{m}})!} \right) \\&\quad =\sum _{{\textbf{m}}\le {\textbf{p}}} u_{{\textbf{m}}}({\textbf{x}}) \odot _L \left( \frac{(1-1)^{|{\textbf{p}}-{\textbf{m}}|} V_{{\textbf{p}}-{\textbf{m}}}({\textbf{x}}) }{({\textbf{p}}-{\textbf{m}})!} \right) =u_{{\textbf{p}}}({\textbf{x}}), \\ \end{aligned} \end{aligned}$$

which implies that the composition is the identity. We can similarly show that the composition of (4.1) and (4.2) is the identity. $\square$

Now we study continuous homomorphism from $A_{\rho _1}$ and $A_{\rho _2}$ and those from $A_{\rho _1,+0}$ and $A_{\rho _2,+0}$ where $\rho _i(r)$ ($i=1,\,2$) are two proximate orders for positive orders $\rho _i=\lim _{r\rightarrow \infty }\rho _i(r)>0$, satisfying

$$\begin{aligned} r^{\rho _1(r)}=O(r^{\rho _2(r)}),\quad \text {as } r\rightarrow \infty . \end{aligned}$$

(4.3)

Definition 4.2

Let $\rho _i$ ($i=1,\, 2$) be two proximate orders for orders $\rho _i>0$ satisfying (4.3). We take normalization $\hat{\rho _1}$ of $\rho _1$ as in Definition 2.8 and $G_{\hat{\rho }_1,q}$ by (2.2). We denote by $\textbf{D}_{\rho _1\rightarrow \rho _2}$ and by $\textbf{D}_{\rho _1\rightarrow \rho _2,0}$ the sets of all formal right linear differential operator P of the form

$$\begin{aligned} P=\sum _{\textbf{m}\in {\mathbb {N}}^n}u_\textbf{m}(\textbf{x}) \odot _L \partial _{\textbf{x}}^{\textbf{m}} \end{aligned}$$

where the multisequence $(u_\textbf{m}(\textbf{x}))_{\textbf{m}\in {\mathbb {N}}^n_0}\subset A_{\rho _2}$ satisfies

$$\begin{aligned} \forall \lambda>0,\, \exists \sigma>0,\, \exists C>0,\, \forall \textbf{m}\in {\mathbb {N}}^n_0,\, \Vert u_\textbf{m}\Vert _{\rho _2,\sigma }\le C\frac{G_{\hat{\rho }_1,|\textbf{m}|}}{\textbf{m}!}\lambda ^{|\textbf{m}|} \end{aligned}$$

(4.4)

and

$$\begin{aligned} \forall \sigma>0,\, \exists \lambda>0,\, \exists C>0,\, \forall \textbf{m}\in {\mathbb {N}}^n_0,\, \Vert u_\textbf{m}\Vert _{\rho _2,\sigma }\le C\frac{G_{\hat{\rho }_1,|\textbf{m}|}}{\textbf{m}!}\lambda ^{|\textbf{m}|}, \end{aligned}$$

(4.5)

respectively. Note that in the latter case, each $u_\textbf{m}$ belongs to $A_{\rho _2,+0}$.

For the following remark see also [11, Remark 4.4].

Remark 4.3

By adding $\ln (c)/\ln (r)$ for a constant $c>0$ (with a suitable modification near $r=0$) to a proximate order $\rho _2(r)$ with order $\rho _2$, we get a new proximate order $\tilde{\rho }_2(r)$ for the same order $\rho _2$ satisfying

$$\begin{aligned} \tilde{\rho }_2(r)=\rho _2(r)+\ln (c)/\ln (r) \end{aligned}$$

for $r>1$, that is,

$$\begin{aligned} r^{\tilde{\rho }_2(r)}=cr^{\rho _2(r)}, \end{aligned}$$

eventually. Then $\Vert \cdot \Vert _{\tilde{\rho }_2,\sigma }$ and $\Vert \cdot \Vert _{\rho _2,c\sigma }$ become equivalent norms for $\sigma >0$, and the spaces $A_{\tilde{\rho }_2}$ and $A_{\tilde{\rho }_2,+0}$ are homeomorphic to $A_{\rho _2}$ and $A_{\rho _2,+0}$ respectively. By taking c sufficiently large, we can take $\tilde{\rho }_2$ as

$$\begin{aligned} \rho _1(r)\le \tilde{\rho }_2(r), \quad \text { for } r\ge r_0 \end{aligned}$$

for a suitable $r_0$, and we can choose normalizations $\hat{\rho }_1$ of $\rho _1$ and $\hat{\rho }_2$ of $\tilde{\rho }_2$ as

• (i) $\hat{\rho }_1(r)\le \hat{\rho }_2(r)$ for $r\ge 0$.

  Since a proximate order and its normalization define equivalent norms, we have

• (ii) $\Vert \cdot \Vert _{\hat{\rho }_2,\sigma }$ and $\Vert \cdot \Vert _{\rho _2,c\sigma }$ are equivalent for any $c>0$,

• (iii) $\textbf{D}_{\rho _1\rightarrow \rho _2}=\textbf{D}_{\hat{\rho }_1\rightarrow \hat{\rho }_2}$, $\textbf{D}_{\rho _1\rightarrow \rho _2, 0}=\textbf{D}_{\hat{\rho }_1\rightarrow \hat{\rho }_2,0}$.

Note further that Theorems 4.4 and 4.5 below are not affected by the replacement of $\rho _1$ and $\rho _2$ by $\hat{\rho }_1$ and $\hat{\rho }_2$.

Now we will prove:

Theorem 4.4

Let $\rho _i(r)$ $(i=1,\, 2)$ be two proximate orders for orders $\rho _i>0$ satisfying (4.3).

• (1) Suppose that $P=\sum _{\textbf{m}\in {\mathbb {N}}^n}u_\textbf{m}(\textbf{x}) \odot _L \partial _{\textbf{x}}^{\textbf{m}}\in \textbf{D}_{\rho _1\rightarrow \rho _2}$. For a left monogenic entire function $f\in A_{\rho _1}$,

  $$\begin{aligned} Pf=\sum _{\textbf{m}\in {\mathbb {N}}^n}u_\textbf{m}(\textbf{x}) \odot _L \partial _{\textbf{x}}^{\textbf{m}} f \end{aligned}$$

  converges and $Pf\in A_{\rho _2}$. Moreover, $f\mapsto Pf$ defines a continuous right linear homomorphism $P: A_{\rho _1}\rightarrow A_{\rho _2}$.

• (2) Let $F:A_{\rho _1}\rightarrow A_{\rho _2}$ be a continuous right linear homomorphism. Then there is a unique $P\in \textbf{D}_{\rho _1\rightarrow \rho _2}$ such that $Ff=Pf$ holds for any $f\in A_{\rho _1}$.

Proof

We can replace $\rho _i$ by $\hat{\rho }_i$ ($i=1,\, 2$) as in Remark 4.3, and we may assume from the beginning that $\rho _i$ $(i=1,\, 2)$ are normalized proximate orders satisfying

$$\begin{aligned} \rho _1(r)\le \rho _2(r),\quad \text {for } r\ge 0, \end{aligned}$$

which implies

$$\begin{aligned} \Vert f\Vert _{\rho _2,\tau }\le \Vert f\Vert _{\rho _1,\tau } \end{aligned}$$

(4.6)

for any f and $\tau >0$.

(1) We fix $\eta >0$ and $C_\eta >0$ in a such a way that

$$\begin{aligned} (cr)^{\rho _2(cr)} \le (1+\eta ) c^{\rho _2} r^{\rho _2(r)}+C_\eta \end{aligned}$$

see [28, p. 16, Proposition 1.20]. Using Lemma 3.9, Lemma 3.13, Remark 3.10 and the estimate (4.4) for $(u_{\textbf{m}}(\textbf{x}))_{\textbf{m}\in {\mathbb {N}}^n}$ in Definition 4.2, we have a constant $k=k(\rho _1)$ depending only on $\rho _1$ such that for any $\epsilon >0$, $\tau >0$ there exists positive constants $\sigma (\epsilon )$, $C(n,\eta ,\sigma (\epsilon ), k\tau )$ and $C'(n,\eta ,\sigma (\epsilon ), k\tau )$ with the estimate

$$\begin{aligned}{} & {} \sum _{\textbf{m}\in {\mathbb {N}}^n_0}\Vert u_{\textbf{m}} \odot _L \partial ^{\textbf{m}}_{\textbf{x}} f\Vert _{\rho _2, (1+\eta )(n+\delta +1)^{\rho _2}(\sigma (\epsilon )+k\tau )}\nonumber \\{} & {} \quad \le C(n,\eta ,\sigma (\epsilon ), k\tau ) \sum _{\textbf{m}\in {\mathbb {N}}^n_0} \Vert u_{\textbf{m}}\Vert _{\rho _2,\sigma (\epsilon )} \Vert \partial ^{\textbf{m}}_{\textbf{x}} f\Vert _{\rho _2,k\tau }\nonumber \\{} & {} \quad \le C(n,\eta ,\sigma (\epsilon ), k\tau ) \sum _{\textbf{m}\in {\mathbb {N}}^n_0} \Vert u_{\textbf{m}}\Vert _{\rho _2,\sigma (\epsilon )} \Vert \partial ^{\textbf{m}}_{\textbf{x}} f\Vert _{\rho _1,k\tau }\nonumber \\{} & {} \quad \le C'(n,\eta ,\sigma (\epsilon ), k\tau ) \sum _{\textbf{m}\in {\mathbb {N}}^n_0}\frac{G_{\rho _1,|\textbf{m}|}}{\textbf{m}!} \epsilon ^{|\textbf{m}|} \Vert f\Vert _{\rho _1,\tau }\nonumber \\{} & {} \qquad \times \frac{\textbf{m}!}{G_{\rho _1,|\textbf{m}|}}(2k(c(n,{\textbf{m}}))^{\rho _1/|{\textbf{m}}|} \tau )^{|\textbf{m}|/\rho _1}\nonumber \\{} & {} \quad = C'(n,\eta ,\sigma (\epsilon ), k\tau ) \Vert f\Vert _{\rho ,\tau } \sum _{q\in {\mathbb {N}}_0} \sum _{|\textbf{m}|=q} \epsilon ^q(2k(C(n))^{\rho _1/q} \tau )^{q/\rho _1}\nonumber \\{} & {} \quad \le C'(n,\eta ,\sigma (\epsilon ), k\tau ) \Vert f\Vert _{\rho ,\tau } \sum _{q=0}^{+\infty } (q+1)^{n-1} \epsilon ^q(2k(C(n))^{\rho _1/q} \tau )^{q/\rho _1}. \end{aligned}$$

(4.7)

In view of Remark 3.10, the last sum converges if $\epsilon <(C(n)(2k\tau )^{1/\rho _1})^{-1}$. For such a choice of $\epsilon >0$ depending on $\rho _1$ and $\tau$, we set

$$\begin{aligned} \tau '(\tau ):= (1+\eta )(n+\delta +1)^{\rho _2}(\sigma (\epsilon )+k\tau ) \end{aligned}$$

and

$$\begin{aligned} C''= C'(n,\eta ,\sigma (\epsilon ), k\tau ) \sum _{q=0}^{+\infty } (q+1)^{n-1} \epsilon ^q(2k(C(n))^{\rho _1} \tau )^{q/\rho _1}. \end{aligned}$$

Then, $\sum _{|\textbf{m}|} u_{\textbf{m}}(\textbf{x})\odot _L \partial ^{\textbf{m}}_{\textbf{x}} f$ converges in $A_{\rho _2,\tau '(\tau )}$ and defines an element $Pf\in A_{\rho _2,\tau '(\tau )}$ satisfying

$$\begin{aligned} \Vert Pf\Vert _{\rho _2,\tau '(\tau )}\le C''\Vert f\Vert _{\rho _1,\tau }. \end{aligned}$$

Since $f\in A_{\rho _1,\tau }$ was chosen arbitrarily, this implies the well-definedness and the continuity of $P:A_{\rho _1,\tau }\rightarrow A_{\rho _2,\tau '(\tau )}$. Also since $\tau >0$ was chosen arbitrarily, we get the well-definedness and the continuity of $P:A_{\rho _1}\rightarrow A_{\rho _2}$ by the definition of the inductive limit of locally convex spaces.

(2) Let $F: A_{\rho _1}\rightarrow A_{\rho _2}$ be a continuous right linear homomorphism. Then, thanks to the theory of locally convex spaces, we can conclude, using Lemma 3.1, that for any $\tau >0$, there exists $\tau '=\tau '(\tau )>0$ such that $F(A_{\rho _1,\tau })\subset A_{\rho _2,\tau '(\tau )}$ and that

$$\begin{aligned} F: A_{\rho _1,\tau }\rightarrow A_{\rho _2,\tau '(\tau )} \end{aligned}$$

is continuous. Therefore, we can in particular take $C(\tau )$ depending on $\tau >0$ for which

$$\begin{aligned} \Vert F f\Vert _{\rho _2,\tau '(\tau )}\le C(\tau )\Vert f\Vert _{\rho _1,\tau },\quad \text {for any } f\in A_{\rho _1,\tau }. \end{aligned}$$

(4.8)

Let us define a multi-sequence of entire functions $(u_{{\textbf{m}}}({\textbf{x}}))_{{\textbf{m}}\in {\mathbb {N}}^n_0}$ by

$$\begin{aligned} u_{{\textbf{m}}}({\textbf{x}})=\sum _{{\textbf{s}}\le {\textbf{m}}}\frac{F(V_{{\textbf{s}}}({\textbf{x}})) \odot _L V_{{\textbf{m}}-{\textbf{s}}}(-{\textbf{x}})}{({\textbf{m}}-{\textbf{s}})! {\textbf{m}}!}, \end{aligned}$$

whose convergence in $A_{\rho _2}$ will be proved together with their estimates. We define a formal differential operator $P\in \hat{D}$ of infinite order by

$$\begin{aligned} P=\sum _{{\textbf{m}}} u_{{\textbf{m}}}({\textbf{x}})\odot _L\partial ^{\textbf{m}}_{\textbf{x}}. \end{aligned}$$

(4.9)

First we show that $P\in \textbf{D}_{\rho _1\rightarrow \rho _2}$. For any fixed $\tau _0$ and $\tau _1$ with $0<\tau _0<\tau _1$, we have

$$\begin{aligned}{} & {} \Vert u_{{\textbf{m}}} \Vert _{\rho _2, (1+\eta )(n+\delta +1)^{\rho _1}(\tau _1+\tau '(\tau _1))}\nonumber \\{} & {} \quad \le C(\eta , n,\tau _1,\tau '(\tau _1)) \sum _{{\textbf{s}}\le {\textbf{m}}}\frac{\Vert V_{{\textbf{m}}-{\textbf{s}}} ({\textbf{x}})\Vert _{\rho _2,\tau _1} \Vert F(V_{{\textbf{s}}}({\textbf{x}}))\Vert _{\rho _2,\tau '(\tau _1)}}{({\textbf{m}}-{\textbf{s}})!{\textbf{s}}!}\nonumber \\{} & {} \quad \le C(\eta , n,\tau _1,\tau '(\tau _1)) \sum _{{\textbf{s}}\le {\textbf{m}}}\frac{\Vert V_{{\textbf{m}}-{\textbf{s}}} ({\textbf{x}})\Vert _{\rho _1,\tau _1} C(\tau _1) \Vert V_{{\textbf{s}}}({\textbf{x}})\Vert _{\rho _1,\tau _1}}{({\textbf{m}}-{\textbf{s}})!{\textbf{s}}!}\nonumber \\{} & {} \quad \le C(\tau _1)C(\eta , n,\tau _1,\tau '(\tau _1)) \sum _{{\textbf{s}}\le {\textbf{m}}} \nonumber \\{} & {} \qquad \times \frac{C(\tau _0,\tau _1)\tau _0^{-|{\textbf{m}}-{\textbf{s}}|/\rho _1} G_{\rho _1,|{\textbf{m}}-{\textbf{s}}|} C(\tau _0,\tau _1) \tau _0^{-|{\textbf{s}}|/\rho _1} G_{\rho _1,|{\textbf{s}}|}}{({\textbf{m}}-{\textbf{s}})! {\textbf{s}}!}\nonumber \\{} & {} \quad \le C(\tau _1) C(\eta , n,\tau _1,\tau '(\tau _1)) C(\tau _0,\tau _1)^2 \sum _{{\textbf{s}}\le {\textbf{m}}} \left( {\begin{array}{c}{\textbf{m}}\\ {\textbf{s}}\end{array}}\right) \frac{G_{\rho _1,|{\textbf{m}}|}}{{\textbf{m}}!}\tau _0^{-|{\textbf{m}}|/\rho _1}\nonumber \\{} & {} \quad = C(\tau _1) C(\eta , n,\tau _1,\tau '(\tau _1)) C(\tau _0,\tau _1)^2 \frac{G_{\rho _1,|{\textbf{m}}|}}{{\textbf{m}}!} 2^{|{\textbf{m}}|} \tau _0^{-|{\textbf{m}}|/\rho _1}. \end{aligned}$$

(4.10)

Here we used Lemma 3.13 at the first inequality, (4.6) and (4.8) at the second inequality, Lemma 3.8 with $0<\tau _0<\tau _1$ at the third inequality, and Lemma 2.10 at the fourth inequality. For a given $\epsilon >0$, we can take $\tau _0>0$ large enough such that

$$\begin{aligned} 2\tau _0^{-1/\rho _1}<\epsilon . \end{aligned}$$

Then, by choosing $\tau _1$ as $\tau _1>\tau _0$ and by putting

$$\begin{aligned} \sigma :=(1+\eta )(n+\delta +1)^{\rho _1}(\tau _1+\tau '(\tau _1)), \end{aligned}$$

we have

$$\begin{aligned} \Vert u_{\textbf{m}}\Vert _{\rho _2,\sigma }\le C'\frac{G_{\rho _1, |{\textbf{m}}|}}{{\textbf{m}}!}\epsilon ^{|{\textbf{m}}|} \end{aligned}$$

for any ${\textbf{m}}$, which implies $P\in \textbf{D}_{\rho _1\rightarrow \rho _2}$.

Now we show that $Pf=Ff$ for any $f\in A_{\rho _1}$. First, we show that the equality holds for $f\in {\mathbb {R}}_n [{\textbf{x}}]$. This will imply the equality holds for any $f\in A_{\rho _1}$ since ${\mathbb {R}}_n[{\textbf{x}}]$ is dense in $A_{\rho _1}$ by Proposition 3.11 and P and F are continuous. If $f\in {\mathbb {R}}_n [{\textbf{x}}]$ then there exists $m\in {\mathbb {N}}_0$ and $b_{{\textbf{s}}}\in {\mathbb {R}}_n$ for $|{\textbf{s}}|\le m$ such that $f({\textbf{x}})=\sum _{|{\textbf{s}}|\le m} V_{{\textbf{s}}}({\textbf{x}}) b_{\textbf{s}}$. We have that

$$\begin{aligned} F(f({\textbf{x}}))=\sum _{|{\textbf{s}}|\le m} F(V_{{\textbf{s}}} ({\textbf{x}})) b_{\textbf{s}}\end{aligned}$$

and

$$\begin{aligned} P(f({\textbf{x}}))= & {} \sum _{|{\textbf{m}}|\le m} u_{\textbf{m}}({\textbf{x}})\odot _L \partial _{\textbf{x}}^{\textbf{m}}f({\textbf{x}})\nonumber \\= & {} \sum _{|{\textbf{m}}|\le m} u_{\textbf{m}}({\textbf{x}})\odot _L \partial _{\textbf{x}}^{\textbf{m}}\left( \sum _{|{\textbf{s}}|\le m} V_{{\textbf{s}}}({\textbf{x}}) b_{\textbf{s}}\right) \nonumber \\= & {} \sum _{|{\textbf{s}}|\le m} \sum _{{\textbf{m}}\le {\textbf{s}}} u_{{\textbf{m}}}({\textbf{x}})\odot _L \partial ^{\textbf{m}}_{\textbf{x}}(V_{\textbf{s}}({\textbf{x}})) b_{\textbf{s}}\nonumber \\= & {} \sum _{|{\textbf{s}}|\le m} \sum _{{\textbf{m}}\le {\textbf{s}}} \sum _{{\textbf{p}}\le {\textbf{m}}} \frac{{\textbf{s}}! F(V_{\textbf{p}}({\textbf{x}}))\odot _L V_{{\textbf{m}}-{\textbf{p}}}(-{\textbf{x}})\odot _L V_{{\textbf{s}}-{\textbf{m}}}({\textbf{x}})}{({\textbf{m}}-{\textbf{p}})!({\textbf{s}}-{\textbf{m}})!{\textbf{p}}!} b_{\textbf{s}}\nonumber \\= & {} \sum _{|{\textbf{s}}|\le m} \sum _{{\textbf{p}}\le {\textbf{s}}} F(V_{{\textbf{p}}}({\textbf{x}})) \odot _L\sum _{0\le {\textbf{m}}-{\textbf{p}}\le {\textbf{s}}-{\textbf{p}}} \frac{{\textbf{s}}! ({\textbf{s}}-{\textbf{p}})! (-1)^{|{\textbf{m}}-{\textbf{p}}|} V_{{\textbf{s}}-{\textbf{p}}}({\textbf{x}})}{({\textbf{s}}-{\textbf{p}})!({\textbf{m}}-{\textbf{p}})! {\textbf{p}}!({\textbf{s}}-{\textbf{m}})!} b_{\textbf{s}}\nonumber \\= & {} \sum _{|{\textbf{s}}|\le {\textbf{m}}} \sum _{{\textbf{p}}\le {\textbf{s}}} F(V_{\textbf{p}}({\textbf{x}}))\odot _L \frac{{\textbf{s}}! (1-1)^{|{\textbf{s}}-{\textbf{p}}|}}{({\textbf{s}}-{\textbf{p}})! {\textbf{p}}!} V_{{\textbf{s}}-{\textbf{p}}}({\textbf{x}}) b_{\textbf{s}}\nonumber \\= & {} \sum _{|{\textbf{s}}|\le m} F(V_{\textbf{s}}({\textbf{x}})) b_{\textbf{s}}\end{aligned}$$

(4.11)

thus $F(f(\textbf{x}))=P(f(\textbf{x})).$ $\square$

Theorem 4.5

Let $\rho _i(r)$ $(i=1,\, 2)$ be two proximate orders for orders $\rho _i>0$ satisfying (4.3).

• (1) Suppose that $P=\sum _{\textbf{m}\in {\mathbb {N}}^n_0}u_\textbf{m}(\textbf{x}) \odot _L \partial _{\textbf{x}}^{\textbf{m}}\in \textbf{D}_{\rho _1\rightarrow \rho _2,0}$. For a left monogenic entire function $f\in A_{\rho _1,+0}$,

  $$\begin{aligned} Pf=\sum _{\textbf{m}\in {\mathbb {N}}^n_0}u_\textbf{m}(\textbf{x}) \odot _L \partial _{\textbf{x}}^{\textbf{m}} f \end{aligned}$$

  converges and $Pf\in A_{\rho _2,+0}$. Moreover, $f\mapsto Pf$ defines a continuous homomorphism $P: A_{\rho _1,+0}\rightarrow A_{\rho _2,+0}$.

• (2) Let $F:A_{\rho _1,+0}\rightarrow A_{\rho _2,+0}$ be a continuous right linear homomorphism. Then there is a unique $P\in \textbf{D}_{\rho _1\rightarrow \rho _2,0}$ such that $Ff=Pf$ holds for any $f\in A_{\rho _1,+0}$.

Proof

Again we can make a substitution of proximate orders as in Remark 4.3, and we may assume from the beginning that $\rho _i$ $(i=1,\, 2)$ are normalized proximate orders satisfying (4.6).

(1) We assume condition (4.5) for $(u_{\textbf{m}}({\textbf{x}}))_{{\textbf{m}}\in {\mathbb {N}}^n_0}$ and denote by $\lambda _\sigma$ the constant $\lambda$ given there according to $\sigma$. Similar computations as in (4.7) yield

$$\begin{aligned}{} & {} \sum _{\textbf{m}\in {\mathbb {N}}^n_0}\Vert u_{\textbf{m}} \odot _L \partial ^{\textbf{m}}_{\textbf{x}} f\Vert _{\rho _2, (1+\eta )(n+\delta +1)^{\rho _2}(\sigma +k\tau )}\nonumber \\{} & {} \quad \le C(n,\eta ,\sigma , k\tau ) \sum _{\textbf{m}\in {\mathbb {N}}^n_0} \Vert u_{\textbf{m}}\Vert _{\rho _2,\sigma } \Vert \partial ^{\textbf{m}}_{\textbf{x}} f\Vert _{\rho _2,k\tau }\nonumber \\{} & {} \quad \le C(n,\eta ,\sigma , k\tau ) \sum _{\textbf{m}\in {\mathbb {N}}^n_0} \Vert u_{\textbf{m}}\Vert _{\rho _2,\sigma } \Vert \partial ^{\textbf{m}}_{\textbf{x}} f\Vert _{\rho _1,k\tau }\nonumber \\{} & {} \quad \le C'(n,\eta ,\sigma , k\tau ) \sum _{\textbf{m}\in {\mathbb {N}}^n_0} \frac{G_{\rho _1,|\textbf{m}|}}{\textbf{m}!} \lambda _\sigma ^{|\textbf{m}|} \Vert f\Vert _{\rho _1,\tau } \frac{\textbf{m}!}{G_{\rho _1,|\textbf{m}|}}(2k(c(n,{\textbf{m}}))^{\rho _1/|{\textbf{m}}|} \tau )^{|\textbf{m}|/\rho _1}\nonumber \\{} & {} \quad = C'(n,\eta ,\sigma , k\tau ) \Vert f\Vert _{\rho ,\tau } \sum _{q\in {\mathbb {N}}_0} \sum _{|\textbf{m}|=q} \lambda _\sigma ^q(2k(c(n,{\textbf{m}}))^{\rho _1/q} \tau )^{q/\rho _1}\nonumber \\{} & {} \quad \le C'(n,\eta ,\sigma , k\tau ) \Vert f\Vert _{\rho ,\tau } \sum _{q=0}^{+\infty } (q+1)^{n-1} ( C(n) (2k\tau )^{1/\rho _1} \lambda _\sigma )^{q}, \end{aligned}$$

(4.12)

for any $\sigma ,\, \tau >0,$ $f\in A_{\rho _1,+0}$. Here $k=k(\rho _1)$ depends only on $\rho _1$, and $C'(n,\eta ,\sigma , k\tau )$ is independent of f. Note that the last sum is finite if $C(n) (2k\tau )^{1/\rho _1} \lambda _\sigma <1$. In fact, we may first choose $\sigma$ as

$$\begin{aligned} 0<\sigma <\frac{\epsilon }{2(1+\eta )(n+\delta +1)^{\rho _2}}, \end{aligned}$$

which determines $\lambda _\sigma$, and then secondly choose $\tau >0$ as

$$\begin{aligned} k\tau<\frac{\epsilon }{2(1+\eta )(n+\delta +1)^{\rho _2}}, \quad C(n) (2k\tau )^{1/\rho _1} \lambda _\sigma <1. \end{aligned}$$

Therefore, for any $\epsilon >0$, there exist $C''(\epsilon )>0$ and $\tau (\epsilon )>0$ such that

$$\begin{aligned} \Vert P(f)\Vert _{\rho _2, \epsilon }\le C''(\epsilon ) \Vert f\Vert _{\rho _1, \tau (\epsilon )},\quad \text {for } f\in A_{\rho ,+0}, \end{aligned}$$

which implies that $P:A_{\rho _1,+0}\rightarrow A_{\rho _2,+0}$ is continuous.

(2) For a given continuous homomorphism $F:A_{\rho _1,+0}\rightarrow A_{\rho _2,+0}$, we construct

$$\begin{aligned} P=\sum _{{\textbf{m}}} u_{\textbf{m}}({\textbf{x}})\odot _L\partial _{\textbf{x}}^{\textbf{m}}\end{aligned}$$

via (4.9) in the same way as in the proof of Theorem 4.4 (2). Let us show the convergence of P in $A_{\rho _2,+0}$ together with the estimates. For any $\sigma '>0$, by continuity of F there exists a $\tau _1>0$ and a $C(\tau _1)>0$ such that

$$\begin{aligned} \Vert F(f)\Vert _{\rho _2,\sigma '/2} \le C(\tau _1)\Vert f\Vert _{\rho _1,\tau _1},\quad \text {for } f\in A_{\rho _1,+0}. \end{aligned}$$

(4.13)

We fix $\sigma >0$. Take $\tau _0$ with $0<\tau _0<\min \{\tau _1,\sigma '/2\}$ where $\sigma '=\frac{\sigma }{(1+\eta )(n+\delta +1)^{\rho _1}}$. Similarly to the proof of (4.12), we have

$$\begin{aligned} \Vert u_{{\textbf{m}}}\Vert _{\rho _2,\sigma }\le & {} \Vert u_{{\textbf{m}}} \Vert _{\rho _2, (1+\eta )(n+\delta +1)^{\rho _1} \sigma '}\nonumber \\\le & {} C(\eta , n,\tau _1,\tau '(\tau _1)) \sum _{{\textbf{s}}\le {\textbf{m}}}\frac{\Vert V_{{\textbf{m}}-{\textbf{s}}} ({\textbf{x}})\Vert _{\rho _2,\sigma '/2} \Vert F(V_{{\textbf{s}}}({\textbf{x}}))\Vert _{\rho _2,\sigma '/2}}{({\textbf{m}}-{\textbf{s}})!{\textbf{s}}!}\nonumber \\\le & {} C(\eta , n,\tau _1,\tau '(\tau _1)) \sum _{{\textbf{s}}\le {\textbf{m}}}\frac{\Vert V_{{\textbf{m}}-{\textbf{s}}} ({\textbf{x}})\Vert _{\rho _1,\sigma '/2} C(\tau _1) \Vert V_{{\textbf{s}}}({\textbf{x}})\Vert _{\rho _1,\tau _1}}{({\textbf{m}}-{\textbf{s}})!{\textbf{s}}!}\nonumber \\\le & {} C(\eta , n,\tau _1,\tau '(\tau _1))\nonumber \\{} & {} \times \sum _{{\textbf{s}}\le {\textbf{m}}} \frac{C(\tau _0,\sigma '/2)\tau _0^{-|{\textbf{m}}-{\textbf{s}}|/\rho _1} G_{\rho _1,|{\textbf{m}}-{\textbf{s}}|} C(\tau _1) C(\tau _0,\tau _1) \tau _0^{-|{\textbf{s}}|/\rho _1} G_{\rho _1,|{\textbf{s}}|}}{({\textbf{m}}-{\textbf{s}})! {\textbf{s}}!}\nonumber \\\le & {} C(\eta , n,\tau _1,\tau '(\tau _1)) C(\tau _1) C(\tau _0,\sigma '/2) C(\tau _0,\tau _1) \sum _{{\textbf{s}}\le {\textbf{m}}} \left( {\begin{array}{c}{\textbf{m}}\\ {\textbf{s}}\end{array}}\right) \frac{G_{\rho _1,|{\textbf{m}}|}}{{\textbf{m}}!}\tau _0^{-|{\textbf{m}}|/\rho _1}\nonumber \\= & {} C(\eta , n,\tau _1,\tau '(\tau _1)) C(\tau _1) C(\tau _0,\sigma '/2) C(\tau _0,\tau _1) \frac{G_{\rho _1,|{\textbf{m}}|}}{{\textbf{m}}!} 2^{|{\textbf{m}}|} \tau _0^{-|{\textbf{m}}|/\rho _1}.\nonumber \\ \end{aligned}$$

(4.14)

Here we used Lemma 3.13 at the first inequality, (4.6) and (4.13) at the second, Lemma 3.8 twice with $0<\tau _0<\sigma '/2$ and $0<\tau _0<\tau _1$ at the third, and Lemma 2.10 at the fourth.

Therefore, by defining

$$\begin{aligned} \lambda :=2\tau _0^{-1/\rho _1}, \end{aligned}$$

we have

$$\begin{aligned} \Vert u_{\textbf{m}}({\textbf{x}})\Vert _{\rho _2,\sigma } \le C'\frac{G_{\rho _1,|{\textbf{m}}|}}{{\textbf{m}}!} \lambda ^{|{\textbf{m}}|} \end{aligned}$$

for any ${\textbf{m}}$. Since $\sigma >0$ can be chosen arbitrarily, this implies $P\in \textbf{D}_{\rho _1\rightarrow \rho _2,0}$.

The remaining thing is to show $F(f)=P(f)$ for $f\in A_{\rho _1,+0}$. This can be done completely in the same way as in the case of normal type, that is, the equality for polynomial f due to (4.11) can be extended to $A_{\rho _1,+0}$ by continuity, since polynomials form a dense subset of $A_{\rho _1,+0}$, for which we again refer to Proposition 3.11. $\square$