About a conjecture on difference equations in quasianalytic Carleman classes

TL;DR

This paper investigates the solvability of linear difference equations within quasianalytic Carleman classes, providing partial progress towards a longstanding conjecture in the field.

Contribution

It offers a partial solution to a conjecture concerning the solvability of difference equations in quasianalytic Carleman classes, advancing understanding in this area.

Findings

Partial answer to the conjecture on difference equations

Insights into the structure of quasianalytic Carleman classes

Progress towards full resolution of the conjecture

Abstract

In this paper we obtain a partial answer to a conjecture on the solvabilty of linear difference equations in quasianalytic Carleman classes.

Equations

$$(E):\stackrel{q}{\sum _{j=1}}{a}_i(x)\phi (x+{\alpha }_i)=\chi (x)$$

$$\left\{\begin{array}[]{c}\mathbb{R}_{\rho}:=\{z\in\mathbb{C}:|\mathop{\mathrm{I}m}(z)|<\rho\},\text{ }\mathbb{R}_{\rho}^{\pm}:=\{z\in\mathbb{R}_{\rho}:\pm\mathop{\mathrm{R}e}(z)>\rho\}\\ \mathbb{R}_{\rho,a}:=\{z\in\mathbb{R}_{\rho}:|\mathop{\mathrm{R}e}(z)|\leq a\}\\ \Delta_{\rho}:=\{z\in\mathbb{C}:|z|<\rho\},\text{ }\Delta_{\rho}^{\pm}:=\{z\in\Delta_{\rho}:\pm\mathop{\mathrm{R}e}(z)\leq 0\}\\ \Gamma_{\rho}:=\{z\in\mathbb{C}:|z|=\rho\},\ \Gamma_{\rho}^{\pm}:=\{z\in\Gamma_{\rho}:\pm\mathop{\mathrm{R}e}(z)\leq 0\}\end{array}\right.\text{ }$$

$$\left\{\begin{array}[]{c}V^{\text{ }(0)}:=V,\text{ }V^{\text{ }(n)}:=\{u_{1}+...+u_{n}\text{ }:u_{j}\in V,\text{ }j=1,...,n\text{ }\},\text{ }n\geq 1\\ z+V:=\{z+u:u\in V\text{ }\},\text{ }z-V:=\{z-u:u\in V\text{ }\}\end{array}\right.$$

$$\overline{\partial }F(z):=\frac{1}{2}[\frac{\partial F}{\partial x}(z)+i\frac{\partial F}{\partial y}(z)]$$

$$||f^{(n)}|{|}_{\infty ,I}\le C_I{\rho }_I^n{M}_n,n\in \mathbb{N}$$

$$\left\{\begin{array}[]{c}\left(\frac{M_{n+1}}{(n+1)!}\right)^{2}\leq\frac{M_{n}}{n!}\frac{M_{n+2}}{(n+2)!},\text{ }n\in\mathbb{N}\\ \underset{n\in\mathbb{N}}{\sup}\left(\frac{M_{n+1}}{(n+1)M_{n}}\right)^{\frac{1}{n}}<+\infty\\ \underset{n\rightarrow+\infty}{\lim}M_{n}^{\frac{1}{n}}=+\infty\end{array}\right.$$

$$H_M(x):=\underset{n\in \mathbb{N}}{\inf }\left(\frac{M_n}{n!}{x}^n\right),x>0$$

$$|\overline{\partial }{F}_I(z)|\le A_I{H}_M(B_I|\mathrm{Im}(z)|),z\in \mathbb{C}$$

$$\underset{z\in {\mathbb{R}}_{\delta }}{\sup }\left(\left(\sum _{j=2}^q\left|\frac{a_j(z)}{a_1(z)}\right|+\sum _{j=1}^{q-1}\left|\frac{a_j(z)}{a_q(z)}\right|+\frac{1}{|a_1(z)|}+\frac{1}{|a_q(z)|}\right)e^{-e^{C|\mathrm{Re}z|}}\right)<+\infty$$

$$(E):\stackrel{q}{\sum _{j=1}}{a}_i(x)\phi (x+{\alpha }_i)=\chi (x)$$

$$f(x)$$

$$|f_{+}(z)|$$

$$|f_{-}(z)|$$

$$|\overline{\partial }F(z)|\le A{H}_M(B|\mathrm{Im}(z)|),z\in \mathbb{C}$$

$$\left\{\begin{array}[]{c}\begin{array}[]{c}f_{+}(z)=\frac{1}{2i\pi}e^{-e^{C_{0}\text{ }z}-e^{-C_{0}z}}\int_{\Gamma_{\rho}^{+}}\frac{e^{e^{C_{0}\zeta}+e^{-C_{0}\zeta}}F(\zeta)}{\zeta-z}d\zeta-\\ -\frac{1}{\pi}e^{-e^{C\text{ }z}-e^{-C\text{ }z}}\iint_{\Delta_{\rho}^{+}}\frac{e^{e^{C\text{ }_{0}\zeta}+e^{-C_{0}\zeta}}\overline{\partial}F(\zeta)}{\zeta-z}dm(\zeta)\end{array}\\ f_{-}(z)=\frac{1}{2i\pi}e^{-e^{C_{0}\text{ }z}-e^{-C_{0}\text{ }z}}\int_{\Gamma_{\rho}^{-}}\frac{e^{e^{C_{0}\zeta}+e^{-C_{0}\zeta}}F(\zeta)}{\zeta-z}d\zeta-\\ -\frac{1}{\pi}e^{-e^{C_{0}\text{ }z}-e^{-C_{0}\text{ }z}}\iint_{\Delta_{\rho}^{-}}\frac{e^{e^{C_{0}\text{ }\zeta}+e^{-C_{0}\zeta}}\overline{\partial}F(\zeta)}{\zeta-z}dm(\zeta)\end{array}\right.$$

$$\left\{\begin{array}[]{c}\beta_{j}:=\alpha_{j}-\alpha_{1},\text{ }j=2,...,q\\ b_{j}(z):=-\frac{a_{j}(z)}{a_{1}(z)},\text{ }z\in\mathbb{R}_{\delta},\text{ }j=2,...,q\\ \gamma_{j}:=\alpha_{q}-\alpha_{j},\text{ }j=1,...,q-1\\ c_{j}(z):=-\frac{a_{j}(z)}{a_{q}(z)},\text{ }z\in\mathbb{R}_{\delta},\text{ }j=1,...,q-1\end{array}\right.$$

$$\left\{\begin{array}[]{c}\chi(x)=\chi_{+}(x)+\chi_{-}(x),\text{ }x\in[-\delta_{0},\delta_{0}]\\ |\chi_{+}(z)|\leq D_{1}e^{-\cos(C_{1}\delta_{0})e^{C_{1}|\mathop{\mathrm{R}e}(z\text{ })|}},\text{ }z\in\mathbb{R}_{\delta_{0}}^{+}\\ |\chi_{-}(z)|\leq D_{1}e^{-\cos(C_{1}\delta_{0})e^{C_{1}|\mathop{\mathrm{R}e}(z\text{ })|}},\text{ }z\in\mathbb{R}_{\delta_{0}}^{-}\text{ }\end{array}\right.$$

$$\left\{\begin{array}[]{c}g_{{}_{0}}(z):=\frac{\chi_{+}(z)}{a_{1}(z)},\text{ }g_{n+1}(z):=\underset{j=2}{\overset{q}{\sum}}b_{j}(z)g_{n}(z+\beta_{j})\\ h_{{}_{0}}(z):=\frac{\chi_{-}(z)}{a_{q}(z)},\text{ }h_{n+1}(z):=\overset{q-1}{\underset{j=1}{\sum}}c_{j}(z)h_{n}(z-\gamma_{j})\end{array}\right.$$

$$\left\{\begin{array}[]{c}K_{1}:=\{\beta_{j}:j\in 2,...,q\}\\ K_{2}:=\{\gamma_{j}:j\in 1,...,q-1\}\end{array}\right.$$

$$\left\{\begin{array}[]{c}|g_{n+1}(z)|\text{ }\leq e^{Le^{C|\mathop{\mathrm{R}e}(z\text{ })|}}\underset{u\text{ }\in\text{ }z\text{ }+K}{\max}|g_{n}(u)|\\ |h_{n+1}(z)|\text{ }\leq e^{Le^{C|\mathop{\mathrm{R}e}(z\text{ })|}}\underset{u\text{ }\in\text{ }z\text{ }-K}{\max}|h_{n}(u)|\end{array}\right.$$

$$\left\{\begin{array}[]{c}\begin{array}[]{c}g_{n}(z)|\text{ }\leq e^{\underset{j=0}{\overset{n-1}{\sum}}Le^{C\text{ }(|\mathop{\mathrm{R}e}(z\text{ })|+j\beta_{q})}}\underset{u\text{ }\in\text{ }z\text{ }\mathbf{+}K_{1}^{(n)}}{\max}|g_{\text{ }0}(u)|\\ \leq\text{ }e^{nLe^{C\text{ }(|\mathop{\mathrm{R}e}(z\text{ })|+n\text{ }\beta_{q})}}\underset{u\text{ }\in\text{ }z\text{ }\mathbf{+}K_{1}^{(n)}}{\max}|\chi_{\text{ }+}(u)|\end{array}\\ |h_{n}(z)|\text{ }\leq e^{\underset{j=0}{\overset{n-1}{\sum L}}e^{C\text{ }(|\mathop{\mathrm{R}e}(z\text{ })|+j\gamma_{1})}}\underset{u\text{ }\in\text{ }z\text{ }\mathbf{-}K_{2}^{(n)}}{\max}|h_{\text{ }0}(u)|\\ \leq\text{ }e^{nLe^{C\text{ }(|\mathop{\mathrm{R}e}(z\text{ })|+n\text{ }\gamma_{1})}}\underset{u\text{ }\in\text{ }z\text{ }\mathbf{-}K_{2}^{(n)}}{\max}|\chi_{\text{ }-}(u)|\end{array}\right.$$

$$\left\{\begin{array}[]{c}z+K_{1}^{(n)}\subset\mathbb{R}_{\delta_{0}}^{+},\text{ }n\geq N_{a},\text{ }z\in\mathbb{R}_{\delta_{0},a}\\ z-K_{2}^{(n)}\subset\mathbb{R}_{\delta_{0}}^{-},\text{ }n\geq N_{a},\text{ }z\in\mathbb{R}_{\delta_{0},a}\end{array}\right.$$

$$\left\{\begin{array}[]{c}\begin{array}[]{c}\underset{u\text{ }\in\text{ }z\text{ }\mathbf{+}K_{1}^{(n)}}{\max}|\chi_{\text{ }+}(u)|\leq D_{1}e^{-\cos(C_{1}\delta_{0})e^{C_{1}\underset{u\text{ }\in\text{ }z\text{ }\mathbf{+}K_{1}^{(n)}}{\min}|\mathop{\mathrm{R}e}(u\text{ })|}}\\ \leq D_{1}e^{-\cos(C_{1}\delta_{0})e^{\text{ }C_{1}(-a\text{ }+n\text{ }\beta_{2})}}\end{array}\\ \underset{u\text{ }\in\text{ }z\text{ }\mathbf{-}K_{2}^{(n)}}{\max}|\chi_{\text{ }-}(u)|\leq D_{1}e^{-\cos(C_{1}\delta_{0})e^{C_{1}\underset{u\text{ }\in\text{ }z\text{ }\mathbf{-}K_{2}^{(n)}}{\min}|\mathop{\mathrm{R}e}(u)|}}\\ \leq D_{1}e^{-\cos(C_{1}\delta_{0})e^{C_{1}(-a\text{ }+n\text{ }\gamma_{q-1})}}\end{array}\right.$$

$$\left\{\begin{array}[]{c}|g_{n}(z)|\leq D_{1}e^{nLe^{C\text{ }(a\text{ }+n\text{ }\beta_{q})}\text{ }]-\cos(C_{1}\delta_{0})e^{C_{1}(-a\text{ }+n\text{ }\beta_{2})}}\\ |h_{n}(z)|\leq D_{1}e^{nLe^{C\text{ }(a\text{ }+n\text{ }\gamma_{1})}\text{ }]--\cos(C_{1}\delta_{0})e^{C_{1}(-a\text{ }+n\text{ }\gamma_{q-1})|}}\end{array}\right.$$

$$\left\{\begin{array}[]{c}nLe^{C\text{ }(a\text{ }+n\text{ }\beta_{q})}=\underset{n\rightarrow+\infty}{o}\left(\cos(C_{1}\delta_{0})e^{\text{ }C_{1}(-a\text{ }+n\text{ }\beta_{2})}\right)\\ nLe^{C\text{ }(a\text{ }+n\text{ }\gamma_{1})}=\underset{n\rightarrow+\infty}{o}\left(\cos(C_{1}\delta_{0})e^{\text{ }C_{1}(-a\text{ }+n\text{ }\gamma_{q})|}\right)\end{array}\right.$$

$$\left\{\begin{array}[]{c}|g_{n}(z)|\text{ }\leq D_{a}e^{-E_{a}e^{\text{ }C_{1}(-a\text{ }+n\text{ }\beta_{2})}},\text{ }z\in\mathbb{R}_{\delta_{0},a}\text{ },\text{ }n\geq P_{a}\\ |h_{n}(z)|\text{ }\leq D_{a}e^{-E_{a}e^{\text{ }C_{1}(-a\text{ }+n\text{ }\gamma_{q})|}},\text{ }z\in\mathbb{R}_{\delta_{0},a}\text{ },\text{ }n\geq P_{a}\end{array}\right.$$

$$\left\{\begin{array}[]{c}\underset{j=1}{\overset{q}{\sum}}a_{j}(x)g_{+}(x+\alpha_{j})=\chi_{+}(x),\text{ }x\in\mathbb{R}\\ \underset{j=1}{\overset{q}{\sum}}a_{j}(x)g_{-}(x+\alpha_{j})=\chi_{-}(x),\text{ }x\in\mathbb{R}\end{array}\right.$$

Full text

1. Introduction

In the paper ([1]), G. Belitskii, E. M. Dyn’kin and V. Tkachenko have formulated the following conjecture :

**”**Let $\chi,\ a_{i},\ i=1,...,q,$ be functions in a Carleman class $C_{M}(\mathbb{R})$ such that $a_{1}$and $a_{q}$ are nowhere vanishing on $\mathbb{R},$ and $\alpha_{1}<...<\alpha_{q}$ some real numbers. Then the difference equation $:$

[TABLE]

is solvable in the Carleman class $C_{M}\{\mathbb{R}\}."$

In that paper the authors, relying on a result of decomposition in Carleman classes, have proved the conjecture in the particular cases where the coefficients $a_{j}$ are constants or when the coefficients are variables with $q=2.$ They have also suggested that the same method could be used to show the solvability of the equation $(E)$ in a quasianalytic Carleman class $C_{M}\{\mathbb{R}\},$ if we assume that the functions $\frac{1}{a_{1}},$ $\frac{1}{a_{q}},$ $\frac{a_{2}}{a_{1}},...,\frac{a_{q}}{a_{1}},$ $\frac{a_{1}}{a_{q}},...,$ $\frac{a_{q-1}}{a_{q}}$ $(q\geq 3)$ can be continued in some strip $\mathbb{R}_{\delta}:=\{z\in\mathbb{C}:|\mathop{\mathrm{I}m}(z)|<\delta\}$ as analytic functions increasing on $\mathbb{R}_{\delta}$ not too fast at infinity. As an example of such coefficients, they have mentioned the class of rational functions. In this paper, our purpose is to give a precise meaning to this assertion, by proving that the result is true even if the functions $\frac{1}{a_{1}},$ $\frac{1}{a_{q}},$ $\frac{a_{2}}{a_{1}},...,\frac{a_{q}}{a_{1}},\frac{a_{1}}{a_{q}},...,\frac{a_{q-1}}{a_{q}}$ have more rapide increase at infinity provided that it is of the form $e^{e^{C|\mathop{\mathrm{R}e}(z\text{ })|}}$ where $C>0$ is a constant.

2. Notations, definitions and statement of the main result

We set for every $\rho>0,$ $a\geq 0:$

[TABLE]

For every non empty subset $V$ of $\mathbb{C}$ and every $z\in\mathbb{C}$ and $n\in\mathbb{N}^{\ast}$we set :

[TABLE]

$dm(\zeta)$ denotes the Lebesgue mesure on $\mathbb{C}.$

Let $S$ be a nonempty subset of $\mathbb{C}.$ By $O(S$ $)$ we denote the set of holomorphic functions on some neighborhood of $S$.

Let $F:U\subset\mathbb{C}\rightarrow\mathbb{C}$ be a function of class $C^{1}$ on an open subset $U$ of $\mathbb{C}.$ We set for all $z\in U:$

[TABLE]

$\overline{\partial}$ is called the operator of Cauchy-Riemann.

Let $M:=(M_{n})_{n\in\mathbb{N}}$ be a sequence of strictly positive numbers. Let $M:=(M_{n})_{n\text{ }\geq 0\text{ }}$be a sequence of strictly positive real numbers. The Carleman class $C_{M}$ $\{\mathbb{R}\}$ is the set of all functions $f:W\rightarrow\mathbb{C}$ of class $C^{\infty}$such that

[TABLE]

for every compact interval $I$   contained in $\mathbb{R}$ with some constants $C_{I},\rho_{I}>0.$

The Carleman class $C_{M}$ $\{\mathbb{R}\}$ is said to be quasinalytic if every function $f\in C_{M}$ $\{\mathbb{R}\}$ such that $f^{\text{ }(n)}(u)=0$ for some $u$ $\in\mathbb{R}$   and every $n\in\mathbb{N}$ is identically equal to $0.$

The Carleman class $C_{M}$ $\{\mathbb{R}\}$ is called regular if the following conditions hold :

[TABLE]

To the Carleman $C_{M}$ $\{\mathbb{R}\}$ we associate its weight $H_{M}$ defined by the following relation :

[TABLE]

In this paper the following result will play a crucial role.

Theorem 1**.**

$($[2]$)$ We assume that the Carleman class $C_{M}\{\mathbb{R}\}$ is regular. A function $f:\mathbb{R}\rightarrow\mathbb{C}$ belongs to $C_{M}\{\mathbb{R}\}$ if and only if there exists for every compact interval $I$ of $\mathbb{R}$ a compactly supported function $F_{I}:\mathbb{C}\rightarrow\mathbb{C}$ of class $C^{1}$ such that $F_{I}$ is an extension of $f$ and satisfies the following estimate $:$

[TABLE]

where $A_{I},$ $B_{I}>0$ are constants.

Throughout this paper, we assume that the Carleman class $C_{M}\{\mathbb{R}\}$ is regular and quasianalytic.

Our main result in this paper is the following.

Theorem 2**.**

Let $q\in\mathbb{N}^{\ast}\backslash\{1,2\},\ \delta>0,$ $\chi\in C_{M}\{\mathbb{R}\}$* and $a_{j}$ $\in O(\mathbb{R}_{\delta})$ $(j=1,...,q)$ such that $a_{1}$* and $a_{q}$ are nowhere vanishing on $\mathbb{R}_{\delta}.$ We assume that the following growth condition holds $:$

[TABLE]

for some constant $C>0.$ Then the difference equation $:$

[TABLE]

is solvable in the Carleman class $C_{M}\{\mathbb{R}\}.$

3. Proof of the main result

Let us first prove the following lemma.

Lemma 3**.**

Given $f\in C_{M}\{\mathbb{R}\},$ $C_{0}>0$ and $\rho\in$ $\left]0,\frac{\pi}{2C_{0}}\right[,$ there exist two functions$f_{\pm}:$ $(\mathbb{C}\backslash\Delta_{\rho}^{\pm})\cup\mathbb{R}\longrightarrow\mathbb{C}$ which are holomorphic on $\mathbb{C}\backslash(\Gamma_{\rho}^{\pm}\cup\Delta_{\rho}^{\pm}),$ whose restrictions to $\mathbb{R}$ belong to $C_{M}\{\mathbb{R}\},$and such that the following conditions hold $:$

[TABLE]

where $D_{0}>0$ is a constant.

Proof.

Since $f$ belongs to $C_{M}\{\mathbb{R}\}$ there exists, according to Dyn’kin’s theorem ([2]), a compactly supported function $F:\mathbb{C}\rightarrow\mathbb{C}$ of class $C^{1}$ such that $F$ is an extension of the restriction of $f$ to the interval $[-\rho,\rho]$ and satisfies the following estimate :

[TABLE]

where $A,$ $B>0$ are constants. Following the same approach as that of ([1], pages $34,35),$ but using the Cauchy-Pompeiu formula on the disk $\Delta_{\rho},$ for the function $e^{e^{C\text{ }_{0}z}+e^{-C_{0}\text{ }z}}f(z),$ we show that the functions :

[TABLE]

satisfy the required conditions.

Now we set :

[TABLE]

Let $C_{1}>C$  and $\delta_{0}\in\left]0,\min\left(\delta,\frac{\pi}{2C_{1}}\right)\right[.$ Then according to the lemma above, there exist onstant $D_{1}>0$ and two functions $\chi_{\pm}:(\mathbb{C}\backslash\Delta_{\delta_{0}}^{\pm})\cup\mathbb{R}\longrightarrow\mathbb{C}$  which are holomorphic on $\mathbb{C}\backslash(\Gamma_{\delta_{0}}^{\pm}\cup\Delta_{\delta_{0}}^{\pm}),$ whose restrictions to $\mathbb{R}$ belong to $C_{M}\{\mathbb{R}\},$ and such that the following conditions hold :

[TABLE]

Let $(g_{n})_{n\in\mathbb{N}\text{ }}$and $(h_{n})_{n\in\mathbb{N}\text{ }}$ be the sequences of complex valued functions defined on the strip $\mathbb{R}_{\delta_{0}}$ by the formulas :

[TABLE]

It is clear that all the functions $g_{n|\mathbb{R}}$ and $h_{n|\mathbb{R}}$ belong to $C_{M}$ $(\mathbb{R}).$ Let us set :

[TABLE]

It follows from (2.1) that we have for evey $n\in\mathbb{N}$ , $z\in\mathbb{R}_{\delta_{0}}:$

[TABLE]

where $L>1$ is a constant. Thence we have for all $n\in\mathbb{N}^{\ast},$ $z\in\mathbb{R}_{\delta_{0}}:$

[TABLE]

Let $a>0.$ There exists $N_{a}\in\mathbb{N}^{\ast}$ such that $(\beta_{2}+\gamma_{q-1})N_{a}\geq a$ and :

[TABLE]

It follows then from (LABEL:split) that we have for all $n\geq N_{a},$ $z\in\mathbb{R}_{\delta_{0},a}:$

[TABLE]

Consequently we have for all $n\geq N_{a},$ $z\in\mathbb{R}_{\delta_{0},a}:$

[TABLE]

On the other hand we have :

[TABLE]

Thence there exist real constants $D_{a}>0$ and $E_{a}>0$ and an integer $P_{a}\geq N_{a}$ such that the following inequalities hold :

[TABLE]

It follows that the function series $\sum g_{n|\mathbb{R}_{\delta_{0}}}$ and $\sum h_{n|\mathbb{R}_{\delta_{0}}}$ are uniformly convergent on every compact subset of $\mathbb{R}_{\delta_{0}}$ and that the functions \overset{+\infty}{\underset{n=N_{a}}{\sum}}g_{n}\and $\overset{+\infty}{\underset{n=N_{a}}{\sum}}h_{n}$ are holomorphic on $\mathbb{R}_{\delta_{0},a}$ for every $a>0.$ Let $G_{+}$ and $G_{-}$ be respectively the sums of $\sum g_{n|\mathbb{R}_{\delta_{0}}}$ and $\sum h_{n|\mathbb{R}_{\delta_{0}}}.$ Since all the functions $g_{n|\mathbb{R}}$ and $h_{n|\mathbb{R}}$ belong to $C_{M}\{\mathbb{R}\},$ it follows then that the functions $g_{+}:=G_{+}|_{\mathbb{R}}$ and $g_{-}:=G_{-}|_{\mathbb{R}}$ belong to $C_{M}\{\mathbb{R}\}.$ Elementary computations show that :

[TABLE]

Then it follows from (3.1) that the function $g:=g_{+}+g_{-}$ is a solution on the interval $[-\delta_{0},\delta_{0}]$ of the difference equation $(E$ $).$ But the function $x\mapsto\underset{j=1}{\overset{q}{\sum}}a_{j}(x)g(x+\alpha_{j})-\chi$ $(x)$ belongs to the quasianalytic Carleman class $C_{M}\{\mathbb{R}\}.$ Consequently the function $g\in C_{M}\{\mathbb{R}\}$ is a solution on $\mathbb{R}$ of the difference equation $(E).$ Thence the proof of the main result is complete.

$\square$