Asymptotic Behavior of Solutions of the Dirac System with an Integrable Potential

Abstract

We consider the Dirac system on the interval [0, 1] with a spectral parameter $\mu \in {\mathbb {C}}$ and a complex-valued potential with entries from $L_p[0,1]$, where $1\le p$. We study the asymptotic behavior of its solutions in a strip $|\mathrm{Im}\,\mu |\le d$ for $\mu \rightarrow \infty$. These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system.

1 Introduction

Consider a Cauchy problem

$$\begin{aligned} D'(x)+J(x) D(x)=A_\mu D(x),\quad D(0)=I, \end{aligned}$$

(1.1)

where $x \in [0,1],$ $A_\mu =i\mu J_0$, and

$$\begin{aligned} J_0=\left[ \begin{array}{cc} 1 &{} 0\\ 0 &{}-1 \end{array} \right] ,\quad J(x)=\left[ \begin{array}{cc} 0 &{} \sigma _1(x)\\ \sigma _2(x) &{} 0 \end{array} \right] ,\quad I:=\left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} 1 \end{array}\right] , \end{aligned}$$

(1.2)

$\mu \in {\mathbb {C}}$ is a spectral parameter, and $\sigma _j \in L_p[0,1],\,\, j = 1, 2$ are complex-valued functions where $1\le p < 2$. We study the asymptotic behavior of its solutions $D(x)=D(x,\mu )$ with respect to $\mu$ from a horizontal strip

$$\begin{aligned} P_d:=\{\mu \in {\mathbb {C}}:\,|\mathrm{Im}\,\mu |\le d\}. \end{aligned}$$

and $\mu \rightarrow \infty$.

The solution of (1.1) is a matrix D with entries from the space of absolutely continuous functions on [0, 1] (i.e. from the Sobolev space $W_1^1[0,1]$) satisfying (1.1) for a.e. $x\in [0,1]$. In our case, this definition together with the equation imply that D has entries from $W^1_p[0,1]$.

This article is an addendum to the paper [8], where the problem (1.1) was analyzed for $\sigma _j\in L_2[0,1]$, $j=1,2$. In that text one can find a background on Dirac systems and their connection with Sturm–Liouville problems.

Here, we use all advantages of the method from [8] to obtain sharp asymptotic formulas for D and consequently for solutions of spectral problems associated with (1.1). In the case when $\sigma _j\in L_p[0,1]$, $j=1,2$, $p > 2$ one can use the results from [8] due to the obvious embedding between $L_p[0,1]$ spaces. Thus, in this text we restrict our reasoning only to $1\le p < 2$.

We are interested in the following spectral problem:

$$\begin{aligned} Y'(x)+J(x) Y(x)=A_\mu Y(x),\quad x\in [0,1], \end{aligned}$$

(1.3)

where $Y=[y_1,y_2]^T$ and

$$\begin{aligned} y_1(0)=y_2(0), \ \ y_1(1)=y_2(1). \end{aligned}$$

(1.4)

Conditions (1.4) are an example of strongly regular boundary conditions. The Dirac-type systems or equation (1.3) with a general formulation of regular or strongly regular conditions have been studied for many years and by different methods.

The classical results can be found in [4], where G. D. Birkhoff and R. E. Langer obtained refined asymptotic formulas for general $n\times n$ system of the form

$$\begin{aligned} Ly&=-iBy'+Q(x)y=\lambda y,\nonumber \\ B&= \mathop {\mathrm {diag}}(b_1,\ldots ,b_n), \ \ y=\mathop {\mathrm {col}}(y_1,\ldots ,y_n), \ \ x\in [0,1], \end{aligned}$$

(1.5)

where $Q\in C^m[0,1] \times {\mathbb {C}}^{n\times n}$, $m\ge 1$. These formulas were used to obtain asymptotic formulas for eigenvalues and eigenfunctions of regular boundary value problem and to prove a pointwise convergence result for spectral decompositions. It is worth to mention that these results were generalized by V. S. Rykhlov in [18] for $Q\in L_1[0,1] \times {\mathbb {C}}^{n\times n}$.

For $2\times 2$ Dirac systems with potential matrix $Q\in C[0,1] \times {\mathbb {C}}^{2\times 2}$ the existence of a triangle transformation operator has been proved for the first time by Gasymov and Levitan (see [9]). Application to asymptotic formulas for solutions and eigenvalues can also be found there in chapter vii. In this case, V. A. Marchenko proved completeness property of regular BVP for Dirac operator in his classical monograph [16] (see exercises in §1.3).

Whereas, for $Q\in L_1[0,1] \times {\mathbb {C}}^{n\times n}$ there is a series of paper [1, 12, 14, 15] where asymptotic formulas for solutions were obtained in a special sectors and has been applied to establish the completeness property of regular and non-regular boundary value problems. In $2\times 2$ case more refined asymptotic formulas were derived using transformation operators.

The Riesz basis property of the root vectors system of the Dirac operator with strictly regular boundary conditions and summable potential matrix $Q\in L_1[0,1] \times {\mathbb {C}}^{2\times 2}$ was established by A. A. Lunyov and M. M. Malamud [11, 13] on the one hand, and A. M. Savchuk and A. A. Shkalikov in [25] on the other hand, independently by different methods and at the same time. In both papers basic asymptotic formula for solutions to the Dirac system were established in the strip $P_d$. This result was used to prove the following asymptotic formula

$$\begin{aligned} \mu _n=\mu _n^0+o(1) \end{aligned}$$

for the eigenvalues, which was one of the main ingredients for proving a Riesz basis property.

Furthermore A. A. Lunyov and M. M. Malamud have recently presented a preprint [10] where Lipschitz dependence on Q in $l^p$-norms of the sequences of eigenvalues and eigenfunctions of BVPs for Dirac operator was proved on compacts and balls in $L_p[0,1]\times {\mathbb {C}}^{2\times 2}$.

Moreover, in [25] A. M. Savchuk and A. A. Shkalikov obtained for $p\ge 1$ basic asymptotic formulas for eigenvalues and for fundamental solutions of the Dirac-type system only with the leading term and the remainders expressed by $\gamma$ and $\gamma _q$ given by (2.21) and (2.22). They got their results by applying Prüfer’s substitution.

The works of A. M. Savchuk and I. V. Sadovnichaya: [19, 21] and [22] may be regarded as articles which building up on the method from [20, 25] and its application for $p=1$ to problems from the fields of asymptotics formulas and basis properties (see also [23, 24]). Almost all aforementioned works prove or use the same type of results as mentioned before since they deal with the Riesz basis property and very detailed formulas are not needed.

In order to study inverse spectral problems S. Albeverio, R. Hryniv and Y. Mykytyuk in [2] investigated a direct spectral problem for the Dirac system in the form

$$\begin{aligned} BZ'(x)+Q(x)Z(x) =\mu Z(x),\quad x\in [0,1], \end{aligned}$$

(1.6)

where

$$\begin{aligned} B=\left[ \begin{array}{cc} 0 &{} 1\\ -1 &{} 0 \end{array} \right] ,\quad Q(x)=\left[ \begin{array}{cc} q_{1}(x) &{} q_{2}(x)\\ q_{2}(x) &{} -q_{1}(x)\end{array} \right] ,\quad q_{j}\in L_p[0,1], \ \ j=1,2, \end{aligned}$$

with $p\ge 1$. They proved also short formulas for fundamental system of solutions, where remainders were expressed in terms of Fourier coefficients of unknown functions from $L_p$. Furthermore, for the operators associated with the system (1.6) with two kinds of conditions

$$\begin{aligned} z_j(1)=z_2(0)=0, \ \ j=1,2, \end{aligned}$$

(1.7)

they presented basic formulas for eigenvalues with the same type of remainders.

More results concerning different type of problems for the Dirac system may be found in the series of paper of P. Djakov and B. Mityagin: [5, 6] and [7] or D. V. Puyda [17].

The main result of this paper concerning Dirac systems is

Theorem 1.1

Let $1 \le p \le 2 \, and\, 1/p+1/q=1$. Then for every $d>0$ there exist constants $C_j=C_j(d,\sigma _1,\sigma _2)$, $j=0,1,2$ such that for all $x\in [0,1]$ and $\mu \in P_d,$ the solutions of (1.1) admit the following representation

$$\begin{aligned} D(x,\mu )=e^{x A_\mu }+R(x,\mu ), \end{aligned}$$

(1.8)

where

$$\begin{aligned} \Vert R(x,\mu )\Vert _{{\mathbb {C}}^{2\times 2}}\le C_1(\gamma _q(\mu )+\gamma (x,\mu ))\le C_0, \ \ x\in [0,1]. \end{aligned}$$

Moreover,

$$\begin{aligned} D(x,\mu )=e^{xA_\mu } +D_0(x,\mu )+R_0(x,\mu ), \end{aligned}$$

(1.9)

where

$$\begin{aligned} D_{0}(x,\mu ):=\int _0^x e^{(x-2t)A_\mu }(-J(t)+{\tilde{J}}(x,t))\,dt \end{aligned}$$

and

$$\begin{aligned} \Vert R_0(x,\mu )\Vert _{{\mathbb {C}}^{2\times 2}}\le C_2(\gamma _q(\mu )\gamma (x,\mu )+{\widetilde{\gamma }}(\mu )),\quad x\in [0,1], \end{aligned}$$

where $\gamma ,\gamma _{0} \,\, and\,\, {\widetilde{\gamma }}$ are given by (2.21)-(2.23) and ${\tilde{J}}$ by (2.8).

Whereas, asymptotic representation for solutions of the spectral problem (1.3)-(1.4) are contained in the following theorem.

Theorem 1.2

The eigenvalues of the spectral problem (1.3)-(1.4) lie in a certain strip $P_d$ and admit the representation

$$\begin{aligned} \mu _n=\pi n + \mu _{0,n}+ \rho _n, \ \ n\in {\mathbb {Z}}, \end{aligned}$$

(1.10)

with

$$\begin{aligned} \mu _{0,n}&=\frac{1}{2i}\int _0^1 e^{-2\pi i n t} \sigma _1(t)dt -\frac{1}{2i}\int _0^1 e^{2\pi i n t}\sigma _2(t)dt\nonumber \\&-i\int _0^1\int _0^t\sigma _1(t)\sigma _2(\xi )e^{-2\pi i n t}e^{2\pi i n \xi } d \xi dt \end{aligned}$$

(1.11)

and for $p=1$ it holds that

$$\begin{aligned} |\rho _n|<c \Gamma ^2(\pi n), \end{aligned}$$

(1.12)

where $\Gamma$ is defined in (2.24), whereas for $1<p<2$ it is true that

$$\begin{aligned} \sum _{n\in {\mathbb {Z}}}|\rho _n|^{q/2}&< \infty . \end{aligned}$$

(1.13)

The result by A. M. Savchuk and A. A. Shkalikov from [25] is equivalent to first thesis (1.8) of Theorem 1.1. Note that the next statement (1.9) is a significant extension of the previous result. Its version for $p=1$ may be found in Remark 2.4. The most general result is the content of Lemma 2.2.

Using our method it is also possible to obtain very detailed formulas for eigenvalues and eigenfunctions. In case of the spectral problem associated with (1.4) the eigenvalues admit the representation (1.10)-(1.11) with remainders satisfying (1.12) and (1.13) for $p=1$ and $1<p<2$ respectively. In literature (for instance in [25]) for $1<p<2$ one may find results which state that eigenvalues are of the form $\pi n + r_n$, where $(r_n)\in l_q$, and q is conjugated to p. Here it is worth to underline that beside the leading term in our asymptotic formulas there occur Fourier coefficients of known functions and the remainder, which belongs to $l_{q/2}$. Additionally, for $p=1$ we extend known formulas with $|r_n|<c \Gamma (\pi n)$ (where $\Gamma$ is defined in (2.24)) into more detailed one with the remainder satisfying $|r_n|<c \Gamma ^2(\pi n)$.

In the same spirit Theorem 3.2 and Corollary 3.3 generalize significantly the results on eigenfunctions from literature.

Our method is applicable not only to the spectral problem (1.3)-(1.4) but it works as well for different cases of strongly regular boundary conditions. Moreover, it may be used to deal with the class of regular boundary conditions (in the sense of Birkhoff).

Results by S. Albeverio, R. Hryniv and Y. Mykytyuk from [2] can be directly derived from our approach with the help of transformation $Z=UY$, where

$$\begin{aligned} U=\left[ \begin{array}{cc} 1 &{} -i\\ -i &{} 1 \end{array}\right] . \end{aligned}$$

It leads to the system (1.1) with $\sigma _1=q_1+iq_2$ and $\sigma _2=q_1-iq_2$ with appropriate conditions. The relation between different formulations of Dirac systems is explained deeper in [8].

We start our presentation with the section concerning asymptotic behavior for solutions of Dirac system. Next, in section 3 we apply these results to the aforementioned spectral problem. For the clarity of exposition some technical results are placed at the end in appendix.

2 Dirac System and its Solutions

In this section we study the matrix Cauchy problem (1.1) and the behavior of its solution in a special integral form. The idea of this approach was taken from [16, Ch. 1, $\S 24$] and developed in [8]. We follow it here directly for similar operators but in different function spaces.

First, we introduce necessary notation. We use throughout the text a standard symbol $L_p[0,1]$, $p\ge 1$ to denote the space of measurable complex functions integrable with p-th power with the classical norm

$$\begin{aligned} \Vert f\Vert _{L_p}=\Big (\int _{0}^1|f(x)|^p dx\Big )^{1/p}. \end{aligned}$$

We write $l_p$, $p\ge 1$ for the space of complex sequences summable with p-th power and endowed with the norm

$$\begin{aligned} \Vert (x_n)\Vert _p=\Big (\sum _{n\in {\mathbb {Z}}}|x_n|^p\Big )^{1/p}. \end{aligned}$$

$W_p^1[0,1]$ is a standard Sobolev space with the derivative in $L_p[0,1]$.

If X is a Banach space, then M(X) stands for the Banach space of $2\times 2$ matrices with entries from X and the norm

$$\begin{aligned} \Vert Q\Vert _{M(X)}:=\sum _{k,j=1}^2 \Vert Q_{jk}\Vert _X,\quad Q=[Q_{jk}]_{j,k=1}^2. \end{aligned}$$

We assume throughout the text that $1\le p<2$. Moreover, let q and p be conjugate exponents and r be a number from Young’s convolution inequality i.e.

$$\begin{aligned} \frac{1}{p}+\frac{1}{q}=1\ \ \text{ and } \ \ r=\frac{p}{2-p}. \end{aligned}$$

(2.1)

Let

$$\begin{aligned} \Delta :=\{(x,t)\in {\mathbb {R}}^2: \; 0\le t\le x \le 1\} \end{aligned}$$

(2.2)

and

$$\begin{aligned} B:=\{f:[0,1]\times [0,1]\rightarrow {\mathbb {C}}\; a.e.\;&:\forall _{x\in [0,1]} \; f(x,\cdot )\in C([0,1],L_r),\\&\mathrm {supp} f\subset \Delta )\}. \end{aligned}$$

We equip B with the norm

$$\begin{aligned} \Vert f\Vert _B:=\sup _{x\in [0,1]}\Vert f(x,\cdot )\Vert _{L_r[0,x]}, \end{aligned}$$

so that B is a Banach space. In particular, directly from the definition if $f\in B$, then $f(x,t)=0$ for $0\le x < t \le 1$. This comment allows us to underline the property which will be used in the text i.e. for $f\in B$ there it holds that

$$\begin{aligned} \int _0^xf(x,t)dt=\int _0^1f(x,t)dt \in C[0,1]. \end{aligned}$$

(2.3)

We will use the series of constants connected with functions $\sigma _j$, $j=1,2$ in our estimations:

$$\begin{aligned} a:=\Vert \sigma _1\Vert _{L_1}\cdot \Vert \sigma _2\Vert _{L_1},\quad a_1:=\Vert \sigma _1\Vert _{L_1}+\Vert \sigma _2\Vert _{L_1},\quad a_p:=\Vert \sigma _1\Vert _{L_p}+\Vert \sigma _2\Vert _{L_p}.\nonumber \\ \end{aligned}$$

(2.4)

Moreover, let

$$\begin{aligned} \sigma _0(x):=|\sigma _1(x)|+|\sigma _2(x)|\in L_p[0,1]. \end{aligned}$$

(2.5)

Now we are ready to establish a first crucial property of the solutions of (1.1). The proof of the following lemma relays on technical results related to certain integral operators, which are placed in appendix.

Lemma 2.1

Let $\sigma _0\in L_p[0,1]$, $1\le p < 2$.

• a) The unique solution $D=D(x,\mu )$ of Cauchy problem (1.1) can be represented as

  $$\begin{aligned} D(x,\mu )=e^{xA_\mu }+\int _0^xe^{(x-2t)A_\mu }[-J(t)+Q(x,t)]dt, \end{aligned}$$

  (2.6)

  where $Q\in M(B)$ is the unique solution of the integral equation

  $$\begin{aligned} Q(x,t)= {\tilde{J}}(x,t)- \int _0^{x-t}J(t+\xi )Q(t+\xi ,\xi )d\xi , \end{aligned}$$

  (2.7)

  with ${\tilde{J}}\in M(B)$ given by

  $$\begin{aligned} {\tilde{J}}(x,t):= \int _0^{x-t}J(t+\xi )J(\xi )d\xi = \int _t^x J(s)J(s-t)\,ds,\quad (x,t)\in \Delta . \end{aligned}$$

  (2.8)

• b) The following estimates hold:

  $$\begin{aligned} \Vert Q\Vert _{M(B)}\le C_1, \ \ \Vert D\Vert _{M(C[0,1])} \le C_2, \ \ \mu \in P_d \end{aligned}$$

  (2.9)

  with certain constants $C_j=C_j(d,\sigma _1,\sigma _2)$, $j=1,2$.

Proof

Note that the uniqueness of solutions follows from general results on Sturm–Liouville equations (for instance [26, Thm. 1.2.1]). We look for solutions of (1.1) in a special form

$$\begin{aligned} D(x,\mu )=e^{xA_\mu }U(x,\mu ), \quad U(0,\mu )=I. \end{aligned}$$

(2.10)

The identity

$$\begin{aligned} J(x)e^{xA_\mu }=e^{-xA_\mu }J(x),\quad \text{ a. } \text{ e. }\;\;x\in [0,1] \end{aligned}$$

(2.11)

implies that U satisfies the Cauchy problem

$$\begin{aligned} U'(x,\mu )+e^{-2x A_\mu }J(x) U(x,\mu )=0,\qquad x\in [0,1],\quad U(0,\mu )=I, \end{aligned}$$

which is equivalent to the integral equation

$$\begin{aligned} U(x,\mu )=I-\int _0^xe^{-2tA_\mu }J(t)U(t,\mu )dt,\qquad x\in [0,1]. \end{aligned}$$

(2.12)

We will seek for solutions of (2.12) of the form

$$\begin{aligned} U(x,\mu )=I+\int _0^xe^{-2tA_\mu }Q_0(x,t)dt,\quad \end{aligned}$$

(2.13)

where $Q_0\in M(B)$ does not depend on $\mu$. Inserting (2.13) into (2.12), we obtain

$$\begin{aligned} \int _0^x e^{-2tA_\mu }Q_0(x,t)\,dt&= -\int _0^x e^{-2tA_\mu }J(t)\,dt\\&-\int _0^x e^{-2tA_\mu }J(t) \int _0^t e^{-2s A_\mu } Q_0(t,s)\,ds\,dt. \end{aligned}$$

Due to the fact that

$$\begin{aligned} J_0^2=I,\quad J_0 J(x)+J(x)J_0=0,\qquad \text{ a.e. }\;x\in [0,1], \end{aligned}$$

(2.14)

we get

$$\begin{aligned} \int _0^xe^{-2tA_\mu }J(t) \int _0^t e^{-2s A_\mu }&Q_0(t,s)ds\,dt =\int _0^xe^{-2tA_\mu } \int _0^t e^{2s A_\mu } J(t)Q_0(t,s)ds\,dt \\&=\int _0^xe^{-2tA_\mu }\int _0^{x-t} J(t+\xi )Q_0(t+\xi ,\xi )d\xi dt, \end{aligned}$$

thus

$$\begin{aligned} \int _0^xe^{-2tA_\mu }Q_0(x,t)\,dt= -\int _0^xe^{-2tA_\mu }\left( J(t) \!+\!\int _0^{x-t} J(t+\xi ) Q_0(t+\xi ,\xi )d\xi \right) \!dt \end{aligned}$$

for all $x\in [0,1].$ We conclude that U is a solution of (2.12) if and only if $Q_0\in M(B)$ is a solution of

$$\begin{aligned} Q_0(x,t)=-J(t)-\int _0^{x-t}J(t+\xi )Q_0(t+\xi ,\xi )d\xi . \end{aligned}$$

(2.15)

Next, setting

$$\begin{aligned} Q_0(x,t)=-J(t)+Q(x,t),\qquad (x,t)\in \Delta , \end{aligned}$$

and using (2.15), we infer that Q satisfies (2.7). For Q the equation (2.7) can be rewritten in an operator form

$$\begin{aligned} Q={\tilde{J}}+{\widetilde{T}} Q,\quad {\widetilde{T}}= -\left[ \begin{array}{cc} 0 &{} T_{\sigma _1}\\ T_{\sigma _2} &{} 0 \end{array}\right] , \end{aligned}$$

for the operators $T_{\sigma _1}$ and $T_{\sigma _2},$ defined on B by

$$\begin{aligned} (T_{\sigma }f)(x,t)=\int _0^{x-t}\sigma (t+\xi )f(t+\xi ,\xi )d\xi = \int _t^x\sigma (s)f(s,s-t)ds, \end{aligned}$$

(2.16)

where $\sigma \in L_p[0,1]$.

Observe that

$$\begin{aligned} {\tilde{J}}(x,t)=\left( \begin{array}{cc} {\tilde{\sigma }}_1(x,t) &{} 0\\ 0 &{} {\tilde{\sigma }}_2(x,t)\end{array}\right) , \end{aligned}$$

where

$$\begin{aligned} {\tilde{\sigma }}_1(x,t):=\int _0^{x-t}\sigma _1(t+\xi )\sigma _2(\xi )d\xi ,\quad {\tilde{\sigma }}_2(x,t):=\int _0^{x-t}\sigma _2(t+\xi )\sigma _1(\xi )d\xi .\nonumber \\ \end{aligned}$$

(2.17)

According to Lemma 4.1${\tilde{J}}\in M(B)$. What is more, the operators $T_{\sigma }$ are linear and bounded on B due to Lemma 4.3. In particular, we have

$$\begin{aligned} \Vert {\widetilde{T}}F\Vert _{M(B)} \le a_1 \Vert F\Vert _{M(B)},\quad F\in M(B). \end{aligned}$$

Next observe that

$$\begin{aligned} {\widetilde{T}}^{2n}= \left[ \begin{array}{cc} T_{12}^n &{}{} 0\\ 0 &{}{} T_{21}^n \end{array}\right] ,\quad n\in \mathbb {N}, \end{aligned}$$

for bounded linear operators $T_{12}$ and $T_{21}$ on B given by

$$\begin{aligned} T_{12}:=T_{\sigma _1} T_{\sigma _2},\quad T_{21}:=T_{\sigma _2} T_{\sigma _1}. \end{aligned}$$

Therefore by (4.4), we derive

$$\begin{aligned} \Vert {\widetilde{T}}^{2n}F\Vert _{M(B)} \le \frac{a^n}{n!}\Vert F\Vert _{M(B)},\quad F\in M(B), \end{aligned}$$

thus we see that (2.7) has a unique solution $Q\in M(B)$ of the form

$$\begin{aligned} Q=\sum _{n=0}^\infty {\widetilde{T}}^n {\tilde{J}}=\sum _{n=0}^\infty {\widetilde{T}}^{2n}(I+{\widetilde{T}}){\tilde{J}}, \end{aligned}$$

(2.18)

and moreover

$$\begin{aligned} \quad \Vert Q\Vert _{M(B)} \le (1+a_1)e^a\Vert {\tilde{J}}\Vert _{M(B)}. \end{aligned}$$

(2.19)

Then (2.18) and (4.1) yield (2.9).

Note that from (2.3) and (2.6) we get $D\in C[0,1]$. Adding together (2.6) and (2.9), we obtain

$$\begin{aligned} \Vert D\Vert _{M(C[0,1])}\le e^{d} \Big (1+a_1+\Vert Q(x,t)\Vert _{M(B)}\big ),\quad \mu \in P_d. \end{aligned}$$

(2.20)

$\square$

We now proceed to derivation of asymptotic formulas for D with the use of the previous lemma. In what follows we will use different types of estimates for remainders. For fixed $\sigma _j\in L_p$, $p\ge 1$, $j=1,2$, and $\mu \in {\mathbb {C}}$ define

$$\begin{aligned} \gamma _q(\mu )&:=\sum _{j=1}^2\left( \Big \Vert \int _0^xe^{-2i\mu t}\sigma _j(t)dt\Big \Vert _{L_q}+\Big \Vert \int _0^xe^{2i\mu t}\sigma _j(t)dt\Big \Vert _{L_q} \right) , \end{aligned}$$

(2.21)

where $1/q+1/p=1$. We will need also

$$\begin{aligned} \gamma (x,\mu )&:=\sum _{j=1}^2\left( \Big |\int _0^xe^{-2i\mu t}\sigma _j(t)dt\Big |+\Big |\int _0^xe^{2i\mu t}\sigma _j(t)dt\Big |\right) ,\quad x\in [0,1] \end{aligned}$$

(2.22)

and

$$\begin{aligned} {\widetilde{\gamma }}(\mu )&:=\int _0^1 \sigma _0(s)\gamma ^2(s,\mu )\,ds, \end{aligned}$$

(2.23)

$$\begin{aligned} \Gamma (\mu )&:=\sum _{j=1}^2\left( \sup _{x\in [0,1]}\Big |\int _0^xe^{-2i\mu t}\sigma _j(t)dt\Big |+\sup _{x\in [0,1]}\Big |\int _0^xe^{2i\mu t}\sigma _j(t)dt\Big |\right) . \end{aligned}$$

(2.24)

Note that $\Gamma$ is nothing else than $\gamma _q$ for $p=1$ and $q=\infty$.

It is easy to see that if $\mu \in P_d$ then

$$\begin{aligned} \gamma (x,\mu )\le 2e^{2d}a_1, \ \ \Vert \gamma (x,\mu )\Vert _{L_q}\le \gamma _q(\mu ),\quad \gamma _q(\mu )\le 2e^{2d}a_1, \quad x\in [0,1],\nonumber \\ \end{aligned}$$

(2.25)

and

$$\begin{aligned} {\widetilde{\gamma }}(\mu )\le 4a_1^3e^{4d},\quad {\widetilde{\gamma }}(\mu )\le 2a_1e^{2d}\Vert \sigma _0\Vert _{L_p}\gamma _q(\mu ). \end{aligned}$$

(2.26)

In the following lemma we will need

$$\begin{aligned} N(x,t):=({\tilde{J}}+{\widetilde{T}}{\tilde{J}})(x,t)\in B. \end{aligned}$$

(2.27)

Observe that the explicit form of N is

$$\begin{aligned} N(x,t)= \left( \begin{array}{cc} {\tilde{\sigma }}_1(x,t) &{} -(T_{\sigma _1}{\tilde{\sigma }}_2)(x,t)\\ -(T_{\sigma _2}{\tilde{\sigma }}_1)(x,t) &{} {\tilde{\sigma }}_2(x,t) \end{array} \right) . \end{aligned}$$

The very basic but crucial result uses mainly the description of some integrals connected with the operator ${\widetilde{T}}$ and its powers stated in Lemma 4.6.

Lemma 2.2

Let $\sigma _j\in L_p$, $1\le p <2$ for $j=1,2$. If $D(\cdot ,\mu )$ is the solution of (1.1) then $D(\cdot ,\mu )\in C[0,1]$ and

$$\begin{aligned} D(x,\mu )&=e^{x A_\mu } + D^{(0)}(x,\mu )+D^{(1)}(x,\mu ), \end{aligned}$$

(2.28)

where

$$\begin{aligned} D^{(0)}(x,\mu )=-\int _0^x e^{(x-2t)A_\mu } J(t)\,dt +\int _0^x e^{(x-2t)A_\mu } N(x,t)\,dt, \end{aligned}$$

and for all $\mu \in P_d$ and $x \in [0,1],$

$$\begin{aligned} \Vert D^{(1)}(x,\mu )\Vert _{M({\mathbb {C}})}\le C{\widetilde{\gamma }}(\mu ), \end{aligned}$$

(2.29)

where $C=C(d,\sigma _1,\sigma _2)$.

Proof

Note that by using (2.6) and (2.18) for $D=D(x,\mu )$, $x\in [0,1]$, $\mu \in P_d$, we get

$$\begin{aligned} D(x,\mu )&=e^{xA_\mu }-\int _0^xe^{(x-2t)A_\mu }J(t)dt +\int _0^xe^{(x-2t)A_\mu }{\tilde{J}}(x,t)dt \nonumber \\&+\int _0^xe^{(x-2t)A_\mu }({\widetilde{T}}{\tilde{J}})(x,t)dt+D^{(1)}(x,\mu ) \end{aligned}$$

(2.30)

where

$$\begin{aligned} D^{(1)}(x,\mu )=\int _0^xe^{(x-2t)A_\mu }\sum _{n=2}^\infty ({\widetilde{T}}^n{\tilde{J}})(x,t)dt. \end{aligned}$$

Using (2.30) and the inequality (4.12) proved in appendix, we show that

$$\begin{aligned} \Vert D^{(1)}(x,\mu )\Vert _{M({\mathbb {C}})}&\le \sum _{n=2}^\infty \left\| \int _0^xe^{(x-2t)A_\mu }({\widetilde{T}}^n{\tilde{J}})(x,t)dt\right\| _{M({\mathbb {C}})} \\&\le 3 e^d {\widetilde{\gamma }}(\mu )\sum _{n=2}^\infty \frac{e^{2nd}a_1^{n-1}}{(n-2)!}= 3a_1e^{5d}\exp {(e^{2d}a_1)}{\widetilde{\gamma }}(\mu ), \end{aligned}$$

for all $x\in [0,1]$ and $\mu \in P_d$. $\square$

The above lemma leads to sharp asymptotic formulas for D, which are the main result of this section and were stated in Theorem 1.1.

Proof of Theorem 1.1

Let us start with several simple observations. First of all, remark that clearly

$$\begin{aligned} C_1(\gamma _q(\mu )+\gamma (x,\mu )\big )\le C_0. \end{aligned}$$

Furthermore, due to inequalities (2.26), (2.29) and (4.11) we get

$$\begin{aligned} \Vert R_{0}(x,\mu )\Vert _{M({\mathbb {C}})}&\le \left\| \int _0^x e^{i\mu (x-2t)}{\widetilde{T}}{\tilde{J}}(x,t) \,dt\right\| _{M({\mathbb {C}})} +\Vert D^{(1)}(x,\mu )\Vert _{M({\mathbb {C}})} \\&\le e^{3d}(a_2+1)\big (\gamma _q(\mu )\gamma (x,\mu )+\gamma _1(\mu )\big ) +C{\widetilde{\gamma }}(\mu )\\&\le C_2(\gamma _q(\mu )\gamma (x,\mu )+{\widetilde{\gamma }}(\mu )\big )\\&\le 2C_2a_1e^{2d}(\gamma (x,\mu )+a_2\gamma _q(\mu )\big ) \end{aligned}$$

Note also that from

$$\begin{aligned} \left\| \int _0^x e^{(x-2t)A_\mu } J(t)\,dt\right\| _{M({\mathbb {C}})}\le e^d\gamma (x,\mu ),\quad x\in [0,1], \end{aligned}$$

(4.9), and (4.10) it follows that

$$\begin{aligned} \left\| D^{(0)}(x,\mu )\right\| _{M({\mathbb {C}})}\le e^d\gamma (x,\mu )+ 2e^{5d} (1+a_p)a_p \gamma _q(\mu ),\quad x\in [0,1]. \end{aligned}$$

Combining all these inequalities with Lemma 2.2 and the estimates from (2.26), we prove representations for D from Theorem 1.1. $\square$

Remark 2.3

Note that the explicit formula for $D_0$ is the following

$$\begin{aligned}&D_{0}(x,\mu )=\left( \begin{array}{cc} r_1(x,\mu ) &{} q_1(x,\mu ) \\ q_2(x,\mu ) &{} r_2(x,\mu ) \end{array} \right) , \\&q_1(x,\mu ):=-\int _0^x e^{i\mu (x-2t)} \sigma _1(t)\,dt,\quad q_2(x,\mu ):=-\int _0^x e^{-i\mu (x-2t)} \sigma _2(t)\,dt\\&r_1(x,\mu ):=\int _0^x e^{i\mu (x-2t)} {\tilde{\sigma }}_1(x,t)\,dt,\quad r_2(x,\mu ):=\int _0^x e^{-i\mu (x-2t)} {\tilde{\sigma }}_2(x,t)\,dt \end{aligned}$$

and ${\tilde{\sigma }}_j$ are given by (2.17).

Remark 2.4

If $p=1$, then the remainder $R_0$ from (1.9) satisfies

$$\begin{aligned} \Vert R_0(x,\mu )\Vert _{M({\mathbb {C}})}\le C_2\Gamma ^2(\mu ), \end{aligned}$$

where $\Gamma$ is given by (2.24).

Remark 2.5

There are misprints in the paper [8] in the formulation of the analogon of Lemma 2.1. The identities (2.14) and (2.15) from [8] should be the same as (2.6) and (2.7) from this paper. Consequently, it implies changes of signs in (2.28), (2.35), (2.37) and inside Remark 2.6 in [8]. Results concerning the spectral problem remains true.

3 Spectral Problem

We consider a spectral problem

$$\begin{aligned} Y'(x)+J(x) Y(x)=A_\mu Y(x),\quad x\in [0,1], \end{aligned}$$

(3.1)

associated with the matrix problem (1.1) where $Y=[y_1,y_2]^T$ and

$$\begin{aligned} y_1(0)=y_2(0), \ \ y_1(1)=y_2(1). \end{aligned}$$

(3.2)

Let ${\mathbf {c}}={\mathbf {c}}(x,\mu )=[c_1,c_2]^T$ and ${\mathbf {s}}={\mathbf {s}}(x,\mu )=[s_1,s_2]^T$ be the solutions of (3.1) satisfying $c_1(0)=1$, $c_2(0)=0$ and $s_1(0)=0$, $s_2(0)=1$. Then due to conditions (3.2) we find that the eigenvalues are the zeros of

$$\begin{aligned} \Phi (\lambda )=c_1(1,\lambda ) + s_1(1,\lambda ) - c_2(1,\lambda ) - s_2(1,\lambda ). \end{aligned}$$

(3.3)

The eigenfunctions will be of the form:

$$\begin{aligned} Y=\left( \begin{array}{c} y_1 \\ y_2 \end{array} \right) = \left( \begin{array}{c} c_1(\cdot ,\mu _n) + s_1(\cdot ,\mu _n) \\ c_2(\cdot ,\mu _n) + s_2(\cdot ,\mu _n) \end{array} \right) \end{aligned}$$

(3.4)

The analysis of zeros of (3.3) will lead to the characterization of eigenvalues stated in Theorem 1.2.

Proof

(Proof of Theorem 1.2) The standard approach is to obtain first basic formula for eigenvalues and then derive more accurate form using sharp asymptotic results. We thus need results related to functions ${\mathbf {s}}$ and ${\mathbf {c}}$ from (2.6). We derive that

$$\begin{aligned} \Phi (\mu )&=2i\sin \mu + \int _0^1 e^{(1-2t)i\mu } \Big (Q_{11}(1,t)+Q_{12}(1,t)-\sigma _1(t)\Big )dt\nonumber \\&-\int _0^1 e^{-(1-2t)i\mu } \Big (Q_{21}(1,t)+Q_{22}(1,t)-\sigma _2(t)\Big )dt. \end{aligned}$$

(3.5)

Via changing variables in the expression above we obtain

$$\begin{aligned} \Phi (\mu )=2i\sin (\mu )+ V(\mu ), \end{aligned}$$

(3.6)

where

$$\begin{aligned} V(\mu )=\int _{-1}^1e^{i\mu s}h(s)ds \end{aligned}$$

(3.7)

and h is a certain function from $L_p[-1,1]$.

Note that the identities (3.5) and (3.6) are true not only for $\mu \in P_d$ but for all $\mu \in {\mathbb {C}}$. It is a standard procedure (see for instance [3]) to derive using Rouche Theorem that zeros of $\Phi$ are of the form $\mu _n=\pi n + {\widetilde{\mu }}_n$, where $({\widetilde{\mu }}_n)$ is bounded. This conclusion implies that eigenvalues lie in a certain horizontal strip of the complex plane. We may continue and investigate more precisely the behavior of $({\widetilde{\mu }}_n)$.

The formula for $\Phi$ gives us

$$\begin{aligned} \sin ({\widetilde{\mu }}_n)=\frac{(-1)^{n+1}}{2i}V(\pi n+{\widetilde{\mu }}_n). \end{aligned}$$

(3.8)

This expression converges to zero since the convergence of the integral in (3.7) follows from Lebesgue–Riemann Lemma and the fact that ${\widetilde{\mu }}_n$ are bounded. Thus ${\widetilde{\mu }}_n\rightarrow 0$ when $n\rightarrow \infty$. Here ends the reasoning and first claim for $p=1$.

For $1<p<2$ we may continue in order to obtain more information. Using $\sin x= x +\text{ O }(x^3),$ $x\rightarrow 0$, and the fact that ${\widetilde{\mu }}_n\rightarrow 0$ we obtain

$$\begin{aligned} {\widetilde{\mu }}_n&=\frac{(-1)^{n+1}}{2i}\int _{-1}^1e^{i{\widetilde{\mu }}_n s}e^{i\pi n s}h(s)ds +\text{ O }(|{\widetilde{\mu }}_n|^3). \end{aligned}$$

(3.9)

Next, we recall the expansion of the exponential function

$$\begin{aligned} e^{\mu t}=1+\mu t+O(|\mu |^2),\ \ \mu \rightarrow 0, \ \ |t|\le 1. \end{aligned}$$

The above formula and (3.9) yield

$$\begin{aligned} {\widetilde{\mu }}_n\Big (1-ic_{1,n}-O(|{\widetilde{\mu }}_n|)\Big )=c_{0,n}, \end{aligned}$$

(3.10)

where $c_{j,n}=\int _0^1e^{i\pi n s }s^jh(s)ds$ are Fourier coefficients of $s^jh(s)$, $j=0,1$. Since $h\in L_p[0,1]$, then by the Hausdorff–Young theorem $(c_{j,n})\in l_q$, $j=0,1$. Thus $c_{1,n}\rightarrow 0$ and ${\widetilde{\mu }}_n\rightarrow 0$ as $n\rightarrow \infty$. Formula (3.10) now implies that $({\widetilde{\mu }}_n)\in l_q$.

Summarizing, we showed that the eigenvalues $\mu _n$ of our spectral problem satisfy

$$\begin{aligned} \mu _n=\pi n + {\widetilde{\mu }}_n, \ \ ({\widetilde{\mu }}_n)\in l_q, \ \ n\in {\mathbb {Z}}. \end{aligned}$$

(3.11)

This representation for $1<p$ and the fact that for $p=1$ the remainder goes to zero allows us to find in both cases more accurate description of eigenvalues. Recall we showed that eigenvalues lie in $P_d$ for a certain $d>0$, thus we can use asymptotic formulas which are true for $\mu \in P_{d}$. The main tool will be the formulas for ${\mathbf {c}}$ and ${\mathbf {s}}$ from Theorem 1.1 and Remark 2.3 and consequently for $\Phi$.

This way we get

$$\begin{aligned} \Phi (\mu )&=2i\sin \mu - \int _0^1 e^{(1-2t)i\mu } \sigma _1(t)dt+ \int _0^1 e^{(1-2t)i\mu } {\tilde{\sigma }}_1(1,t)dt\nonumber \\&+\int _0^1 e^{-(1-2t)i\mu }\sigma _2(t)dt-\int _0^1 e^{-(1-2t)i\mu }{\tilde{\sigma }}_2(1,t)dt+r(\mu ), \end{aligned}$$

(3.12)

where

$$\begin{aligned} |r(\mu )|\le c(\gamma _q(\mu )\gamma (1,\mu )+{\widetilde{\gamma }}(\mu ))\le c(\gamma _q^2(\mu )+\gamma ^2(1,\mu )+{\widetilde{\gamma }}(\mu )). \end{aligned}$$

We now follow along the same lines as in the discussion about eigenvalues to analyze remainders. The representation (3.11) yield

$$\begin{aligned} \int _0^1 e^{\pm (1-2t)i\mu _n} \sigma (t)dt=(-1)^n\int _0^1 e^{\mp ti \pi n} e^{\pm (1-2t)i{\widetilde{\mu }}_n} \sigma (t)dt \end{aligned}$$

(3.13)

for $\sigma \in L_p$. Using an expansion

$$\begin{aligned} e^{\pm (1-2t)i{\widetilde{\mu }}_n}=1\pm (1-2t)i{\widetilde{\mu }}_n+O(|{\widetilde{\mu }}_n|^2), \ \ n\rightarrow \infty , \end{aligned}$$

(3.14)

and Lemma 4.2 we establish

$$\begin{aligned} 2i{{\widetilde{\mu }}}_n&=\int _0^1 e^{-2\pi i n t} \sigma _1(t)dt -\int _0^1 e^{2\pi i n t}\sigma _2(t)dt\nonumber \\&+2\int _0^1\int _0^t\sigma _1(t)\sigma _2(\xi )e^{-2\pi i n t}e^{2\pi i n \xi } d \xi dt +r(\mu _n)(1+s_n+O(|{\widetilde{\mu }}_n|)), \end{aligned}$$

(3.15)

where $s_n$ are Fourier coefficients of some functions from $L_1[0,1]$. For $p=1$ the last term can be estimated by $\Gamma ^2(\pi n)$.

Our last aim is to prove that for $(r(\mu _n))\in l_{q/2}$. In what follows we will use a basic formula for eigenvalues (3.11), a simple inequality $|e^{iz}-1|\le |z|e^{d}$, $z\in P_d$ and the Hausdorff–Young inequality. We infer for $\sigma \in L_p[0,1]$ that $1<p<2$

$$\begin{aligned} \sum _{n\in {\mathbb {Z}}} \Big |\int _0^xe^{\pm 2i\mu _nt}\sigma (t)dt\Big |^{q}&\le c_q\sum _{n\in {\mathbb {Z}}} \left| \int _0^xe^{\pm 2\pi i n t}\sigma (t)dt\right| ^q\nonumber \\&+ c_q\sum _{n\in {\mathbb {Z}}} \left( \int _0^x|e^{\pm 2i{\tilde{\mu }}_n t}-1||\sigma (t)|dt\right) ^q \nonumber \\&\le c_q \Vert \sigma \Vert ^q_{L_p[0,1]}+c\Vert \sigma \Vert _{L_1[0,1]}^q\sum _{n\in {\mathbb {Z}}} |{\tilde{\mu }}_n|^q\le m<\infty , \end{aligned}$$

(3.16)

for any $x\in [0,1]$. It follows from (3.16) that

$$\begin{aligned} \sup _{x\in [0,1]}\sum _{n\in {\mathbb {Z}}} \gamma ^{q}(x,\mu _n)< \infty , \end{aligned}$$

(3.17)

Note that by (3.17)

$$\begin{aligned} \sum _{n\in {\mathbb {Z}}}\gamma _q^q(\mu _n) \le c \int _0^1 \sum _{n\in {\mathbb {Z}}}\gamma ^q(s,\mu _n) ds < \infty . \end{aligned}$$

and

$$\begin{aligned} \Vert ({\widetilde{\gamma }}(\mu _n))\Vert _{l_{q/2}}&\le \int _0^1 \sigma _0(s)\Vert (\gamma ^2(s,\mu _n))\Vert _{l_{q/2}}\,ds\\&=\int _0^1 \sigma _0(s) \Bigg (\sum _{n\in {\mathbb {Z}}}\gamma ^q(s,\mu _n)\Bigg )^{2/q}\,ds \le c\Vert \sigma _0\Vert _{L_1}. \end{aligned}$$

Finally, we obtain

$$\begin{aligned} \sum _{n\in {\mathbb {Z}}} |r_n|^{q/2}&< \infty . \end{aligned}$$

Summarizing the discussion above we proved Theorem 1.2. $\square$

Remark 3.1

Recall that according to Lemma 4.1 for every $x\in [0,1]$ functions ${\tilde{\sigma }}_j(x,\cdot )$ are from $L_r$. If $1<p\le \frac{4}{3}$, then $1<r\le 2$ and Fourier coefficients of ${\tilde{\sigma }}_j(x,\cdot )$ are from $l_{q/2}$. Then the representation (1.10) with

$$\begin{aligned} \sum _{n\in {\mathbb {Z}}} |\rho _n|^{q/2}&< \infty \end{aligned}$$

is true but with $\mu _{0,n}$ given by

$$\begin{aligned} 2i\mu _{0,n}&=\int _0^1 e^{-2\pi i n t} \sigma _1(t)dt -\int _0^1 e^{2\pi i n t}\sigma _2(t)dt. \end{aligned}$$

(3.18)

Now, we can proceed to eigenfunctions. We are going to combine results from Theorem 1.2 with Lemma 2.2 and Theorem 1.1.

Theorem 3.2

Let $1<p<2$ and

$$\begin{aligned} F_1(x,t)&=-\sigma _1(t)+{\tilde{\sigma }}_1(x,t)-(T_{\sigma _1}{\tilde{\sigma }}_2)(x,t)\\ F_2(x,t)&=-\sigma _2(t)+{\tilde{\sigma }}_2(x,t)-(T_{\sigma _2}{\tilde{\sigma }}_1)(x,t). \end{aligned}$$

The eigenfunctions of the spectral problem (3.1)-(3.2) admit the representation

$$\begin{aligned} y_1(x,\mu _n)&=e^{i\pi n x}(1+i\mu _{0,n}x)\Bigg (1+\int _0^x e^{-2\pi i n t} F_1(x,t)dt\Bigg )\\&-2i\mu _{0,n}e^{i\pi n x}\int _0^x e^{-2\pi i n t} tF_1(x,y)dt + r_1(x,n),\\ y_2(x,\mu _n)&=e^{-i\pi n x}(1-i\mu _{0,n}x)\Bigg (1+\int _0^x e^{2\pi i n t} F_2(x,t)dt\Bigg )\\&+2i\mu _{0,n}e^{-i\pi n x}\int _0^x e^{2\pi i n t} tF_2(x,t)dt + r_2(x,n), \end{aligned}$$

where

$$\begin{aligned} \sum _{n\in {\mathbb {Z}}}\sup _{x\in [0,1]} |r_j(x,n)|^{q/2}&< \infty , \end{aligned}$$

Proof

According to (3.4) eigenfunctions are expressed by solutions ${\mathbf {c}}$ and ${\mathbf {s}}$ in the following way

$$\begin{aligned} y_1(x,\mu _n)=c_1(x,\mu _n) + s_1(x,\mu _n) \end{aligned}$$

and

$$\begin{aligned} y_2(x,\mu _n)=c_2(x,\mu _n) + s_2(x,\mu _n). \end{aligned}$$

The formula (2.28) yields that

$$\begin{aligned} y_1(x,\mu _n)&=e^{i\mu _n x} - \int _0^x e^{(x-2t)i\mu _n} \sigma _1(t)dt\\&+ \int _0^x e^{(x-2t)i\mu _n} {\tilde{\sigma }}_1(x,t)dt-\int _0^xe^{i \mu (x-2t)}(T_{\sigma _1}{\tilde{\sigma }}_2)(x,t)dt+ \alpha (x,\mu _n),\\ y_2(x,\mu _n)&=e^{-i\mu _n x} -\int _0^x e^{-(x-2t)i\mu _n} \sigma _2(t)dt + \int _0^x e^{-(x-2t)i\mu _n} {\tilde{\sigma }}_2(x,t)dt\\&-\int _0^xe^{-i \mu (x-2t)}(T_{\sigma _2}{\tilde{\sigma }}_1)(x,t)dt+ \beta (x,\mu _n), \end{aligned}$$

where

$$\begin{aligned} |\alpha (x,\mu _n)|+|\beta (x,\mu _n)| \le c{\widetilde{\gamma }}(\mu _n). \end{aligned}$$

Repeating once more all arguments used in order to derive formulas for eigenvalues, we obtain the thesis with claimed estimates for remainders. $\square$

It is possible to obtain shorter but less precise formulas for eigenfunctions. This time we use the representation (1.9) and comments from Lemma 4.2 to prove the following fact.

Corollary 3.3

Let $1\le p<2$, then the eigenfunctions of the spectral problem (3.1)-(3.2) admit the representation

$$\begin{aligned} y_1(x,\mu _n)&=e^{i\pi n x}\Big (1+i\mu _{0,n}x-\int _0^x e^{-2\pi i n t} \sigma _1(t)dt\nonumber \\&+\int _0^x\int _0^s\sigma _1(s)\sigma _2(\xi )e^{-2i\mu s}e^{2i\mu \xi } d \xi ds \Big )+ r_1(x,n), \nonumber \\ y_2(x,\mu _n)&=e^{-i\pi n x}\Big (1-i\mu _{0,n}x-\int _0^x e^{2\pi i n t} \sigma _2(t)dt\nonumber \\&+\int _0^x\int _0^s\sigma _1(\xi )\sigma _2(s)e^{2i\mu s}e^{-2i\mu \xi } d \xi ds\Big ) + r_2(x,n), \end{aligned}$$

(3.19)

where for $1<p<2$ we have

$$\begin{aligned} \sup _{x\in [0,1]}\sum _{n\in {\mathbb {Z}}} |r_j(x,n)|^{q/2}&< \infty , \end{aligned}$$

whereas for $p=1$ there holds

$$\begin{aligned} |r_j(x,n)|\le c\Gamma ^2(\pi n). \end{aligned}$$

Appendix

Lemma 4.1

For every $x\in [0,1]$ and $j=1,2$ the functions ${\tilde{\sigma }}_j(x,\cdot )$ belong to $L_r[0,1]$, where r is defined in (2.1). Furthermore, we have ${\tilde{\sigma }}_j\in B$, $j=1,2,$ and ${\tilde{J}}\in M(B)$.

Proof

We take $(x,t)\in \Delta$. Let ${\widehat{\sigma }}_1,{\widehat{\sigma }}_2$ denote the extension of $\sigma _1$ and $\sigma _2$ by zero outside [0, 1]. Note that for every $x\in [0,1]$ we get

$$\begin{aligned} \int _0^{x-t}\sigma _1(t+\xi )\sigma _2(\xi )d\xi&=\int _{-\infty }^\infty {\widehat{\sigma }}_1(t+\xi ){\widehat{\sigma }}_2(\xi )\chi (x-(t+\xi ))d\xi \\&=\int _{-\infty }^\infty {\widehat{\sigma }}_1(t-s){\widehat{\sigma }}_2(-s)\chi (x-(t-s))ds\\&=\Big (({\widehat{\sigma }}_1(\cdot )\chi (x-\cdot ))*({\widehat{\sigma }}_2(-\cdot ))\Big )(t). \end{aligned}$$

We thus have

$$\begin{aligned} \Big \Vert \int _0^{x-t}\sigma _1(t+\xi )\sigma _2(\xi )d\xi \Big \Vert _{L_r[0,1]}&\le \Big \Vert \int _0^{x-t}{\widehat{\sigma }}_1(t+\xi ){\widehat{\sigma }}_2(\xi )d\xi \Big \Vert _{L_r({\mathbb {R}})}\\&\le \Vert ({\widehat{\sigma }}_1(\cdot )\chi (x-\cdot ))\Vert _{L_p({\mathbb {R}})}\Vert {\widehat{\sigma }}_2(-\cdot )\Vert _{L_p({\mathbb {R}})}\\&\le \Vert \sigma _1\Vert _{L_p[0,1]}\Vert \sigma _2\Vert _{L_p[0,1]}, \end{aligned}$$

hence

$$\begin{aligned} \Vert {\tilde{\sigma }}_j\Vert _{B}\le \Vert \sigma _1\Vert _{L_p[0,1]}\Vert \sigma _2\Vert _{L_p[0,1]}, \qquad j=1,2. \end{aligned}$$

(4.1)

Clearly, a similar estimate holds for ${\tilde{\sigma }}_2$ as well.

Therefore, if we consider $\epsilon$ and x such that $0\le t\le x+\epsilon \le 1$, then using again the inequality for convolutions, we obtain

$$\begin{aligned}&\Big \Vert \int _0^{x+\epsilon -t}\sigma _1(t+\xi )\sigma _2(\xi )d\xi -\int _0^{x-t}\sigma _1(t+\xi )\sigma _2(\xi )d\xi \Big \Vert _{L_r[0,1]}\\&\quad \le \Vert ({\widehat{\sigma }}_1(\cdot )\big [\chi (x+\epsilon -\cdot )-\chi (x-\cdot )\big ]\Vert _{L_p({\mathbb {R}})}\Vert {\widehat{\sigma }}_2(-\cdot )\Vert _{L_p({\mathbb {R}})}\\&\quad \le \int _{{\mathbb {R}}}|{\widehat{\sigma }}_1(s)|^p|\chi (x+\epsilon -s)-\chi (x-s)|^pds\Vert {\widehat{\sigma }}_2\Vert _{L_p({\mathbb {R}})}. \end{aligned}$$

The integral in the last line converges to zero, if $\epsilon \rightarrow 0$, because of Lebesgue Theorem, hence the mapping $x\mapsto {\tilde{\sigma }}_j(x,\cdot )\in L_r[0,1]$ is continuous for$j=1,2$. $\square$

Lemma 4.2

The following identity holds

$$\begin{aligned} \int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt + \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt&=\int _0^xe^{-2i\mu \xi }\sigma _1(\xi )d \xi \int _0^x\sigma _2(s)e^{2i\mu s} ds. \end{aligned}$$

Moreover, we have

$$\begin{aligned} \Big |\int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt + \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt\Big |\le c \gamma ^2(x,\mu ) \end{aligned}$$

and

$$\begin{aligned}&\int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt - \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt\\&\quad =-2\int _0^x\int _0^s\sigma _1(\xi )\sigma _2(s)e^{2i\mu s}e^{-2i\mu \xi } d \xi ds + \alpha (\mu ), \end{aligned}$$

where $\alpha (\mu )=O(\gamma ^2(x,\mu ))$.

Proof

Note that

$$\begin{aligned} \int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt&=\int _0^x\int _0^s\sigma _1(s)\sigma _2(\xi )e^{-2i\mu s}e^{2i\mu \xi } d \xi ds\\ \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt&=\int _0^x\int _0^s\sigma _1(\xi )\sigma _2(s)e^{2i\mu s}e^{-2i\mu \xi } d \xi ds. \end{aligned}$$

Observe that the change of variables yields

$$\begin{aligned} \int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt&=\int _0^x\int _\xi ^x\sigma _1(\xi )\sigma _2(s)e^{-2i\mu s}e^{2i\mu \xi } ds d\xi \\&=\int _0^x\int _s^x\sigma _1(s)\sigma _2(\xi )e^{-2i\mu \xi }e^{2i\mu s} d\xi ds, \end{aligned}$$

thus

$$\begin{aligned} \int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt&+ \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt\\&=\int _0^x\int _0^x\sigma _1(\xi )\sigma _2(s)e^{2i\mu s}e^{-2i\mu \xi } d \xi ds\\&=\int _0^xe^{-2i\mu \xi }\sigma _1(\xi )d \xi \int _0^x\sigma _2(s)e^{2i\mu s} ds. \end{aligned}$$

This equality implies that

$$\begin{aligned} \Big |\int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt + \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt\Big |\le c \gamma ^2(x,\mu ). \end{aligned}$$

What is more, then

$$\begin{aligned} \int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt&- \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt\\&=\int _0^xe^{-2i\mu \xi }\sigma _1(\xi )d \xi \int _0^x\sigma _2(s)e^{2i\mu s} ds\\&-2\int _0^x\int _0^s\sigma _1(\xi )\sigma _2(s)e^{2i\mu s}e^{-2i\mu \xi } d \xi ds, \end{aligned}$$

thus

$$\begin{aligned} \int _0^x e^{-2ti\mu } {\tilde{\sigma }}_1(x,t)dt&- \int _0^x e^{2ti\mu } {\tilde{\sigma }}_2(x,t)dt\\&=-2\int _0^x\int _0^s\sigma _1(\xi )\sigma _2(s)e^{2i\mu s}e^{-2i\mu \xi } d \xi ds\\&+ \alpha (\mu ), \end{aligned}$$

where $\alpha (\mu )=O(\gamma ^2(x,\mu ))$. $\square$

Lemma 4.3

The linear operator $T_{\sigma }$

$$\begin{aligned} (T_{\sigma }f)(x,t)=\int _0^{x-t}\sigma (t+\xi )f(t+\xi ,\xi )d\xi = \int _t^x\sigma (s)f(s,s-t)ds, \end{aligned}$$

(4.2)

where $\sigma \in L_p[0,1]$, is bounded in B.

Proof

Note that

$$\begin{aligned} \Big (\int _0^x|(T_\sigma f)(x,t)|^rdt\Big )^{1/r}&=\Big (\int _0^x\Big |\int _0^x\chi (s-t)\sigma (s)f(s,s-t)ds\Big |^rdt\Big )^{1/r}\nonumber \\&\le \int _0^x|\sigma (s)|\Big (\int _0^s|f(s,s-t)|^rdt\Big )^{1/r}ds\nonumber \\&\le \int _0^x|\sigma (s)|\Big (\int _0^s|f(s,\tau )|^rd\tau \Big )^{1/r}ds\nonumber \\&\le \int _0^x|\sigma (s)|ds \sup _{s\in [0,1]}\Big (\int _0^s|f(s,\tau )|^rd\tau \Big )^{1/r}\nonumber \\&\le \Vert \sigma \Vert _{L_1}\Vert f\Vert _B. \end{aligned}$$

(4.3)

For the proof of continuity we take $\epsilon$ and x such that $0\le t\le x+\epsilon \le 1$. Then

$$\begin{aligned} \Big \Vert (T_\sigma f)(x+\epsilon ,\cdot )-(T_\sigma f)(x,\cdot )\Big \Vert _{L_r[0,1]}&\le \!\Big (\int _0^x\Big |\int _x^{x+\epsilon }\sigma (s)f(s,s-t)ds\Big |^rdt\Big )^{1/r}\nonumber \\&\!+\!\Big (\int _x^{x+\epsilon }\Big |\int _t^{x+\epsilon }\sigma (s)f(s,s-t)ds\Big |^rdt\Big )^{1/r}. \end{aligned}$$

First integral can be estimated as follows

$$\begin{aligned} \Big (\int _0^x\Big |\int _x^{x+\epsilon }\sigma (s)f(s,s-t)ds\Big |^rdt\Big )^{1/r}&\le \! \int _x^{x+\epsilon }|\sigma (s)|\Big (\int _0^x|f(s,s-t)|^rdt\Big )^{1/r}ds\\&\le \! \int _x^{x+\epsilon }|\sigma (s)|\Big (\int _0^s|f(s,\tau )|^rd\tau \Big )^{1/r}ds\\&\le \!\Vert f\Vert _B \int _x^{x+\epsilon }|\sigma (s)|ds \end{aligned}$$

and this expression goes to zero whenever $\epsilon$ does.

Second integral can be treated in an analogous way, hence the proof is completed. $\square$

Lemma 4.4

The operators $T_{kj}, k,j=1,2, k\not =j$ satisfy the following estimate

$$\begin{aligned} \Vert T_{kj}^nf\Vert _{B} \le \frac{a^n}{n!}\Vert f\Vert _{B},\qquad f\in B,\quad n\in {\mathbb {N}},\quad k,j=1,2,\quad k\not =j. \end{aligned}$$

(4.4)

Proof

Consider the operator $T_{12}$. Note that directly from the third line of (4.3) we get

$$\begin{aligned} \Big (\int _0^x|(T_{12}f)(x,t)|^rdt\Big )^{1/r}&\le \int _0^x|\sigma _1(s)|\Big (\int _0^s|(T_{\sigma _2}f)(s,\tau )|^rd\tau \Big )^{1/r}ds\nonumber \\&\le \int _0^x|\sigma _1(s)|\int _0^s|\sigma _2(\tau )| \Big (\int _0^\tau |f(\tau ,\xi )|^rd\xi \Big )^{1/r} d\tau ds\nonumber \\&\le \Vert f\Vert _B\int _0^x|\sigma (s)|\int _0^s|\sigma _2(\tau )|ds. \end{aligned}$$

(4.5)

Define $\eta \in C[0,1]$ by

$$\begin{aligned} \eta (x):=\int _0^x |\sigma _1(s)|\left( \int _0^s |\sigma _2(\tau )|\,d\tau \right) \,ds,\qquad x\in [0,1]. \end{aligned}$$

This function is increasing and bounded by $a=\Vert \sigma _1\Vert _{L_1}\Vert \sigma _2\Vert _{L_1}$. It suffices to prove that for all $(x,t)\in \Delta$ and $n=1,2,\dots ,$

$$\begin{aligned} \Big (\int _0^x|(T^n_{12}f)(x,t)|^rdt\Big )^{1/r} \le \frac{\Vert f\Vert _{B}}{n!} \eta ^n(x),\qquad f\in B. \end{aligned}$$

(4.6)

For $n=1$ the estimate (4.6) was shown above. Arguing by induction, suppose that (4.6) holds for some $n\in {\mathbb {N}}$. Then, for $(x,t)\in \Delta$ and $f\in B$, by (4.5) we have

$$\begin{aligned}&\Big (\int _0^x|(T^{n+1}_{12}f)(x,t)|^rdt\Big )^{1/r}\\&\quad \le \int _0^x|\sigma _1(s)|\int _0^s|\sigma _2(\tau )| \Big (\int _0^\tau |(T^{n}_{12}f)(\tau ,\xi )|^rd\xi \Big )^{1/r} d\tau ds\nonumber \\&\quad \le \frac{\Vert f\Vert _{B}}{n!} \int _0^x |\sigma _1(s)| \int _{0}^s |\sigma _2(\tau )| \eta ^n(\tau )\,d\tau \,ds \\&\quad \le \frac{\Vert f\Vert _{B}}{n!} \int _0^x |\sigma _1(s)| \int _0^s |\sigma _2(\tau )|\,d\tau \,\eta ^n (s)\,ds \\&\quad =\frac{\Vert f\Vert _{B}}{n!} \int _0^x \eta ^n(s)\,d\eta (s) =\frac{\Vert f\Vert _{B}}{(n+1)!}\eta ^{n+1}(x). \end{aligned}$$

Therefore (4.6) is true and then after taking supremum over $x\in [0,1]$ we get (4.4). $\square$

Next proposition we state below without a proof, since it can be found in [8, Prop. 6.1].

Proposition 4.5

If $\sigma _j\in L_p[0,1]$, $1\le p <2$ and $F\in M(B)$, then

$$\begin{aligned} \int _0^xe^{-2i\mu t}({\widetilde{T}} F)(x,t)dt =-\int _0^xe^{-2i\mu s}J(s)\int _0^{s}e^{2i\mu \xi }F(s,\xi )d\xi ds. \end{aligned}$$

(4.7)

Moreover,

$$\begin{aligned} \int _0^xe^{-2i\mu t}&({\widetilde{T}}{\tilde{J}})(x,t)\,dt \nonumber \\&=-\int _0^xe^{2i\mu y} \Bigg (\int _y^x J(z)e^{-2i\mu z}\, d z\, \int _0^{y}J^T(\tau )e^{-2i\mu \tau } d\tau \Bigg )J^T(y) dy. \end{aligned}$$

(4.8)

Lemma 4.6

If $\mu \in P_d,$ and $x\in [0,1]$, then the following inequalities hold

$$\begin{aligned}&\left\| \int _0^x e^{-2tA_\mu }{\tilde{J}}(x,t)\,dt\right\| _{M({\mathbb {C}})} \le 2e^{2d}a_p\,\gamma _q(\mu ), \end{aligned}$$

(4.9)

$$\begin{aligned}&\left\| \int _0^x e^{-2tA_\mu }({\widetilde{T}}{\tilde{J}})(x,t)\,dt \right\| _{M({\mathbb {C}})} \le 2 e^{4d} a_p^2\gamma _q(\mu ), \end{aligned}$$

(4.10)

$$\begin{aligned}&\left\| \int _0^x e^{-2tA_\mu }({\widetilde{T}}{\tilde{J}})(x,t)\,dt\right\| _{M( {\mathbb {C}})} \le 2(a_p+1)e^{2d} \Big (\gamma _q(\mu )\gamma (x,\mu )+{\widetilde{\gamma }}(\mu )\Big ),\nonumber \\ \end{aligned}$$

(4.11)

$$\begin{aligned}&\left\| \int _0^x e^{-2tA_\mu }({\widetilde{T}}^n {\tilde{J}})(x,t)\,dt \right\| _{M({\mathbb {C}})} \le 3e^{2n d} \frac{a_1^{n-1}}{(n-2)!} {\widetilde{\gamma }}(\mu ),\quad n\ge 2. \end{aligned}$$

(4.12)

Proof

Note that

$$\begin{aligned} \left\| \int _0^xe^{-2i\mu t} {\tilde{J}}(x,t)dt\right\| _{M({\mathbb {C}})}&= \left\| \int _0^x J(s)e^{-2i\mu s} \int _0^s J(\xi )e^{2i\mu \xi } d \xi ds\right\| _{M( {\mathbb {C}})}\nonumber \\&=\left| \int _0^x e^{-2i\mu s}\sigma _1(s) \int _0^s e^{2i\mu \xi }\sigma _2(\xi ) d \xi ds\right| \nonumber \\&+ \left| \int _0^x e^{-2i\mu s}\sigma _2(s) \int _0^s e^{2i\mu \xi }\sigma _1(\xi ) d \xi ds\right| \nonumber \\&\le e^{2d}\Big \{\Vert \sigma _1\Vert _{L_p}\Big \Vert \int _0^s e^{2i\mu \xi }\sigma _2(\xi )d\xi \Big \Vert _{L_q} \nonumber \\&+\Vert \sigma _2\Vert _{L_p}\Big \Vert \int _0^s e^{2i\mu \xi }\sigma _1(\xi )d\xi \Big \Vert _{L_q}\Big \}\nonumber \\&\le e^{2d}\max \{\Vert \sigma _1\Vert _{L_p},\Vert \sigma _2\Vert _{L_p}\}\,\gamma _q(\mu ),\quad x\in [0,1]. \end{aligned}$$

(4.13)

We thus proved the estimate (4.9).

Next, by (4.7), if $\mu \in P_d$, $x\in [0,1]$ and $F\in M(B)$, then

$$\begin{aligned} \left\| \int _0^xe^{-2i\mu t}({\widetilde{T}} F)(x,t)dt\right\| _{M( {\mathbb {C}})} \le e^{2d}\int _0^x \left\| J(s) \int _0^{s}e^{2i\mu \xi }F(s,\xi )d\xi \right\| _{M({\mathbb {C}})}\,ds,\nonumber \\ \end{aligned}$$

(4.14)

and

$$\begin{aligned} \left\| \int _0^xe^{-2i\mu t}({\widetilde{T}} F)(x,t)dt\right\| _{M({\mathbb {C}})} \le e^{2d}a_1 \left\| \int _0^{s}e^{2i\mu \xi }F(s,\xi )d\xi \right\| _{M({\mathbb {C}})}. \end{aligned}$$

(4.15)

We use (4.15) and (4.13) to obtain that

$$\begin{aligned} \left\| \int _0^x e^{-2i\mu t}({\widetilde{T}}{\tilde{J}})(x,t)\,dt \right\| _{M({\mathbb {C}})}&\le e^{2d}a_1 \left\| \int _0^{s}e^{2i\mu \xi }{\tilde{J}}(s,\xi )\,d\xi \right\| _{M({\mathbb {C}})}\\&\le e^{4d}a_p^2\gamma _q(\mu ), \end{aligned}$$

thus, the estimate (4.10) holds.

Due to the estimate

$$\begin{aligned} \left| \int _0^x \sigma _0(s) \gamma (y,\mu )\,dy\right| \le \Vert \sigma _0\Vert _{L_p}\left\| \gamma (y,\mu )\right\| _{L_q} \le a_2\gamma _q(\mu ), \end{aligned}$$

the inequality (4.11) holds if

$$\begin{aligned} \Bigg \Vert \int _0^xe^{-2i\mu t} ({\widetilde{T}}{\tilde{J}})(x,t)\,dt\Bigg \Vert _{M({\mathbb {C}})} \le e^{2d}\left( {\widetilde{\gamma }}(\mu ) +\gamma (x,\mu ) \int _0^x \sigma _0(y) \gamma (y,\mu )\,dy\right) .\nonumber \\ \end{aligned}$$

(4.16)

Whereas, using (4.8), (4.9), we have

$$\begin{aligned}&\Bigg \Vert \int _0^x e^{-2i\mu t} ({\widetilde{T}}{\tilde{J}})(x,t)\,dt\Bigg \Vert _{M({\mathbb {C}})}\\&\quad \le e^{2d} \int _0^x\left\| \int _{y}^x e^{-2i\mu z}J(z) d z \int _0^y e^{-2i\mu \tau }J^T(\tau )\,d\tau J^T(y)\right\| _{M({{\mathbb {C}}})} dy \\&\quad \le e^{2d} \int _0^x \sigma _0(y) \left\| \int _{y}^x e^{-2i\mu z}J(z) d z\right\| _{M({\mathbb {C}})} \left\| \int _0^y e^{-2i\mu \tau }J(\tau )\,d\tau \right\| _{M({\mathbb {C}})} dy \\&\quad \le e^{2d} \int _0^x \sigma _0(y) \left\| \int _0^y e^{-2i\mu \tau }J(\tau )\,d\tau \right\| ^2_{M({\mathbb {C}})} dy \\&\qquad +e^{2d}\left\| \int _0^x e^{-2i\mu z}J(z) d z\right\| _{M({\mathbb {C}})} \int _0^x \sigma _0(y) \left\| \int _0^y e^{-2i\mu \tau }J(\tau )\,d\tau \right\| _{M({\mathbb {C}})} dy \\&\quad \le e^{2d} {\widetilde{\gamma }}(\mu ) +e^{2d}\gamma (x,\mu ) \int _0^x \sigma _0(y) \gamma (y,\mu )\,dy, \end{aligned}$$

and (4.16) follows.

The estimate (4.12) will be showed, if we prove that for all $n\ge 2$ and any $x\in [0,1],$

$$\begin{aligned} \left\| \int _0^x e^{-2i\mu t}({\widetilde{T}}^n {\tilde{J}})(x,t)\,dt\right\| _{M({\mathbb {C}})} \le \frac{3}{2}\frac{e^{2n d}}{(n-2)!} \left( \int _0^x \sigma _0(s)\,ds\right) ^{n-2}a_1 {\widetilde{\gamma }}(\mu ). \end{aligned}$$

(4.17)

We proceed by induction. Using (4.14) for $F={\widetilde{T}}{\tilde{J}}$ and (4.16), we note that

$$\begin{aligned}&\Big \Vert \int _0^x e^{-2i\mu t}({\widetilde{T}}^2{\tilde{J}}) (x,t)\,dt\Big \Vert _{M({\mathbb {C}})}\\&\quad \le e^{2d}\int _0^x \sigma _0(s) \left\| \int _0^{s}e^{2i\mu \xi }({\widetilde{T}}{\tilde{J}})(s,\xi )d\xi \right\| _{M({\mathbb {C}})}\, ds\\&\quad \le e^{4d}\int _0^x \sigma _0(s) \left( \gamma (s,\mu )\int _0^s\sigma _0(y) \gamma (y,\mu )\,dy+{\widetilde{\gamma }}(\mu )\right) \, ds\\&\quad \le e^{4d}\int _0^x \sigma _0(s)\gamma (s,\mu ) \int _0^{s}\sigma _0(y) \gamma (y,\mu )\,dy\, ds +e^{4d}a_1{\widetilde{\gamma }}(\mu ) \\&\quad \le e^{4d}\frac{\left( \int _0^x \sigma _0(s)\gamma (s,\mu )\,ds\right) ^2}{2} +e^{4d}a_1{\widetilde{\gamma }}(\mu ). \end{aligned}$$

Since $\sigma _0$ is nonnegative we may now apply Cauchy–Bunyakovsky–Schwarz inequality to obtain

$$\begin{aligned} \Big (\int _0^x \sigma _0(s)\gamma (s,\mu ) ds\Big )^2\le \int _0^x \sigma _0(s)ds \int _0^x \sigma _0(s)\gamma ^2(s,\mu ) ds\le a_1{\widetilde{\gamma }}(\mu ). \end{aligned}$$

Therefore, we proved

$$\begin{aligned} \Big \Vert \int _0^x e^{-2i\mu t}({\widetilde{T}}^2{\tilde{J}})&(x,t)\,dt\Big \Vert _{M({\mathbb {C}})} \le \frac{3}{2}e^{4d}a_1{\widetilde{\gamma }}(\mu ), \end{aligned}$$

hence (4.17) holds for $n=2$.

Suppose that (4.17) holds for some $n\ge 2$. We use (4.14) to derive

$$\begin{aligned}&\Big \Vert \int _0^x e^{-2i\mu t}({\widetilde{T}}^{n+1}{\tilde{J}}) (x,t)\,dt\Big \Vert _{M({\mathbb {C}})}\\&\quad \le e^{2d} \int _0^x \sigma _0(s) \left\| \int _0^{s}e^{2i\mu \xi }({\widetilde{T}}^n{\tilde{J}})(s,\xi )d\xi \right\| _{M({\mathbb {C}})} ds \\&\quad \le \frac{3}{2}\frac{e^{2(n+1) d}}{(n-2)!}a_1{\widetilde{\gamma }}(\mu )\int _0^x\sigma _0(s) \left( \int _0^s\sigma _0(\tau )\,d\tau \right) ^{n-2}\,ds \\&\quad =\frac{3}{2}\frac{e^{2(n+1)d}}{(n-1)!}a_1{\widetilde{\gamma }}(\mu ) \left( \int _0^x \sigma _0(\tau )\,d\tau \right) ^{n-1},\quad x\in [0,1], \end{aligned}$$

thus (4.17) holds also for $n+1,$ and the proof of (4.17) is completed. $\square$