Variable order fractional integrals in variable generalized Hölder spaces of holomorphic functions

Abstract

We introduce and study the variable generalized Hölder spaces of holomorphic functions over the unit disc in the complex plane. These spaces are defined either directly in terms of modulus of continuity or in terms of estimates of derivatives near the boundary. We provide conditions of Zygmund type for imbedding of the former into the latter and vice versa. We study mapping properties of variable order fractional integrals in the frameworks of such spaces.

1 Introduction

The purpose of this paper is twofold. We study variable generalized Hölder spaces of holomorphic functions over the unit disc ${\mathbb {D}}$ in the complex plane ${\mathbb {C}}$ and investigate mapping properties of variable order fractional integrals in such spaces.

Variable exponent and generalized Hölder spaces and operators in such spaces attract attention of many authors during last two decades. We refer, for instance, for such a study in the framework of real analysis to the papers [8,9,10] and the books [5, 6] and overview paper [7] for further references. For the investigations in complex analysis settings we refer to the recent papers [1,2,3]. See also books [11, 12] and references therein for the theory of classical (constant order) Hölder spaces of holomorphic functions.

We give two definitions of variable generalized Hölder spaces of holomorphic functions. One, denoted by $A^{\omega (\cdot )}({\mathbb {D}})$ is defined directly via variable modulus of continuity. Another denoted by $B^{\omega (\cdot )}({\mathbb {D}})$ is defined in terms of behavior of derivatives near the boundary. We find conditions in terms of Zygmund type inequalities for the inclusions $B^{\omega (\cdot )}({\mathbb {D}})\hookrightarrow A^{\omega (\cdot )}({\mathbb {D}})$ and $A^{\omega (\cdot )}({\mathbb {D}})\hookrightarrow B^{\omega (\cdot )}({\mathbb {D}}).$

We study mapping properties of the variable order fractional integral

$$\begin{aligned} {{\mathbb {I}}}^{\alpha (\cdot )}_{\lambda } f(z)=\int _{{\mathbb {D}}}\frac{f(w)\mathrm {d}{\mu _\lambda }(w)}{(1-z{\overline{w}})^{2+\lambda -\alpha (z)}},\ \ \mathfrak {R}\alpha (z)\geqslant 0,\ \ \mathfrak {R}\lambda >-1, \end{aligned}$$

between spaces $A^{\omega (\cdot )}({\mathbb {D}})$ and between spaces $B^{\omega (\cdot )}({\mathbb {D}}).$ Our first approach is based on the Complex Analysis technique and admits degeneracy of the variable exponent $\alpha$. In this approach we study, in fact, an operator more general than ${{\mathbb {I}}}^{\alpha (\cdot )}_{\lambda },$ see Theorem 4.7. This admission of the degeneracy of the exponent $\alpha$ illustrates a difference from the Real Analysis situation. In Real Analysis when the exponent degenerates the corresponding integral defining the potential operator becames divergent even on constant functions. In Complex Analysis setting the operator ${{\mathbb {I}}}^{\alpha (\cdot )}_{\lambda }$ is well defined at the points where $\alpha (z)=0$ having a behavior of Bergman projection operator at these points.

Our second approach is based mainly on techniques used in Real Analysis. In this way we do not allow $\alpha$ to be degenerate, but we impose less restrictive conditions on the corresponding modulus of continuity (see Theorem 4.12).

Both results complement each other and allow one to use that our another condition in the further study of such spaces and operators on such spaces.

Note that in [4] we introduced and studied the so-called Hadamard–Bergman convolution operators in Bergman and Hölder spaces of holomorphic functions. Operators of fractional integrals of constant order $\alpha$ are examples of such convolutions, which is not true in the case of variable $\alpha .$ We briefly touch an extension of the class of Hadamard–Bergman convolutions so that to include ${\mathbb I}^{\alpha (\cdot )}_{\lambda }$ into this class.

The paper is organized as follows. In Sect. 2 we collect main definitions and notion from the theory of Hölder spaces and variable exponent analysis. We also present the mentioned above extension of the class of Hadamard–Bergman operators. In Sect. 3 we study the spaces $A^{\omega (\cdot )}({\mathbb {D}})$ and $B^{\omega (\cdot )}({\mathbb {D}}).$ In Sect. 4 we investigate mapping properties of the variable order fractional integrals in the Hölder spaces.

2 Preliminaries

2.1 Definitions

Let ${\mathbb {D}}$ stand for the unit disc in complex plane ${\mathbb {C}},$ and ${\mathbb {T}}$ denote unit circle. We say that $\alpha :{\mathbb {D}}\rightarrow {\mathbb {C}}$ satisfies the log-Hölder condition (or briefly: log-condition) on ${\mathbb {D}}$ if

$$\begin{aligned} |\alpha (z)-\alpha (w)|\leqslant \frac{C}{\ln \frac{1}{|z-w|}},\ \ \ z,w\in {\mathbb {D}},\,\,|z-w|<\frac{1}{2}, \end{aligned}$$

(2.1)

where C does not depend on $z,w\in {\mathbb {D}}$. The following fact is well-known: if the function $\alpha$ satisfies the log-Hölder condition then $\alpha$ is bounded and uniformly continuous on ${\mathbb {D}}.$ Hence, it can be extended to a continuous function over $\overline{{\mathbb {D}}}:=\{z\in {\mathbb {D}}: |z|\leqslant 1\}$. We always use the same notation $\alpha$ for the so extended function. We recall that if $\alpha$ satisfies log-Hölder condition on ${\mathbb {D}}$, then there exists $C>0$ such that

$$\begin{aligned} \frac{1}{C}h^{\mathfrak {R}\alpha (z)}\leqslant h^{\mathfrak {R}\alpha (\zeta )}\leqslant C h^{\mathfrak {R}\alpha (z)},\ \ \ \ \ z,\zeta \in {\mathbb {D}},\ \ |z-\zeta |<\min \left\{ h,\frac{1}{2}\right\} . \end{aligned}$$

By $\mathrm {Lip}({\mathbb {D}})$ we denote the class of functions on ${\mathbb {D}}$ which satisfy Lipschitz condition.

A function $\omega : [0,2]\rightarrow {\mathbb {R}}$ is called almost increasing (almost decreasing) on [0, 2] if there exists $C>0$ such that $\omega (h)\leqslant C \omega (\xi )$ ($\omega (\xi )\leqslant C \omega (h)$) for all $0\leqslant h\leqslant \xi \leqslant 2.$

In what follows we will apply the notion of almost increase (almost decrease) to a function $\omega =\omega (z,h)$ as a function of h with parameter z. In such a case we always assume the property of almost increase (almost decrease) to be uniform in z,  which means that the corresponding constant C in the definition of the property of almost increase (almost decrease) above does not depend on z.

A function $\omega =\omega (z,h):{\mathbb {D}}\times [0,2]\longrightarrow {\mathbb {R}}_+$ is called variable modulus of continuity if

• (1) $\omega (z,h)$ is continuous in h on [0, 2] for all $z\in {\mathbb {D}};$

• (2) $\omega (z,h)$ is almost increasing in h on [0,2] uniformly in $z\in {\mathbb {D}};$

• (3) $\omega (z,h)/h$ is almost decreasing on [0,2] uniformly in $z\in {\mathbb {D}};$

• (4) $\inf _{z\in {\mathbb {D}}}\omega (z,h)>0$ for $h>0$ and $\lim _{h\rightarrow 0}\omega (z,h)=0$ for all $z\in {\mathbb {D}}.$

Given a function f defined in ${\mathbb {D}}$ and a point $z\in {\mathbb {D}},$ the local variable modulus of continuity of the function f at the point z is defined as

$$\begin{aligned} \omega (f,z,h)=\sup _{w\in {\mathbb {D}}: |w-z|<h}|f(z)-f(w)|. \end{aligned}$$

From Lemma 2.10 in [9] we know that for all $z,\zeta \in {\mathbb {D}}$ such that $|z-\zeta |<h$ the following inequality holds:

$$\begin{aligned} \frac{1}{4}\omega (z,f,h)\leqslant \omega (\zeta ,f,h)\leqslant 4\omega (z,f,h). \end{aligned}$$

(2.2)

Recall the following Zygmund type conditions

$$\begin{aligned}&\int _0^h\left( \frac{h}{t}\right) ^{\delta (z)}\frac{\omega (z,t)}{t}dt\leqslant C\omega (z,h),\,\, z\in {\mathbb {D}},\ \ h\in [0,1], \end{aligned}$$

(2.3)

$$\begin{aligned}&\int _h^2\left( \frac{h}{t}\right) ^{\nu (z)}\frac{\omega (z,t)}{t}dt\leqslant C\omega (z,h),\,\, z\in {\mathbb {D}},\ \ h\in [0,1], \end{aligned}$$

(2.4)

with C not depending on z and h. The corresponding (generalized) Zygmund - Bary - Stechkin class $\Phi ^{\delta (\cdot )}_{\nu (\cdot )},$ where $0\leqslant \delta (z)< \nu (z),$ $z\in {\mathbb {D}}$ is defined to consist of all functions $\omega :{\mathbb {D}}\times [0,2]\longrightarrow {\mathbb {R}}_+$ which satisfy both conditions (2.3) and (2.4). By $Z^{\delta (\cdot )}$ we also denote the corresponding class with only the first of the conditions satisfied, and by $Z_{\nu (\cdot )},$ the class with only the second one, so that $\Phi ^{\delta (\cdot )}_{\nu (\cdot )}=Z^{\delta (\cdot )}\cap Z_{\nu (\cdot )}.$ From the definitions of the class $Z_{\nu (\cdot )}$ it easily follows that a function $\omega \in Z_{\nu (\cdot )}$ satisfies the property $\omega (z,h)\geqslant Ch^{\nu (z)}$ with a constant $C>0$ not depending on z and h.

Let $H({\mathbb {D}})$ be the set of functions f,  holomorphic in ${\mathbb {D}},$ equipped with the topology defined by the countable set of norms

$$\begin{aligned} \Vert f\Vert _{(m)}:=\sup _{|z|<\frac{m}{m+1}}\left| \sum _{n=0}^\infty f_nz^n\right| ,\ \ m=1,2,\ldots , \end{aligned}$$

where $f_n$ are Taylor coefficients of the function f. The space $H({\mathbb {D}})$ may be identified with the set of series $\sum _{n=0}^\infty a_n z^n,$ such that $\limsup _{n\rightarrow \infty } |a_n|^{\frac{1}{n}}\leqslant 1.$

2.2 Variable Hadamard–Bergman convolutions and variable order fractional integrals

Let $\mathrm {dA}(z)=\frac{1}{\pi }dxdy,$ $z=x+iy,$ be the normalized Lebesgue measure on ${\mathbb {D}}$. Given $\lambda \in {\mathbb {C}}$ let us denote

$$\begin{aligned} \mathrm {d}{\mu _\lambda }(w)=(\lambda +1)(1-|z|^2)^\lambda \mathrm {dA}(z). \end{aligned}$$

Let $g:{\mathbb {D}}\times {\mathbb {D}}\rightarrow {\mathbb {C}}$ be defined on ${\mathbb {D}}\times {\mathbb {D}}$ and be holomorphic in the second variable, i.e. the function $w\rightarrow g(z,w)$ is holomorphic for every $z\in {\mathbb {D}}$. The class of integral operators called Hadamard–Bergman convolutions was introduced and studied in [4]. Here we slightly extend the definition, with a goal to include the operators of variable integration. Namely,

$$\begin{aligned} {\mathbb {K}} f(z)= & {} \int _{{\mathbb {D}}}g(z,z{\overline{w}})f(w)\mathrm {d}{\mu _\lambda }(w) \nonumber \\= & {} \lim _{r\rightarrow 1-}\int _{|w|<r}g(z,z{\overline{w}})f(w)\mathrm {d}{\mu _\lambda }(w),\ \ \ f\in H({\mathbb {D}}), \end{aligned}$$

(2.5)

where the function g is considered as the kernel of the operator ${\mathbb {K}}.$ Here and in what follows we assume that $\mathfrak {R}\lambda >-1.$

We call the construction in (2.5) variable Hadamard–Bergman convolution operator.

The integral in (2.5) exists as improper integral and

$$\begin{aligned}\lim _{r\rightarrow 1-}\int _{|w|<r}g(z,z{\overline{w}})f(w)\mathrm {d} {\mu _\lambda }(w)=\sum _{k=0}^\infty f_kg_k(z) z^{k}, \end{aligned}$$

where $f_k$ are Taylor coefficients of the function f and

$$\begin{aligned} g_k(z)=\int _{{\mathbb {D}}}g(z,w){\overline{w}}^k\mathrm {d}{\mu _\lambda }(w),\ \ \ z\in {\mathbb {D}}. \end{aligned}$$

Consequently,

$$\begin{aligned} \int _{{\mathbb {D}}}g(z,w)f(z{\overline{w}})\mathrm {d} {\mu _\lambda }(w)= \int _{{\mathbb {D}}}g(z,z{\overline{w}})f(w)\mathrm {d} {\mu _\lambda }(w),\ \ \ z\in {\mathbb {D}}.\end{aligned}$$

(2.6)

We note also the following formula

$$\begin{aligned} \int _{{\mathbb {D}}}f({\overline{w}})g(z,z{\overline{w}})d\mu _\lambda (w)= \int _{{\mathbb {D}}}\overline{f(w)}g(z,z{\overline{w}})d\mu _\lambda (w)=g(z,0)\overline{f(0)}, \end{aligned}$$

(2.7)

holds for every $z\in {\mathbb {D}}.$

The variable order fractional integration operator

$$\begin{aligned}{\mathbb I}^{\alpha (\cdot )}_{\lambda } f(z)=\int _{{\mathbb {D}}}\frac{f(w) \mathrm {d}{\mu _\lambda }(w)}{(1-z{\overline{w}})^{2+\lambda -\alpha (z)}},\ \ \mathfrak {R}\alpha (z)\geqslant 0,\ \ \mathfrak {R}\lambda >-1. \end{aligned}$$

is an example of variable Hadamard–Bergman convolution operator. We note that this operator possesses the realization in the form of the functional series

$$\begin{aligned}{\mathbb {I}}^{\alpha (\cdot )}_{\lambda } f(z)= & {} \frac{\Gamma (2+\lambda )}{\Gamma (2+\lambda -\alpha (z))}\sum _{m=0}^\infty \frac{\Gamma (m+2+\lambda -\alpha (z))}{\Gamma (2+m+\lambda )}f_m z^m, \, \, \, z\in {\mathbb {D}}, \\&\alpha \in H({\mathbb {D}}), {\mathfrak {R}}\alpha (z)\geqslant 0, \lambda \in {\mathbb {C}}, \lambda \ne -2,-3,... . \end{aligned}$$

and the series makes sense for a bigger range of the parameter $\lambda ,$ as specified above.

3 Variable generalized Hölder spaces of holomorphic functions

Let $\omega$ be variable modulus of continuity. The variable generalized holomorpic Hölder space $A^{\omega (\cdot )}({\mathbb {D}})$ is defined as

$$\begin{aligned}A^{\omega (\cdot )}({\mathbb {D}})=\{f\in H({\mathbb {D}}): \omega (f,z,h)\leqslant C\omega (z,h)\}, \end{aligned}$$

where C does not depend on z and h. The semi-norm and the norm of a function $f\in A^{\omega (\cdot )}({\mathbb {D}})$ are given as usual

$$\begin{aligned} \Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {D}})}=\sup _{z\in {\mathbb {D}}, h\in [0,2]}\frac{\omega (f,z,h)}{\omega (z,h)}, \Vert f\Vert _{A^{\omega (\cdot )}({\mathbb {D}})}=\Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {D}})}+\Vert f\Vert _{C({\mathbb {D}})}. \end{aligned}$$

The variable generalized holomorpic Hölder space $B^{\omega (\cdot )}({\mathbb {D}})$ is defined as

$$\begin{aligned}B^{\omega (\cdot )}({\mathbb {D}})= & {} \{f\in H({\mathbb {D}}):|f'(z)|\leqslant C\omega (z,1-|z|)/(1-|z|), \ z\in {\mathbb {D}} \},\end{aligned}$$

where C does not depend on z and h. The semi-norm and the norm of a function $f\in B^{\omega (\cdot )}({\mathbb {D}})$ are given by

$$\begin{aligned} \Vert f\Vert _{\#,B^{\omega (\cdot )}({\mathbb {D}})}= & {} \sup _{z\in {\mathbb {D}}, h\in [0,2]}|f'(z)|\frac{\omega (z,1-|z|)}{1-|z|},\\ \Vert f\Vert _{B^{\omega (\cdot )}({\mathbb {D}})}= & {} \Vert f\Vert _{\#,B^{\omega (\cdot )}({\mathbb {D}})}+\Vert f\Vert _{C({\mathbb {D}})}. \end{aligned}$$

Both $A^{\omega (\cdot )}({\mathbb {D}})$ and $B^{\omega (\cdot )}({\mathbb {D}})$ when equipped with the norms as above, became Banach spaces.

In the case $\omega (z,h)=h^{\gamma (z)}$ we will use the conventional notation $A^{\gamma (\cdot )}({\mathbb {D}})$ and $B^{\gamma (\cdot )}({\mathbb {D}})$ to denote the corresponding variable exponent Hölder spaces of holomorphic functions.

Theorem 3.1

Let $\omega$ be variable modulus of continuity. The following statements hold.

• (1) Let $\omega \in Z_1$, then $A^{\omega (\cdot )}({\mathbb {D}})\hookrightarrow B^{\omega (\cdot )}({\mathbb {D}}),$ besides this, $\Vert f\Vert _{\#,B^{\omega (\cdot )}({\mathbb {D}})}\leqslant C\Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {D}})},$ where C does not depend on f.

• (2) Let $\omega \in Z^0$, then $B^{\omega (\cdot )}({\mathbb {D}})\hookrightarrow A^{\omega (\cdot )}({\mathbb {D}}),$ besides this, $\Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {D}})}\leqslant C\Vert f\Vert _{\#,B^{\omega (\cdot )}({\mathbb {D}})},$ where C does not depend on f.

Proof

The proof follows the lines of proof of Theorems 3 and 8 from [3], where the cases of $\omega (z,h)=\omega (h)$ and $\omega (z,h) = h^{\gamma (z)}$ were treated. One have to observe that the corresponding estimates and properties of almost increase (almost decrease) of $\omega$ are uniform in $z\in {\mathbb {D}}.$ $\square$

Corollary 3.2

Let $\omega$ be variable modulus of continuity and $\omega \in \Phi _1^0$, then the spaces $A^{\omega (\cdot )}({\mathbb {D}})$ and $B^{\omega (\cdot )}({\mathbb {D}})$ coincide up to equivalence of norms.

In the theorem below we show that under certain assumptions the space $A^{\omega (\cdot )}({\mathbb {D}})$ remains the same if the Hölder behavior is checked only on the boundary.

Theorem 3.3

Let $\omega$ be variable modulus of continuity and $\omega \in \Phi _1^0$. Let also $\omega (\frac{z}{|z|},1-|z|)\leqslant C\omega (z,1-|z|)$ with a constant $C>0$ and for all $z\in {\mathbb {D}}$ close to the boundary ${\mathbb {T}},$ i.e. $1-\delta<|z|<1$ with a certain $\delta >0.$ Let f be holomorphic in ${\mathbb {D}}.$ Then $f\in A^{\omega (\cdot )}({\mathbb {D}})$ if and only if

• (1) f is continuous in $\overline{{\mathbb {D}}}$ (i.e. f is in the so-called disc algebra),

• (2) $|f(\tau )-f(\sigma )|\leqslant C\omega (\tau ,|\tau -\sigma |), \ \tau , \sigma \in {\mathbb {T}},$ where C is independent on $\tau , \sigma \in {\mathbb {T}}.$

Proof

The implication that $f\in A^{\omega (\cdot )}({\mathbb {D}})$ guarantees the validity of the conditions $(1)-(2)$ is obvious. To prove the converse we note that

$$\begin{aligned} f'(z)=\frac{1}{2\pi i}\int _{{\mathbb {T}}}\frac{f(\tau )}{(\tau -z)^2}d\tau =\frac{1}{2\pi i}\int _{{\mathbb {T}}}\frac{f(\tau )-f(\frac{z}{|z|})}{(\tau -z)^2}d\tau ,\ z\in {\mathbb {D}}. \end{aligned}$$

Hence, since $\frac{\omega (z,h)}{h}$ is almost decreasing on [0, 2] in view of Lemma 4.5 we have

$$\begin{aligned} |f'(z)|\leqslant & {} \frac{1}{2\pi }\Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {T}})} \int _{{\mathbb {T}}}\frac{\omega (\frac{z}{|z|},|\tau -\frac{z}{|z|}|)}{|\tau -z|^2}|d\tau | \leqslant C\int _{{\mathbb {T}}}\frac{\omega (\frac{z}{|z|},|\tau -z|)}{|\tau -z|^2}|d\tau |\\ {}\leqslant & {} C_1 \frac{\omega (\frac{z}{|z|},1-|z|)}{(1-|z|)}\leqslant C_2 \frac{\omega (z,1-|z|)}{(1-|z|)},\,\,\, z\in {\mathbb {D}}, \end{aligned}$$

where $C_2$ does not depend on $z\in {\mathbb {D}}.$ $\square$

Remark 3.4

The assumed above condition $\omega (\frac{z}{|z|},1-|z|)\leqslant C\omega (z,1-|z|)$ is natural: in the variable exponent analysis for the case of $\omega (z,h)=h^{\gamma (z)}$ the log-Hölder condition for the variable exponent $\gamma (\cdot )$ is usually assumed, which implies the same estimate for this particular case.

4 Mapping properties of variable order fractional integral in the spaces $A^{\omega (\cdot )}({\mathbb {D}})$ and $B^{\omega (\cdot )}({\mathbb {D}})$

4.1 Mapping properties of generalized fractional integral in generalized Hölder spaces with admission of degeneracy $\mathfrak {R}\alpha (z)=0$

Here we present our first approach which is based on the Complex Analysis technique. In this approach we admit a possibility for the generalized fractional integral to behave like Bergman projection operator at some points $z\in {\mathbb {D}},$ i.e., $a(z,1-w)\sim C=C(z_0),$ $z\rightarrow z_0$ for some $z_0\in {\mathbb {D}}.$ In the case $a(z,1-w)=(1-w)^{\alpha (z)}$ this means that we admit the degeneracy of $\mathfrak {R}\alpha$ for some points $z\in {\mathbb {D}}$.

Let the function $a=a(z,1-w)$ be holomorphic in $z\in {\mathbb {D}}$ and in $w\in {\mathbb {D}}.$ We consider generalized fractional integral operator

$$\begin{aligned} {\mathcal I}_\lambda ^af(z)=\int _{{\mathbb {D}}}\frac{a(z,1-w)f(z{\overline{w}})}{(1-w)^{2+\lambda }}\mathrm {d}{\mu _\lambda }(w),\ \ \ \mathfrak {R}\lambda >-1,\ \ \ z\in {\mathbb {D}}. \end{aligned}$$

If $a(z,1-w)=(1-w)^{\alpha (z)},$ then ${\mathcal I}^a_\lambda ={\mathbb {I}}_\lambda ^{\alpha (\cdot )}.$

In what follows we assume that the function a has radial majorant $a^*$ i.e.,

$$\begin{aligned} |a(z,1-w)|\leqslant a^*(z,|1-w|),\ \ z, w\in {\mathbb {D}}.\end{aligned}$$

(4.1)

and the function $a^*=a^*(z,h)$ possesses properties of variable modulus of continuity. In the sequel we will use the condition:

$$\begin{aligned} \left| \frac{\frac{\partial }{\partial z} a(z,\zeta )}{a^*(z,|\zeta |)}{|}_{\zeta =1-z{\overline{w}}}\right| \leqslant C\frac{\omega (z,1-|w|)}{1-|w|},\ \ z,w\in {\mathbb {D}}. \end{aligned}$$

(4.2)

We note that in the case $a(z,1-w)=(1-w)^{\alpha (z)},$ when ${{\mathcal {I}}}^a_\lambda ={\mathbb {I}}_\lambda ^{\alpha (\cdot )},$ the property (4.2) is satisfied with $a^*(z,|1-w|)=C|1-w|^{\mathfrak {R}\alpha (z)}$ provided $\sup _{z\in {\mathbb {D}}}|\alpha '(z)|<\infty$ and

$$\begin{aligned} \ln \frac{e}{h}\leqslant \frac{C}{h}\inf _{z\in {\mathbb {D}}}\omega (z,h),\ \ \ h\in [0,2], \end{aligned}$$

(4.3)

where the constant C is independent of h.

Lemma 4.5

Let $\gamma :{\mathbb {D}}\rightarrow (1,\infty )$ be bounded. Let $\phi :{\mathbb {D}}\times [0,2]\rightarrow {\mathbb {R}}_+$ be bounded and almost increasing on [0, 2] uniformly in $z\in {\mathbb {D}}$ and let $\frac{\phi (z,h)}{h}$ be almost decreasing on [0, 2] uniformly in $z\in {\mathbb {D}}.$ Assume that

$$\begin{aligned}\int _t^2\frac{\phi (z,s)}{s^{\gamma (z)}}ds\leqslant k(z)\frac{\phi (z,t)}{t^{\gamma (z)-1}},\ \ 0<t<2, z\in {\mathbb {D}}, \end{aligned}$$

where k(z) does not depend on t. Then

$$\begin{aligned} \int _{{\mathbb {T}}}\frac{\phi (z,|\tau -z|)}{|\tau -z|^{\gamma (z)}}|d\tau |\leqslant Ck(z)\frac{\phi (z,1-|z|)}{(1-|z|)^{\gamma (z)-1}}, \ z\in {\mathbb {D}}, \end{aligned}$$

where C does not depend on $z\in {\mathbb {D}}.$

Proof

The proof follows the lines of the proof of Lemma 1 from [3]. $\square$

Lemma 4.6

Let $\omega _1$ and $\omega _2$ be variable modulus of continuity such that $\omega _1\in Z_1,$ $\omega _2\in Z^0$, and the product $\omega _1\omega _2\in Z_1$. Then the following estimate is true

$$\begin{aligned} \int _{{\mathbb {D}}} \frac{\omega _1(z,|1-z{\overline{w}}|)}{|1-z{\overline{w}}|^2}\frac{\omega _2(z,1-|w|)}{1-|w|}\mathrm {dA}(w)\leqslant C\frac{\omega _1(z,1-|z|)\omega _2(z,1-|z|)}{1-|z|}, \end{aligned}$$

where the constant C does not depend on $z\in {\mathbb {D}}.$

Proof

Proof follows the lines of the proof of Lemma 6.1 from [4]. $\square$

Theorem 4.7

Let $a^*$ be a radial majorant of a,  satisfying (4.2). Let $a^*$ and $\omega$ be variable modulus of continuity and let $\omega _a =\omega a^*.$ Let also $\omega \in Z^0$, $a^*\in Z_1$, and let $\omega _a\in Z_1$. Then for the operator ${\mathcal I}_\lambda ^a$ the following mapping property holds:

$$\begin{aligned} {{\mathcal {I}}}_\lambda ^a: B^{\omega (\cdot )}({\mathbb {D}})\rightarrow B^{\omega _a(\cdot )}({\mathbb {D}}). \end{aligned}$$

(4.4)

Proof

For all $z\in {\mathbb {D}}$ we have

$$\begin{aligned} \left( {{\mathcal {I}}}_\lambda ^af\right) ^{'}(z)= & {} \int _{{\mathbb {D}}} \frac{a(z,1-w){\overline{w}}f'(z{\overline{w}})}{(1-w)^{2+\lambda }}\mathrm {d}{\mu _\lambda }(w)\\&+\int _{{\mathbb {D}}} \frac{\frac{\partial }{\partial z}a(z,1-w) f(z{\overline{w}})}{(1-w)^{2+\lambda }}\mathrm {d} {\mu _\lambda }(w). \end{aligned}$$

Consequently, in view of (2.6) for $z\ne 0$

$$\begin{aligned} \left( {{\mathcal {I}}}_\lambda ^af\right) ^{'}(z)= & {} \frac{1}{z} \int _{{\mathbb {D}}}\frac{a(z,1-z{\overline{w}})wf'(w)}{(1-z{\overline{w}})^{2+\lambda }}\mathrm {d}{\mu _\lambda }(w)\\&+ \int _{{\mathbb {D}}}\frac{\frac{\partial }{\partial z} a(z,\zeta ){|}_{\zeta =1-z{\overline{w}}} f(w)}{(1-z{\overline{w}})^{2+\lambda }}\mathrm {d}{\mu _\lambda }(w). \end{aligned}$$

Therefore, for $\frac{1}{2}<|z|<1$ we estimate as follows

$$\begin{aligned} \left| \left( {{\mathcal {I}}}_\lambda ^af\right) ^{'}(z)\right|\leqslant & {} \left( 2\Vert f\Vert _{\#,B^{\omega (\cdot )}({\mathbb {D}})}+\Vert f\Vert _{C({\mathbb {D}})}\right) \int _{{\mathbb {D}}} \frac{a^*(z,|1-z{\overline{w}}|)}{|1-z{\overline{w}}|^2}\frac{\omega (z,1-|w|)}{1-|w|}\mathrm {dA}(w), \end{aligned}$$

in view of (4.1). It suffices to apply Lemma 4.6 with $\omega _1=a^*$ and $\omega _2=\omega$ to get (4.4). $\square$

Remark 4.8

We note that the assumptions of Theorem 4.7 on $\omega$ and $\omega _a$ imply that the spaces $A^{\omega (\cdot )}({\mathbb {D}})$ and $B^{\omega (\cdot )}({\mathbb {D}})$ coincide up to norm equivalence, and the spaces $A^{\omega _a(\cdot )}({\mathbb {D}})$ and $B^{\omega _a(\cdot )}({\mathbb {D}})$ coincide up to norm equivalence. This follows from the embeddings between such spaces proved in Theorem 3.1 in view of almost increasing of $a^*.$

Let us single out the following special case

$$\begin{aligned} a(z,1-w)=(1-w)^{\alpha (z)}\ln ^{\beta (z)}\frac{e}{1-w}, \end{aligned}$$

where $\alpha \in H({\mathbb {D}}),$ $\mathfrak {R}\alpha (z) \geqslant 0,$ $\beta \in H({\mathbb {D}}).$ Denote

$$\begin{aligned} {\mathbb I}_\lambda ^{\alpha (\cdot ),\beta (\cdot )}f(z)=\int _{{\mathbb {D}}}\frac{(1-w)^{\alpha (z)}\ln ^{\beta }\frac{e}{(1-w)}}{(1-w)^{2+\lambda }}f(z{\overline{w}})\mathrm {d}{\mu _\lambda }(w),\ \ \ \ z\in {\mathbb {D}}. \end{aligned}$$

Corollary 4.9

Let $\omega$ be variable modulus of continuity, $\omega \in Z^0,$ and condition (4.3) be satisfied. Let $\alpha , \beta \in \ H({\mathbb {D}}),$ $\alpha , \beta \in \mathrm {Lip}({\mathbb {D}}),$ $\mathfrak {R}\alpha (z)\geqslant 0,$ $\sup _{z\in {\mathbb {D}}} \mathfrak {R}\alpha (z)<1,$ $\mathrm {inf}_{z\in {\mathbb {D}}}\mathfrak {R}\beta (z)>-\infty .$ Let also $\omega _{\alpha ,\beta }(z,h)=\left( h^{\mathfrak {R}\alpha (z)}\ln ^\beta \frac{e}{h}\right) \omega (z,h),$ and $\omega _{\alpha ,\beta }\in Z_1$. Then for the operator ${{\mathbb {I}}}_\lambda ^{\alpha (\cdot ),\beta }$ the following mapping property holds:

$$\begin{aligned}&{{\mathbb {I}}}_\lambda ^{\alpha (\cdot ),\beta (\cdot )}: B^{\omega (\cdot )}({\mathbb {D}})\rightarrow B^{\omega _{\alpha ,\beta }(\cdot )}({\mathbb {D}}), \\&\quad {{\mathbb {I}}}_\lambda ^{\alpha (\cdot ),\beta (\cdot )}: A^{\omega (\cdot )}({\mathbb {D}})\rightarrow A^{\omega _{\alpha ,\beta }(\cdot )}({\mathbb {D}}). \end{aligned}$$

Proof

The proof is the direct verification of the condition (4.2) where checking of this condition in the presence of logarithmic factor leads to the condition $\ln \ln \frac{e^2}{h}\leqslant C\frac{1}{h}\inf _{z\in {\mathbb {D}}}\omega (z,h),$ $h\in [0,2],$ which holds under the condition (4.3). $\square$

In the case $\beta (z)=0,$ $z\in {\mathbb {D}},$ in the above corollary the Zygmund conditions for $\omega$ may be written as follows $\omega \in \Phi ^0_{1-\mathfrak {R}\alpha (z)}.$ In particular, in the same case $\beta (z) = 0$ we arrive at the following corollary:

Corollary 4.10

Let $\alpha \in H({\mathbb {D}}),$ $\alpha \in \mathrm {Lip}({\mathbb {D}}),$ $\mathfrak {R}\alpha (z)\geqslant 0,$ let the function $\gamma :{\mathbb {D}}\rightarrow (0,1)$ satisfy log-Hölder condition (2.1), $\inf _{z\in {\mathbb {D}}}\gamma (z)>0,$ and $\sup _{z\in {\mathbb {D}}}(\gamma (z)+\mathfrak {R}\alpha (z))<1$. Then for the operator ${{\mathbb {I}}}_\lambda ^{\alpha (\cdot )}$ the following mapping property holds:

$$\begin{aligned}&{{\mathbb {I}}}_\lambda ^{\alpha (\cdot )}: B^{\gamma (\cdot )}({\mathbb {D}})\rightarrow B^{\gamma (\cdot )+\mathfrak {R}\alpha (\cdot )}({\mathbb {D}}), \\&\quad {{\mathbb {I}}}_\lambda ^{\alpha (\cdot )}: A^{\gamma (\cdot )}({\mathbb {D}})\rightarrow A^{\gamma (\cdot )+\mathfrak {R}\alpha (\cdot )}({\mathbb {D}}). \end{aligned}$$

In Corollary 4.10 we recover the result of [4] for constant $\alpha$ and in the case $\lambda =0.$

4.2 Mapping properties of variable order fractional integral in generalized Hölder spaces in the non degenerate case

We proceed with the study of mapping properties of ${\mathbb {I}}^{\alpha (\cdot )}_\lambda$ in the spaces $A^{\omega (\cdot )}({\mathbb {D}})$ without the usage of the spaces $B^{\omega (\cdot )}({\mathbb {D}}).$ This approach is based mainly on techniques developed in [9, 10]. In this way we do not allow $\mathfrak {R}\alpha$ to be degenerate, but we impose less restrictive conditions on the corresponding modulus of continuity.

Let us denote

$$\begin{aligned} \Pi _\alpha =\{z\in {\mathbb {D}}: \mathfrak {R}\alpha (z)=0\},\ \ \ \alpha _h(z)=\inf _{\zeta \in {\mathbb {D}}: |z-\zeta |<h}\mathfrak {R}\alpha (\zeta ). \end{aligned}$$

Theorem 4.11

Let $\alpha \in H({\mathbb {D}}),$ $0\leqslant \mathfrak {R}\alpha (z)\leqslant 1.$ Let the function $\mathfrak {R}\alpha$ satisfy the log-Hölder condition (2.1) on ${\mathbb {D}}.$ Then for all $z\in {\mathbb {D}}{\setminus }\Pi _\alpha$ such that $\alpha _h(z)\ne 0$ the following Zygmund type estimate is valid:

$$\begin{aligned} \omega ({\mathbb {I}}^{\alpha (\cdot )}_\lambda f,z,h )\leqslant & {} \frac{C}{\alpha _h(z)}h^{\mathfrak {R}\alpha (z)} \omega (f,z,h)+Ch\int _{h}^2\frac{\omega (f,z,t)}{t^{2-\mathfrak {R}\alpha (z)}}dt\nonumber \\&+C \omega (\alpha ,z,h)\int _{h}^2\frac{\omega (f,z,t)}{t^{2-\mathfrak {R}\alpha (z)}}dt+Ch\Vert f\Vert _{C({\mathbb {D}})}, \end{aligned}$$

(4.5)

where the constant C does not depend on f,  z and h.

Proof

Fix the numbers $0<\delta<\delta _1<\delta _0<1$ and assume that h is always less then $\delta _1-\delta$ which guarantees that if $z\in \delta {\mathbb {D}}$ and $|z-\zeta |<h$ then $\zeta \in \delta _1{\mathbb {D}}.$

Let us consider the case $z\in \delta {\mathbb {D}}.$ Let $\delta _0{\mathbb {T}}$ stand for the circle with radius $\delta _0$ centered at the origin. Then for $z\in \delta {\mathbb {D}}$ and arbitrary $\zeta \in {\mathbb {D}}$ such that $|z-\zeta |<h$ we have

$$\begin{aligned}&\left| {\mathbb {I}}^{\alpha (\cdot )}_\lambda f(z)-{\mathbb {I}}^{\alpha (\cdot )}_\lambda f(\zeta )\right| \\&\quad = \bigg {|} \int _{{\mathbb {D}}}f(w)\bigg {(}\frac{1}{(1-z{\overline{w}})^{2+\lambda -\alpha (z)}} - \frac{1}{(1-\zeta {\overline{w}})^{2+\lambda -\alpha (\zeta )}}\bigg {)}\mathrm {d} {\mu _\lambda }(w)\bigg {|}\\&\quad = \bigg {|} \int _{{\mathbb {D}}}f(w)\bigg {(} \frac{1}{2\pi i}\int _{\delta _0{\mathbb {T}}}\frac{(1-\tau {\overline{w}})^{-2-\lambda +\alpha (\tau )}}{\tau -z}d\tau \\&\qquad -\frac{1}{2\pi i} \int _{\delta _0{\mathbb {T}}}\frac{(1-\tau {\overline{w}})^{-2-\lambda +\alpha (\tau )}}{\tau -\zeta } d\tau \bigg {)}\mathrm {d} {\mu _\lambda }(w)\bigg {|} \\&\quad = |z-\zeta |\bigg {|} \int _{{\mathbb {D}}}f(w)\left( \frac{1}{2\pi i} \int _{\delta _0{\mathbb {T}}}\frac{(1-\tau {\overline{w}})^{-2-\lambda +\alpha (\tau )}}{(\tau -z)(\tau -\zeta )}d\tau \right) \mathrm {d}{\mu _\lambda }(w)\bigg {|}\\&\quad \leqslant h\sup _{(\tau ,w)\in \delta _0{\mathbb {T}}\times {\mathbb {D}}}\left| (1-\tau {\overline{w}})^{-2-\lambda +\alpha (\tau )} \right| \left| \frac{1}{2\pi }\int _{\delta _0{\mathbb {T}}}\frac{1}{(\tau -z)(\tau -\zeta )}d\tau \right| \Vert f\Vert _{C({\mathbb {D}})}\\&\quad \leqslant C h \Vert f\Vert _{C({\mathbb {D}})}, \end{aligned}$$

where C does not depend on f, z and h.

Now let $z\in {\mathbb {D}}{\setminus }\delta {\mathbb {D}}.$ Note that the following cancellation property

$$\begin{aligned} \int _{\mathbb {D}}\left( \frac{1}{(1-z{\overline{w}})^{2+\lambda -\alpha (z)}}-\frac{1}{(1-\zeta {\overline{w}})^{2+\lambda -\alpha (\zeta )}}\right) \mathrm {d}{\mu _\lambda }(w)=0,\,\,\, z,\zeta \in {\mathbb {D}}, \end{aligned}$$

directly follows from (2.7). Taking into account the cancellation property we have

$$\begin{aligned}&{\mathbb {I}}^{\alpha (\cdot )}_\lambda f(z)-{\mathbb {I}}^{\alpha (\cdot )}_\lambda f(\zeta )\\&\quad =\int _{|z-w|<2h}\frac{f(w)-f(z)}{(1-z{\overline{w}})^{2+\lambda -\alpha (z)}}\mathrm {d} {\mu _\lambda }(w)-\int _{|z-w|<2h}\frac{f(w)-f(z)}{(1-\zeta {\overline{w}})^{2+\lambda -\alpha (\zeta )}}\mathrm {d} {\mu _\lambda }(w)\\&\qquad +\int _{|z-w|>2h} (f(w)-f(z))\left( \frac{1}{(1-z{\overline{w}})^{2+\lambda -\alpha (\zeta )}}-\frac{1}{(1-\zeta {\overline{w}})^{2+\lambda -\alpha (\zeta )}}\right) \mathrm {d} {\mu _\lambda }(w)\\&\qquad +\int _{|z-w|>2h} (f(w)-f(z))\left( \frac{1}{(1-z{\overline{w}})^{2+\lambda -\alpha (z)}}-\frac{1}{(1-z{\overline{w}})^{2+\lambda -\alpha (\zeta )}}\right) \mathrm {d} {\mu _\lambda }(w)\\&\quad =I_1+I_2+I_3+I_4. \end{aligned}$$

For the integral $I_1$ we have

$$\begin{aligned} |I_1|\leqslant & {} \omega (f,z,2h)\int _{|z-w|<2h}\frac{|\mathrm {d} \mu _\lambda (w)|}{\left| (1-z{\overline{w}})^{2+\lambda -\alpha (z)}\right| } \\\leqslant & {} C \omega (f,z,2h)\int _{|z-w|<2h}\frac{|\mathrm {d}\mu _\lambda (w)|}{\left| 1-z{\overline{w}}\right| ^{2+\mathfrak {R}\lambda -\mathfrak {R}\alpha (z)}} \\\leqslant & {} C_1 \omega (f,z,2h)\int _{|z-w|<2h}\frac{\mathrm {d A}(w)}{\left| 1-z{\overline{w}}\right| ^{2-\mathfrak {R}\alpha (z)}}\\\leqslant & {} C_1 \omega (f,z,2h)\int _{|z-w|<2h}\frac{\mathrm {d A}(w)}{|z-w|^{2-\mathfrak {R}\alpha (z)}} \\= & {} 2C_1 \omega (f,z,2h)\int _0^{2h}\frac{dt}{t^{1-\mathfrak {R}\alpha (z)}} \\\leqslant & {} C_2\frac{h^{\mathfrak {R}\alpha (z)} \omega (f,z,h)}{\mathfrak {R}\alpha (z)}, \end{aligned}$$

where $C_2$ does not depend on f, z and h. Here we also used the property $\omega (f,z,2h)\leqslant C\omega (f,z,h)$ with C not depending on z, h. It follows from (2.2):

$$\begin{aligned} \omega (z,f,2h)= & {} \sup _{w\in {\mathbb {D}}:|w-z|<2h}|f(z)-f(w)|\\ {}\leqslant & {} \sup _{w\in {\mathbb {D}}:|w-z|<2h} \left( |f(z)-f\left( \frac{z+w}{2}\right) |+|f\left( \frac{z+w}{2}\right) -f(w)|\right) \\ {}\leqslant & {} C\omega (z,f,h) \end{aligned}$$

with a constant C not depending on z and h.

Since $\{w\in {\mathbb {D}}:|z-w|<2h\}\subset \{w\in {\mathbb {D}}:|\zeta -w|<3h\}$, for the integral $I_2$ we similarly have

$$\begin{aligned} |I_2|\leqslant & {} \omega (f,z,2h)\int _{|z-w|<2h}\frac{|\mathrm {d} \mu _\lambda (w)|}{\left| (1-\zeta {\overline{w}})^{2+\lambda -\alpha (\zeta )}\right| } \\\leqslant & {} \omega (f,z,2h)\int _{|\zeta -w|<3h}\frac{|\mathrm {d} \mu _\lambda (w)|}{\left| (1-\zeta {\overline{w}})^{2+\lambda -\alpha (\zeta )}\right| } \\\leqslant & {} C \frac{h^{\mathfrak {R}\alpha (z)} \omega (f,z,h)}{\mathfrak {R}\alpha (\zeta )}\leqslant C \frac{h^{\mathfrak {R}\alpha (z)} \omega (f,z,h)}{\alpha _h(z)}, \end{aligned}$$

where C does not depend on f, z and h.

To estimate $I_3$ we first note that for $z,\zeta \in {\mathbb {D}}$ and $\gamma \in {\mathbb {C}}$ the following estimate holds

$$\begin{aligned} \left| \frac{1}{(1-z{\overline{w}})^{\gamma }}-\frac{1}{(1-\zeta {\overline{w}})^{\gamma }}\right|= & {} \left| \gamma \int _\zeta ^z \frac{{\overline{w}}d\xi }{(1-\xi {\overline{w}})^{1+\gamma }} \right| \\ {}\leqslant & {} \frac{|\gamma ||z-\zeta |}{ \inf _{\xi \in [\zeta ,z]}\left| (1-\xi {\overline{w}})^{-1-\gamma }\right| }. \end{aligned}$$

Here we took into account that $\delta \in (0,1)$ is sufficiently close to 1. Therefore,

$$\begin{aligned} |I_3|\leqslant & {} C|2+\lambda -\alpha (\zeta )||z-\zeta |\int _{|z-w|>2h}\frac{\omega (f,z,|z-w|)}{\inf _{\xi \in [\zeta ,z]}\left| (1-\xi {\overline{w}})^{3+\lambda -\alpha (\zeta )}\right| } |\mathrm {d}\mu _\lambda (w)|\\ {}\leqslant & {} C_1 h\int _{|z-w|>2h}\frac{\omega (f,z,|z-w|)}{\left| 1-z{\overline{w}}\right| ^{3+\lambda -\mathfrak {R}\alpha (\zeta )}} |\mathrm {d}\mu _\lambda (w)|\\ {}\leqslant & {} C_2 h\int _{|z-w|>2h}\frac{\omega (f,z,|z-w|)}{\left| 1-z{\overline{w}}\right| ^{3-\mathfrak {R}\alpha (\zeta )}} \mathrm {d A}(w)\\ {}\leqslant & {} C_2 h\int _{h}^2\frac{\omega (f,z,t)}{t^{2-\mathfrak {R}\alpha (\zeta )}}dt\leqslant C_3 h\int _{h}^2\frac{\omega (f,z,t)}{t^{2-\mathfrak {R}\alpha (z)}} dt \end{aligned}$$

where $C_3$ does not depend on f, z and h.

Similarly we have

$$\begin{aligned} |I_4|\leqslant & {} \int _{|z-w|>2h}\frac{\omega (f,z,|z-w|)}{\left| (1-z{\overline{w}})^{2+\lambda -\alpha (\zeta )}\right| } \left| \frac{1}{(1-z{\overline{w}})^{\alpha (\zeta )-\alpha (z)}}-1\right| |\mathrm {d} \mu _\lambda (w)|\\ {}\leqslant & {} C |\alpha (z)-\alpha (\zeta )| \int _{|z-w|>2h}\frac{\omega (f,z,|z-w|)}{|1-z{\overline{w}}|^{3+\lambda -\mathfrak {R}\alpha (\zeta )}} |\mathrm {d}\mu _\lambda (w)|\\ {}\leqslant & {} C \omega (\alpha ,z,h)\int _{h}^2\frac{\omega (f,z,t)}{t^{2-\mathfrak {R}\alpha (\zeta )}}dt\leqslant C_1 \omega (\alpha ,z,h)\int _{h}^2\frac{\omega (f,z,t)}{t^{2-\mathfrak {R}\alpha (z)}}dt, \end{aligned}$$

where $C_1$ does not depend on f, z and h.

Gathering the above estimates we arrive at (4.5). $\square$

Finally we prove the following theorem on mapping of variable order fractional integral in the spaces $A^{\omega (\cdot )}({\mathbb {D}})$.

Theorem 4.12

Let $\omega$ be variable modulus of continuity, $\omega \in Z_{1-\mathfrak {R}\alpha (\cdot )}.$ Let $\alpha \in H({\mathbb {D}}), \alpha \in \mathrm {Lip}({\mathbb {D}}),$ $0<\inf _{{\mathbb {D}}}\mathfrak {R}\alpha (z),$ $\sup _{{\mathbb {D}}}\mathfrak {R}\alpha (z)<1,$ $\omega _{\alpha }(h)=h^{\mathfrak {R}\alpha (z)}\omega (z,h).$ Then for the operator ${{\mathbb {I}}}_\lambda ^{\alpha (\cdot )}$ the following mapping property holds:

$$\begin{aligned} {{\mathbb {I}}}_\lambda ^{\alpha (\cdot )}: A^{\omega (\cdot )}({\mathbb {D}})\rightarrow A^{\omega _{\alpha }(\cdot )}({\mathbb {D}}). \end{aligned}$$

Proof

In view of (4.5) and the assumptions on $\alpha$ and $\omega$ we have

$$\begin{aligned} \omega ({\mathbb {I}}^{\alpha (\cdot )}_\lambda f,z,h )\leqslant & {} \bigg {(}\frac{C}{\alpha _h(z)} +C_1 \bigg {)}h^{\mathfrak {R}\alpha (z)}\omega (z,h)\Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {D}})}+Ch\Vert f\Vert _{C({\mathbb {D}})}\\ {}\leqslant & {} C_2 h^{\mathfrak {R}\alpha (z)}\omega (z,h)\Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {D}})}+Ch\Vert f\Vert _{C({\mathbb {D}})}, \end{aligned}$$

where $C_2$ does not depend on f, z and h. Therefore,

$$\begin{aligned} \frac{\omega ({\mathbb {I}}^{\alpha (\cdot )}_\lambda f,z,h)}{h^{\mathfrak {R}\alpha (z)}\omega (z,h)}\leqslant C_2 \Vert f\Vert _{\#,A^{\omega (\cdot )}({\mathbb {D}})}+C\Vert f\Vert _{C({\mathbb {D}})}\leqslant C_3 \Vert f\Vert _{A^{\omega (\cdot )}({\mathbb {D}})},\ \ \ z\in {\mathbb {D}}, \end{aligned}$$

where $C_3$ does not depend on f, z and h. The above estimate implies the statement of the theorem. $\square$