1 Introduction

Let \(\Gamma \subset \mathbb {C} ~~\)be a Jordan rectifiable curve and let \(\omega :\Gamma \rightarrow \left[ 0,\infty \right] \) be a weight function, that is a positive almost everywhere (a.e.) and integrable function on \(\Gamma .~\)For \(1<p<\infty ~\)we define a class \(L^{p}(\Gamma ,\omega )~\)of Lebesgue measurable functions f on \(\Gamma ~\)satisfying the condition

$$\begin{aligned} \left( \frac{1}{\left| \Gamma \right| }\int \limits _{\Gamma } \left| f(z)\right| ^{p}\omega (z)\left| \mathrm{d}z\right|<\infty \right) ^{\frac{1}{p}}<\infty , \end{aligned}$$

where \(\left| \Gamma \right| ~\) is the length of \(\ \Gamma .~\) We denote by \(L^{p),\theta }(\Gamma ,\omega \mathbf {)},\) \(\theta \ge 0,\) the Lebesgue space of all measurable functions f  on \(\Gamma ,~\)that is, the space of all such functions for which

$$\begin{aligned} \left\| f\right\| _{L^{p),\theta }(\Gamma ,\omega )}:=\underset{ 0<\varepsilon<p-1}{\sup }\left( \frac{\varepsilon ^{\theta }}{\left| \Gamma \right| }\int \limits _{\Gamma }\left| f(z)\right| ^{p-\varepsilon }\omega \left( z\right) \left| \mathrm{d}z\right| \right) ^{ \frac{1}{p-\varepsilon }}<\infty .~ \end{aligned}$$

The space \(L^{p),\theta }(\ \Gamma ,\omega \mathbf {)}\) is called the generalized grand Lebesgue space. \(L^{p),\theta }(\Gamma ,\omega \mathbf {)~}\) is Banach function space, nonreflexive and nonseparable. The grand and generalized grand Lebesgue space were introduceed in the works [13, 26], respectively. If \(\ \theta _{1}<\theta _{2}\) then for \(0<\varepsilon <p-1~\)the embeddings:

$$\begin{aligned} L^{p}\left( \Gamma ,\omega \right) \subset L^{p),\theta _{1}}(\Gamma ,\omega ) \subset L^{p),\theta _{2}}(\Gamma ,\omega )\subset L^{p-\varepsilon }\left( \Gamma ,\omega \right) , \quad 1<p<\infty \end{aligned}$$

hold. Note that the information about properties and applications of the grand Lebesgue spaces can be found in [11, 13, 26, 33, 35, 36].

A Jordan curve \(\Gamma ~\)is called Ahlfors 1-regular [37], if there exists a number \(c>0~\)such that for every \(r>0,~\sup \left\{ \left| \Gamma \cap D(z,r)\right| :z\in \Gamma \right\} \le cr,~\), where D(z, r) is an open disk with radius r and centered at z and \( \left| \Gamma \cap D(z,r)\right| ~\)is the length of the set \(\Gamma \cap D(z,r).\)

Let \(\omega ~\)be a weight function on \(\Gamma .~\omega ~\)is said to satisfy Muckenhoupt’s \(A_{p}\)-condition on \(\Gamma ~\)if

$$\begin{aligned} \sup _{z\in \Gamma }\sup _{r>0}\left( \frac{1}{r}\int \limits _{\Gamma \cap D\left( z,r\right) }\omega (\zeta )\left| \mathrm{d}\zeta \right| \right) \left( \frac{1}{r}\int \limits _{\Gamma \cap D\left( z,r\right) }\left[ \omega (\zeta )\right] ^{-\frac{1}{p-1}}\left| \mathrm{d}\zeta \right| \right) ^{p-1}<\infty \end{aligned}$$

Let us further assume that B is a simply connected domain with a rectifiable Jordan boundary \(\Gamma \) and \(B^{-}:=\mathrm{ext}\Gamma \). Without loss of generality we assume that \(0\in B.~\)Let

$$\begin{aligned} \mathbb {T}=\left\{ w\in \mathbb {C}:\left| w\right| =1\right\} ,\quad D:=\mathrm{int}\mathbb {T}, \quad D^{-}:=\mathrm{ext}\mathbb {T}. \end{aligned}$$

Also, \(\phi ^{*}\) stand for the conformal mapping of \(B^{-}\) onto \(D^{-}\) normalized by

$$\begin{aligned} \phi ^{*}(\infty )=\infty \end{aligned}$$

and

$$\begin{aligned} \lim _{z\rightarrow \infty }\frac{\phi ^{*}(z)}{z}>0, \end{aligned}$$

and let \(\psi ^{*}\) be the inverse of \(\phi ^{*}\). Let \(\phi _{1}^{*}\) be the conformal mapping of B onto \(D^{-},\) normalized by

$$\begin{aligned} \phi _{1}^{*}(0)=\infty \end{aligned}$$

and

$$\begin{aligned} \lim _{z\rightarrow 0}z\phi _{1}^{*}(z)>0. \end{aligned}$$

The inverse mapping of \(\phi _{1}^{*}\) will be denoted by \(\psi _{1}^{*}.\)

Note that the mappings \(\psi ^{*}\) and \(\psi _{1}^{*}~\)have in some deleted neighborhood of \(\infty ~\)representations:

$$\begin{aligned} \psi ^{*}(w)=\alpha w+\alpha _{0}+\frac{\alpha _{1}}{w}+\frac{\alpha _{2}}{w^{2}}+\cdots +\frac{\alpha _{k}}{w^{k}}+\cdots ,\quad \alpha >0 \end{aligned}$$

and

$$\begin{aligned} \psi _{1}^{*}(w)=\frac{\beta _{1}}{w}+\frac{\beta _{2}}{w^{2}}+\cdots +\frac{ \beta _{k}}{w^{k}}+\cdots ,\quad \beta _{1}>0. \end{aligned}$$

For \(1<p<\infty \) and \(0<\varepsilon <p-1~\)the functions:

$$\begin{aligned} \frac{\left( \frac{\mathrm{d}\psi ^{*}(w)}{\mathrm{d}w}\right) ^{1-\frac{1}{p-\varepsilon } }}{\psi ^{*}(w)-z},\quad z\in B \end{aligned}$$

and

$$\begin{aligned} \frac{w^{-\frac{2}{p-\varepsilon }}\left( \frac{\mathrm{d}\psi _{1}^{*}(w)}{\mathrm{d}w} \right) ^{1-\frac{1}{p-\varepsilon }}}{\psi _{1}^{*}(w)-z},\quad z\in B^{-}. \end{aligned}$$

are analytic in the domain \(D^{-}.~\)The following expansions hold:

$$\begin{aligned} \frac{\left( \frac{\mathrm{d}\psi ^{*}(w)}{\mathrm{d}w}\right) ^{1-\frac{1}{p-\varepsilon } }}{\psi ^{*}(w)-z}=\sum \limits _{k=0}^{\infty }\frac{\Phi _{k,p-\varepsilon }(z)}{w^{k+1}}, \quad z\in B, \quad w\in D^{-} \end{aligned}$$

and

$$\begin{aligned} \frac{w^{-\frac{2}{p-\epsilon }}\left( \frac{\mathrm{d}\psi _{1}^{*}(w)}{\mathrm{d}w} \right) ^{1-\frac{1}{p-\varepsilon }}}{\psi _{1}^{*}(w)-z} =\sum \limits _{k=1}^{\infty }-\frac{F_{k,p-\varepsilon }(\frac{1}{z})}{ w^{k+1}}, \quad z\in B^{-}, \quad w\in D^{-}, \end{aligned}$$

where \(\Phi _{k,p-\varepsilon }(z)\) and \(F_{k,p-\varepsilon }(\frac{1}{z})\) are the \(p-\varepsilon ~~\)Faber polynomials of degree k with respect to z  and \(\frac{1}{z}~\)for the continuums \(\overline{B\text {~}}\) and \( \overline{B}\backslash B,\) respectively (see also [5, 20, 23] and ([34], pp. 255–257).

Let \(E^{1}(B)~\)be a classical Smirnov class of analytic functions in B. The set \(\ E^{p),\theta }(B,\omega ):=\left\{ f\in E^{1}(B):f\in L^{p),\theta }(\Gamma ,\omega )\right\} \) is called the \(\omega \)-weighted generalized grand Smirnov class in B.

Let \(\omega \in A_{p}(\mathbb {T})\). For \(f\in L^{p),\theta }(\Gamma ,\omega )\) we define the operator

$$\begin{aligned} \left( \nu _{h}^{r}f\right) \left( w\right) :=\dfrac{1}{h} \int \limits _{-0}^{h}\left| \Delta _{t}^{r}f\left( w\right) \right| \mathrm{d}t, \quad h>0, \end{aligned}$$

where

$$\begin{aligned} \Delta _{t}^{r}f\left( w\right) :=\sum \limits _{k=0}^{r}\left( -1\right) ^{r+k+1}\left( \begin{array}{c} r \\ k \end{array} \right) f\left( w\mathrm{e}^{ikt}\right) ,\quad r\in \mathbb {N=}\left\{ 1,2,\ldots \right\} , \quad w\in \mathbb {T}, \quad t>0. \end{aligned}$$

If \(\omega \in A_{p}(\mathbb {T})\mathbb {~}\)and  \(f\in L^{p}({\mathbb {T}}, ~\omega \mathbf {),~}\)then the operator \(\nu _{h}\) is a bounded on \( L^{p),\theta }\left( \mathbb {T},\omega \right) \)  [24]:

$$\begin{aligned} \underset{\left| h\right| \le \delta }{\sup }\left\| \nu _{h}^{r}\left( f\right) \right\| _{L^{p),~\theta }\left( \mathbb {T}, \omega \right) }\le c_{1}\left\| f\right\| _{L^{p),~\theta }\left( \mathbb {T}\text {,}\omega \right) }. \end{aligned}$$

Let \(1<p<\infty ,~\) \(\omega \in A_{p}(\mathbb {T})\) and  \(f\in L^{p),\theta }( \mathbb {T}\),\(~\omega \mathbf {)},\theta >0.\) The function

$$\begin{aligned} \Omega _{p),\theta ,\omega }^{r}\left( f,\delta \right) :=\sup \limits _{\left| h\right| \le \delta }\left\| \nu _{h}^{r}f\left( w\right) \right\| _{L^{p),~\theta }\left( \mathbb {T}, \omega \right) }, \quad \delta >0 \end{aligned}$$

is called the r-th mean modulus of \(f\in L^{p),~\theta }(\mathbb {T}\),\(~\omega \mathbf {)}\).

It can be easily shown that \(\Omega _{p),\theta ,\omega }^{r}\left( f,\cdot \right) \) is a continuous, non-negative and nondecreasing function satisfying the conditions:

$$\begin{aligned} \lim \limits _{\delta \rightarrow 0}\Omega _{p),\theta ,\omega ,}^{r}\left( f,\delta \right) =0,\;\Omega _{p),\theta ,\omega }^{r}\left( f+g,\delta \right) \le \Omega _{p),\theta ,\omega }^{r}\left( f,\delta \right) +\Omega _{p),\theta ,\omega }^{r}\left( g,\delta \right) , \quad \delta >0 \end{aligned}$$

for \(f,g\in L^{p),\theta }(\mathbb {T}\), \(\omega \mathbf {)}\).

Let G be a doubly connected domain in the complex plane \({\mathbb { C}}\), bounded by the rectifiable Jordan curves \(\Gamma _{1}\) and \(\Gamma _{2} \) (the closed curve \(\Gamma _{2}\) is in the closed curve \(\Gamma _{1}\)). Without loss of generality we assume \(0\in \) int\(\Gamma _{2}\). Let \( G_{1}^{0} \): = int\(\Gamma _{1}\), \(G_{1}^{\infty }\): =ext\(\Gamma _{1}\), \( G_{2}^{0}\): =int\(\Gamma _{2}\), \(G_{2}^{\infty }\):=ext\(\Gamma _{2}\).

We denote by \(w=\phi \left( z\right) \) the conformal mapping of \( G_{1}^{\infty }\) onto domain \(D^{-}\) normalized by the conditions:

$$\begin{aligned} \phi \left( \infty \right) =\infty ,\quad \lim _{z\rightarrow \infty }\,\frac{ \phi \left( z\right) }{z}>0 \end{aligned}$$

and let \(\psi \) be the inverse mapping of \(\phi \).

We denote by \(w=\phi _{1}\left( z\right) \) the conformal mapping of \( G_{2}^{0}\) onto domain \(\,D^{-}\,\) normalized by the conditions:

$$\begin{aligned} \phi _{1}\left( 0\right) =\infty ,\quad \lim _{z\rightarrow 0}(z.\phi _{1}\left( z\right) )>0 \end{aligned}$$

and let \(\psi _{1}\) be the inverse mapping of \(\phi _{1}.\)

Let us take

$$\begin{aligned} C_{\rho _{0}}:=\left\{ z:\left| \phi \left( z\right) \right| =\rho _{0}>1\right\} ,\quad \Gamma _{r_{0}}:=\left\{ z:\left| \phi _{1}\left( z\right) \right| =r_{0}>1\right\} . \end{aligned}$$

For \(\Phi _{k,p-\varepsilon }\left( z\right) \) and \( F_{k,p-\varepsilon }\left( \frac{1}{z}\right) \) the following integral representations hold [5, 20, 23, 34], pp. 255–257:

  1. (1)

    If \(z\in intC_{\rho _{0}},\) then

    $$\begin{aligned} \Phi _{k,p-\varepsilon }\left( z\right) =\frac{1}{2\pi i}\int \limits _{C_{ \rho _{0}}}\frac{\left[ \phi \left( \zeta \right) \right] ^{k}\left( \phi ^{\prime }(\zeta )\right) ^{\frac{1}{p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta . \end{aligned}$$
    (1.1)
  2. (2)

    If \(z\in \mathrm{ext}C_{\rho _{0}}\), then

    $$\begin{aligned}&\Phi _{k,p-\varepsilon }\left( z\right) \nonumber \\&\quad =\left[ \phi \left( z\right) \right] ^{k}\left( \phi ^{\prime }(z)\right) ^{\frac{1}{p-\varepsilon }}+\frac{1}{2\pi i}\int \limits _{C_{\rho _{0}}}\frac{ \left[ \phi \left( \zeta \right) \right] ^{k}\left( \phi ^{\prime }(\zeta )\right) ^{ \frac{1}{p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$
    (1.2)
  3. (3)

    If \(z\in intC_{r_{0}}\), then

    $$\begin{aligned}&F_{k,p-\varepsilon }(\frac{1}{z}) \nonumber \\&\quad =\left[ \phi _{1}\left( z\right) \right] ^{k-\frac{2}{p-\varepsilon } }\left( \phi ^{\prime }(z)\right) ^{\frac{1}{p-\varepsilon }}-\frac{1}{2\pi i }\int \limits _{C_{r_{0}}}\frac{\left[ \phi _{1}\left( \zeta \right) \right] ^{k-\frac{2}{p-\varepsilon }}(\phi _{1}^{\prime }(\zeta ))^{\frac{1}{ p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta .\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \end{aligned}$$
    (1.3)
  4. (4)

    If \(z\in \mathrm{ext}C_{r_{0}},~\) then

    $$\begin{aligned} F_{k,p-\varepsilon }\left( \frac{1}{z}\right) =-\frac{1}{2\pi i} \int \limits _{C_{r_{0}}}\frac{\left[ \phi _{1}\left( \zeta \right) \right] ^{k-\frac{2}{p-\varepsilon }}\left( \phi _{1}^{\prime }(\zeta )\right) ^{\frac{1}{ p-\varepsilon }}}{\zeta -z}\mathrm{d}\zeta . \end{aligned}$$
    (1.4)

    If a function \(f\left( z\right) \) is analytic in the doubly connected domain bounded by the curves \(C_{\rho _{0}}\) and \(\Gamma _{r_{0}}\), then the following series expansion holds:

    $$\begin{aligned} f\left( z\right) =\sum \limits _{k=0}^{\infty }a_{k}\Phi _{k,p-\varepsilon }\left( z\right) +\sum \limits _{k=1}^{\infty }b_{k}F_{k,p-\varepsilon }\left( \frac{1}{z}\right) , \end{aligned}$$
    (1.5)

    where

$$\begin{aligned} a_{k}=\frac{1}{2\pi i}\int \limits _{\left| w\right| =\rho _{1}}\frac{f \left[ \psi \left( w\right) \right] \left( \psi ^{\prime }(w)\right) ^{\frac{1}{p-\varepsilon }}}{w^{k+1}}dw,~~\left( 1<\rho _{1}<\rho _{0}\right) ,~k=0,1,2,\ldots \end{aligned}$$

and

$$\begin{aligned} b_{k}=\frac{1}{2\pi i}\int \limits _{\left| w\right| =r_{1}}\frac{f \left[ \psi _{1}(w)\right] (\psi _{1}^{\prime }(w))^{\frac{1}{p-\varepsilon } }w^{\frac{2}{p-\varepsilon }}}{w^{k+1}}\mathrm{d}w,\quad \left( 1<r_{1}<r_{0}\right) ,\quad k=1,2,\ldots \end{aligned}$$

The series (1.5) is called the \(p-\varepsilon ~\) Faber–Laurent series of f,  and the coefficients \(a_{k}\) and \(b_{k}\) are said to be the \( p-\varepsilon ~\) Faber–Laurent coefficients of f . For \(z\in G\) by Cauchy’s integral formulae we have

$$\begin{aligned} f(z)=\frac{1}{2\pi i} \int \limits _{\Gamma _{1} }\frac{f(\zeta )}{\zeta -z} \mathrm{d}\zeta -\frac{1}{2\pi i} \int \limits _{\Gamma _{2} }\frac{f(\xi )}{\xi -z} \mathrm{d}\xi . \end{aligned}$$

If \(z\in \mathrm{int}\Gamma _{2} \) and \(z\in \mathrm{ext}\Gamma _{1} \), then

$$\begin{aligned} \frac{1}{2\pi i} \int \limits _{\Gamma _{1}}\frac{f(\zeta )}{\zeta -z}\mathrm{d}\zeta -\frac{1}{2\pi i} \int \limits _{\Gamma _{2}}\frac{f(\xi )}{\xi -z}\mathrm{d}\xi =0. \end{aligned}$$
(1.6)

Let us consider

$$\begin{aligned} I_{1}(z):=\frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{f(\zeta )}{\zeta -z} \mathrm{d}\zeta ,\quad I_{2}(z):=\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{f(\xi )}{\xi -z}\mathrm{d}\xi . \end{aligned}$$

The function \(I_{1}(z) \) determines the functions \(I_{1}^{+}(z) \) and \( I_{1}^{-}(z) \), while the function \(I_{2}(z) \) determines the functions \( I_{2}^{+}(z) \) and \(I_{2}^{-}(z) \). The functions \(I_{1}^{+}(z) \) and \( I_{1}^{-}(z) \) are analytic in \(\mathrm{int}\Gamma _{1} \) and \(\mathrm{ext}\Gamma _{1} \), respectively. The functions \(I_{2}^{+}(z) \) and \(I_{2}^{-}(z) \) are analytic in \(\mathrm{int}\Gamma _{2} \) and \(\mathrm{ext}\Gamma _{2}\), respectively.

Let B be a finite domain in the complex plane bounded by a rectifiable Jordan curve \(\Gamma \) and \(f\in L_{1}\left( \Gamma \right) \). Then the functions \(f^{+}\) and \(f^{-}\) defined by

$$\begin{aligned} f^{+} \left( z\right) =\frac{1}{2\pi i} \int \limits _{\Gamma }\frac{ f\left( \zeta \right) }{\zeta -z} \mathrm{d}\zeta ,\quad z\in B \end{aligned}$$

and

$$\begin{aligned} f^{-} \left( z\right) =\frac{1}{2\pi i} \int \limits _{\Gamma }\frac{ f\left( \zeta \right) }{\zeta -z} \mathrm{d}\zeta ,\quad z\in B^{-} \end{aligned}$$

are analytic in B and \(B^{-} \)respectively, and \(f^{-} \left( \infty \right) =0.\) Thus the limit

$$\begin{aligned} S_{\Gamma }\left( f\right) \left( z\right) :=\lim _{\varepsilon \rightarrow \infty }\frac{1}{2\pi i}\int \limits _{\Gamma \cap \left\{ \zeta :\left| \zeta -z\right| >\varepsilon \right\} }\frac{f(\zeta )}{\zeta -z}\mathrm{d}\zeta \end{aligned}$$

exists and is finite for almost all \(z\in \Gamma \).

The quantity \(S_{\Gamma } (f)(z)\) is called the Cauchy singular integral of f at \(z\in \Gamma \).

According to the Privalov theorem ([12], p. 431), if one of the functions \( f^{+}\) or \(f^{-}\) has the non-tangential limits a.e. on \(\Gamma \), then \( S_{\Gamma }(f)(z)\) exists a.e. on \(\Gamma \) and also the other one has the non-tangential limits a.e. on \(\Gamma \). Conversely, if \(S_{\Gamma }(f)(z)\) exists a.e. on \(\Gamma \), then the functions \(f^{+}\left( z\right) \) and \( f^{-}\left( z\right) \) have non-tangential limits a.e. on \(\Gamma \). In both cases, the formulae

$$\begin{aligned} f^{+} (z)=S_{\Gamma } (f)(z)+\frac{1}{2} f(z), \quad f^{-} (z)=S_{\Gamma } (f)(z)-\frac{1}{2} f(z) \end{aligned}$$

and hence

$$\begin{aligned} f=f^{+}-f^{-} \end{aligned}$$
(1.7)

holds a.e. on \(\Gamma \). From the results given in [33], it follows that if \( \ \Gamma \) is an Ahlfors 1- regular curve, then \(S_{\Gamma }\) is bounded on \( L^{p),\theta }(\Gamma ,\omega ).\)

We will say that the doubly connected domain G is bounded by the Ahlfors 1-regular curve if the domains \(G_{1}^{0}\) and \(G_{2}^{0}\) are bounded by the closed Ahlfors 1-regular curves.

Let \(\Gamma _{i}\) \((i=1,2)\) be a regular curve and let \(f_{0}:=f\left[ \psi \left( w\right) \right] \psi ^{\prime }(w)^{\frac{1}{p-\varepsilon }}\) for \( f\in L^{p),\theta }(\Gamma _{1},\omega )\) and let \(f_{1}(w):=f\left[ \psi _{1}(w)\right] \) \((\psi _{1}^{\prime }(w))^{\frac{1}{p-\varepsilon }}w^{ \frac{2}{p-\varepsilon }~\ }\)for \(f\in L^{p),\theta }(\Gamma _{2},\omega ).\ \)We also set \(\omega _{0}(w):=\omega \left[ \psi (w)\right] \) , \(\omega _{1}(w):=\omega \left[ \psi _{1}(w)\right] .~\)Then , if \(f\in L^{p),\theta }(\Gamma _{1},\omega )\) and \(f\in L^{p),\theta }(\Gamma _{2},\omega )~\)we obtain \(f_{0}\in L^{p),\theta }(\mathbb {T\,}\),\(~\omega _{0})\) and \(f_{1}\in L^{p),\theta }(\mathbb {T}\),\(~\omega _{1})\).

Moreover, \(f_{0}^{-}(\infty )=~f_{1}^{-}(\infty )=0\) and by (1.7)

$$\begin{aligned} \left. \begin{array}{l} {\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad f_{0}(w)=f_{0}^{+}(w)-f_{0}^{-}(w)} \\ {\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad f_{1}(w)=f_{1}^{+}(w)-f_{1}^{-}(w)} \end{array} \right\} \quad \quad \quad \end{aligned}$$
(1.8)

a.e. on \(\mathbb {T}\).

Now, in the doubly connected domain we define the \(\omega \)-weighted generalized grand Smirnov class . Let \(E^{1}(G)~\)be a classical Smirnov class of analytic functions in G. The set \(E^{p),\theta }(G,\omega ):=\left\{ f\in E^{1}(G): f\in L^{p),\theta }(\Gamma ,\omega )\right\} \) is called the \(\omega \)-weighted generalized grand Smirnov class in \(G.\ \)We denote by \(\mathcal {E}^{p),\theta }\left( G,\omega \right) ~\)the closure of Smirnov class \(E^{p}(G,\omega )\) in the space \(E^{p),\theta }(G,\omega ).\)

Lemma 1.1

[23, 24]. Let \(g\in \mathcal {E}^{p),\theta }(D,\omega )\) , \(\omega \in A_{p}(\mathbb {T} ),~1<p<\infty ~\)and \(\theta >0.~\)If\(~\sum \nolimits _{k=0}^{n}d_{k}(g)w^{k}\) is the nth partial sum of the Taylor series of g  at the origin, then there exists a constant \(c_{2}>0\) such that

$$\begin{aligned} \left\| g(w)-\sum \limits _{k=0}^{n}d_{k}(w)w^{k}\right\| _{L^{p),\theta }(\mathbb {T}\text {,}\omega )}\le c_{2}\Omega _{p),\theta ,\omega }^{r}\left( g,\frac{1}{n}\right) , \quad r\in \mathbb {N} \end{aligned}$$

for every natural number n.

We set

$$\begin{aligned} R_{n}(f,z):=\sum \limits _{k=0}^{n}a_{k}\Phi _{k,p-\varepsilon }(z)+\sum \limits _{k=1}^{n}b_{k}F_{k,p-\varepsilon }\left( \frac{1}{z}\right) . \end{aligned}$$

The rational function \(R_{n}(f,z)\) is called the \(p-\varepsilon ~\) Faber–Laurent rational function of degree n of f.

Since series of Faber polynomials are a generalization of Taylor series to the case of a simply connected domain, it is natural to consider the construction of a similar generalization of Laurent series to the case of a doubly-connected domain.

The problems of approximation of the functions in the non-weighted and weighted grand Lebesgue spaces were investigated in [6,7,8,9,10, 23, 24]. In this study the approximation problems of the functions by Faber–Laurent rational functions in the weighted generalized grand Smirnov classes \(\mathcal {E}^{p),\theta }\left( G,\omega \right) \), \(\theta >0\), defined in the doubly connected domains with the regular boundaries are studied. Similar problems in the different spaces were investigated by several authors (see for example, [1,2,3,4,5, 14,15,16,17,18,19,20,21,22,23, 25, 27,28,32, 38, 39]).

Our main result can be formulated as following.

Theorem 1.2

Let G be a finite doubly connected domain with the Ahlfors 1-regular boundary \(\Gamma =\Gamma _{1}\cup \Gamma _{2}.\) If \(\ \omega \in A_{p}(\Gamma ),\omega _{0},\omega _{1}\in A_{p}(\mathbb {T}),~1<p<\infty ~\)and \(f\in \mathcal {E}^{p),\theta }\left( G,\omega \right) \), \(\theta >0,~\) then there is a constant \(c_{3}\) \(>0~\)such that  for any \(n=1,2,3,\ldots \)

$$\begin{aligned} \left\| f-R_{n}\left( \cdot ,f\right) \right\| _{L^{p),\theta }\left( \Gamma ,\omega \right) }\le c_{3}\left\{ \Omega _{p),\theta ,\omega _{0}}^{r}\left( f_{0},{1/n}\right) +\Omega _{p),\theta ,\omega _{1}}^{r}\left( f_{1},{1/n}\right) \right\} , \end{aligned}$$

where \(R_{n}\left( .,f\right) \) is the \(p-\varepsilon ~\) Faber–Laurent rational function of degree n of f.

2 Proof of main result

Proof of Theorem 1.1

We take the curves \(\Gamma _{1}\), \(\Gamma _{2}\) and \(\mathbb {T}:=\left\{ w\in {\mathbb {C}}:\,\left| w\right| =1\right\} \) as the curves of integration in the formulas (1.2)–(1.5) and (1.6), respectively. (This is possible due to the conditions of Theorem 1.2). Let \(f\in \mathcal {E}^{p),\theta }(G,\omega ).\) Then \(f_{0}\in L^{p),\theta }(\mathbb {T},\omega _{0}),~f_{1}\in L^{p),\theta }(\mathbb {T},\omega _{1}).\) According to (1.8)

$$\begin{aligned} \left. \begin{array}{r} {\ \ \ f(\zeta )=[f_{0}^{+}(\phi (\zeta ))-f_{0}^{-}(\phi (\zeta ))](\phi (\zeta ))^{\frac{1}{p-\varepsilon }}} \\ { \,f(\xi )=[f_{1}^{+}(\phi _{1}(\xi ))-f_{1}^{-}(\phi _{1}(\xi ))](\phi _{1}(\xi ))^{-\frac{2}{p-\varepsilon }}(\phi _{1}^{\prime }(\xi ))^{ \frac{1}{p-\varepsilon }}.} \end{array} \right\} \end{aligned}$$
(2.1)

Let \(z\in \mathrm{ext}\Gamma _{1}.\) Using (1.2) and (2.1) we have

$$\begin{aligned}&\sum \limits _{k=0}^{n}a_{k}\Phi _{k,p}(z) \nonumber \\&\quad ={ \sum \limits _{k=0}^{n}a_{k}\left[ \phi \left( z\right) \right] ^{k}(\phi ^{\prime }(z))^{\frac{1}{p-\varepsilon }}}+\frac{1}{2\pi i} \int \limits _{\Gamma _{1}}\frac{(\phi ^{\prime }(\zeta ))^{\frac{1}{ p-\varepsilon }}\sum \nolimits _{k=0}^{n}a_{k}\left[ \phi \left( \zeta \right) \right] ^{k}}{\zeta -z}\mathrm{d}\zeta \nonumber \\&\quad ={ \sum \limits _{k=0}^{n}a_{k}\left[ \phi \left( z\right) \right] ^{k}(\phi ^{\prime }(z))^{\frac{1}{p-\varepsilon }}} \nonumber \\&\qquad { +\frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{(\phi ^{\prime }(\zeta ))^{\frac{1}{p-\varepsilon }}\sum \nolimits _{k=0}^{n}a_{k}\left[ \phi \left( \zeta \right) \right] ^{k}-f_{0}^{+}\left[ \phi \left( \zeta \right) \right] }{\zeta -z}\mathrm{d}\zeta } \nonumber \\&\qquad +{ \frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{f\left( \zeta \right) }{\zeta -z}\mathrm{d}\zeta -f_{0}^{-}\left[ \phi \left( z\right) \right] } (\phi ^{\prime }(z))^{\frac{1}{p}}~.~ \end{aligned}$$
(2.2)

For \(z\in \mathrm{ext}\Gamma _{2}\), the relations(1.4) and (2.1) imply that

$$\begin{aligned}&{\sum \limits _{k=1}^{n}b_{k}F_{k}\left( \frac{1}{z}\right) } \nonumber \\&\quad =-\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{(\phi _{1}^{\prime }(\xi ))^{\frac{1}{p-\varepsilon }}\phi _{1}(\xi )^{-\frac{2}{ p-\varepsilon }}\sum \nolimits _{k=1}^{n}b_{k}\left[ \phi _{1}\left( \xi \right) \right] ^{k}}{\xi -z}\mathrm{d}\xi \nonumber \\&\qquad -\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{\sum \nolimits _{k=0}^{n}b_{k} \left[ \phi _{1}\left( \xi \right) \right] ^{k}}{\xi -z}\mathrm{d}\xi \nonumber \\&\quad =\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{((\phi _{1}(\xi ))^{-\frac{2}{p-\varepsilon }}(\phi _{1}^{\prime }(\xi ))^{\frac{1}{p-\varepsilon }} \left[ f_{1}^{+}(\phi _{1}\left( \xi \right) ) -\sum \nolimits _{k=0}^{n}b_{k}\left[ \phi _{1}\left( \xi \right) \right] ^{k} \right] }{\xi -z}\mathrm{d}\xi \nonumber \\&\qquad {-\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{f\left( \xi \right) }{\xi -z}\mathrm{d}\xi .} \end{aligned}$$
(2.3)

For \(z\in \mathrm{ext}\Gamma _{1}\), by virtue (2.2), (2.3) we obtain

$$\begin{aligned}&\sum \limits _{k=0}^{n}a_{k}\left[ \Phi _{k}\left( z\right) \right] ^{k}+\sum \limits _{k=1}^{n}a_{k}F_{k}\left( \frac{1}{z}\right) \\&\quad =\sum \limits _{k=0}^{n}a_{k}\left[ \phi \left( z\right) \right] ^{k}(\phi ^{\prime }(z))^{\frac{1}{p-\varepsilon }}+\frac{1}{2\pi i} \int \limits _{\Gamma _{1}}\frac{(\phi ^{\prime }(\zeta ))^{\frac{1}{ p-\varepsilon }}\sum \nolimits _{k=0}^{n}a_{k}\left[ \phi \left( \zeta \right) \right] ^{k}-f_{0}^{+}\left[ \phi \left( \zeta \right) \right] }{\zeta -z}\mathrm{d}\zeta \\&\qquad \ { -f_{0}^{-}\left[ \phi \left( z\right) \right] +\frac{1}{2\pi i} \int \limits _{\Gamma _{2}}\frac{((\phi _{1}(\xi ))^{-\frac{2}{p-\varepsilon } }(\phi _{1}^{\prime }(\xi ))^{\frac{1}{p-\varepsilon }}\left[ f_{1}^{+}(\phi _{1}\left( \xi \right) )-\sum \nolimits _{k=0}^{n}b_{k}\left[ \phi _{1}\left( \xi \right) \right] ^{k}\right] }{\xi -z}\mathrm{d}\xi .} \end{aligned}$$

Taking limit as \(z\rightarrow z^{*}\in \Gamma _{1}\) along all non-tangential paths outside \(\Gamma _{1}\), it appears that

$$\begin{aligned}&{ f\left( z^{*}\right) -\sum \limits _{k=0}^{n}a_{k}\Phi _{k}\left( z^{*}\right) -\sum \limits _{k=1}^{n}b_{k}F_{k}\left( \frac{1}{ z^{*}}\right) } \nonumber \\&\quad =f_{0}^{+}\left[ \phi \left( z^{*}\right) \right] -\sum \limits _{k=0}^{n}a_{k}\left[ \phi \left( z^{*}\right) \right] ^{k}(\phi ^{\prime }(z^{*}))^{\frac{1}{p-\varepsilon }} \nonumber \\&\qquad +\frac{1}{2}(\phi ^{\prime }(z^{*}))^{\frac{1}{p-\varepsilon }}\left( f_{0}^{+}\left[ \phi \left( z^{*}\right) \right] -\sum \limits _{k=0}^{n}a_{k}\left[ \phi \left( z^{*}\right) \right] ^{k}\right) \nonumber \\&\qquad +S_{\Gamma _{1}}\left[ (\phi ^{\prime })^{\frac{1}{p-\varepsilon } }(f_{0}^{+}\circ \phi -\sum \limits _{k=0}^{n}a_{k}\phi ^{k})\right] (z^{*}) \nonumber \\&\qquad { -\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{f_{1}^{+}\left[ \phi _{1}\left( \xi \right) \right] -\sum \nolimits _{k=1}^{n}b_{k}\left[ \phi _{1}\left( \xi \right) \right] ^{k}}{\xi -z^{*}}\mathrm{d}\xi } \end{aligned}$$
(2.4)

a.e. on \(\Gamma _{1} \).

Now using (2.4), Minkowski’s inequality and the boundedness of \(S_{\Gamma _{1}}\) in \({ L}^{p),\theta }(\Gamma _{1},\omega )~\ \) [33] we get

$$\begin{aligned}&\left\| \!f\!-\!R_{n}\!\!\left( f,z\right) \!\right\| _{L^{p),\theta }\!\left( \Gamma _{1}\!,~\omega \right) }\! \nonumber \\&\quad \le c_{4}\!\left\| \!f_{0}^{+}\!\left( \omega \right) \!-\!\!\sum \limits _{k=0}^{n}a_{k}\!\omega ^{k}\!\right\| _{L^{p),\theta }\!\left( \!\mathbb {T}\text {,}\omega _{0}\right) }\!\!\!+\!c_{5}\!\left\| \!f_{1}^{+}\!\left( \!w\!\right) \!-\!\!\sum \limits _{k=0}^{n}b_{k}\!\!\omega ^{k}\!\right\| _{L^{p),\theta }\left( \mathbb {T}\text {,}\omega _{1}\right) }. \end{aligned}$$
(2.5)

That is, the Faber–Laurent coefficients \(a_{k}\) and \(b_{k}\) of the function f are the Taylor coefficients of the functions \( f_{0}^{+}\) and \(f_{1}^{+}\), respectively. Then by (2.5), Lemma 1 and [23] we obtain

$$\begin{aligned} \left\| f-R_{n}\left( .,f\right) \right\| _{L^{p}),\theta }\left( \Gamma _{1},~\omega \right) \le c_{6}(p)\left\{ \Omega _{p),~\theta ,\omega _{0}}^{r}\left( f_{0},{1/n}\right) +\Omega _{p),~\theta ,\omega _{1}}^{r}\left( f_{1},{1/n}\right) \right\} . \end{aligned}$$

Let \(\,z\in \mathrm{int}\Gamma _{2}.\) Then from (1.3) and (2.1) we have

$$\begin{aligned}&\sum \limits _{k=1}^{n}b_{k}F_{k,p}\left( \frac{1}{z}\right) \nonumber \\&\quad ={(\phi _{1}^{\prime }(z))^{\frac{1}{p-\varepsilon }}(\phi _{1}(z))^{-\frac{2}{p-\varepsilon }}\sum \nolimits _{k=1}^{n}b_{k}[\phi _{1}\left( z\right) ^{k}} \nonumber \\&\qquad { -\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{(\phi _{1}^{\prime }(\zeta ))^{\frac{1}{p-\varepsilon }}(\phi _{1}(\zeta ))^{-\frac{2}{ p-\varepsilon }}\sum \nolimits _{k=1}^{n}b_{k}\left[ \phi _{1}\left( \xi \right) \right] ^{k}}{\xi -z}\mathrm{d}\xi } \nonumber \\&\quad ={ (\phi _{1}^{\prime }(z))^{\frac{1}{p-\varepsilon }}(\phi _{1}(z))^{-\frac{2}{p-\varepsilon }}\sum \nolimits _{k=1}^{n}b_{k}[\phi _{1}\left( z\right) ]^{k}} \nonumber \\&\qquad {-\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{(\phi _{1}^{\prime }(\zeta ))^{\frac{1}{p-\varepsilon }}(\phi _{1}(\zeta ))^{-\frac{2}{ p-\varepsilon }}\left( \sum \nolimits _{k=1}^{n}b_{k}\left[ \phi _{1}\left( \xi \right) \right] ^{k}-f_{1}^{+}\left[ \phi _{1}\left( \xi \right) \right] \right) }{\xi -z}\mathrm{d}\xi } \nonumber \\&\qquad { -\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{f\left( \xi \right) }{\xi -z}\mathrm{d}\xi -f_{1}^{-}\left[ \phi _{1}\left( z\right) \right] } (\phi _{1}^{\prime }(z))^{\frac{1}{p-\varepsilon }}(\phi _{1}(z))^{-\frac{2}{ p-\varepsilon }} \end{aligned}$$
(2.6)

For \(z\in \mathrm{int}\Gamma _{1},\) using (1.1) and (2.1) we obtain

$$\begin{aligned}&\sum \limits _{k=1}^{n}a_{k}\Phi _{k}\left( z\right) \nonumber \\&\quad =\frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{(\phi ^{\prime }(\zeta ))^{ \frac{1}{p}}\sum \nolimits _{k=1}^{n}a_{k}\left[ \phi \left( \zeta \right) \right] ^{k}}{\zeta -z}\mathrm{d}\zeta . \nonumber \\&\quad =\frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{(\phi ^{\prime }(\zeta ))^{ \frac{1}{p}}\left( \sum \nolimits _{k=1}^{n}a_{k}\left[ \phi \left( \zeta \right) \right] ^{k}-f_{0}^{+}\left[ \phi \left( \zeta \right) \right] \right) }{\zeta -z}\mathrm{d}\zeta \nonumber \\&\qquad { +\frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{f\left( \zeta \right) }{\zeta -z}\mathrm{d}\zeta .} \end{aligned}$$
(2.7)

Now, by virtue of (2.6) and (2.7) for \(z\in \mathrm{int}\Gamma _{2}\) , we conclude that

$$\begin{aligned}&\sum \limits _{k=0}^{n}a_{k}\Phi _{k}\left( z\right) \\&\qquad +\sum \limits _{k=1}^{n}b_{k}F_{k}\left( \frac{1}{z}\right) \\&\quad ={ \frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{(\phi ^{\prime }(\zeta ))^{\frac{1}{p-\varepsilon }}\left( \sum \nolimits _{k=0}^{n}a_{k}\left[ \phi \left( \zeta \right) \right] ^{k}-f_{0}^{+}\left[ \phi \left( \zeta \right) \right] \right) }{\zeta -z}\mathrm{d}\zeta } \\&\qquad +{ (\phi _{1}^{\prime }(z))^{\frac{1}{p-\varepsilon }}(\phi _{1}(z))^{-\frac{2}{p-\varepsilon }}\sum \nolimits _{k=1}^{n}b_{k}[\phi _{1}\left( z\right) ]^{k}\ } \\&\qquad \ { -\frac{1}{2\pi i}\int \limits _{\Gamma _{2}}\frac{(\phi _{1}^{\prime }(\zeta ))^{\frac{1}{p-\varepsilon }}(\phi _{1}(\zeta ))^{- \frac{2}{p-\varepsilon }}\left( \sum \nolimits _{k=1}^{n}b_{k}\left[ \phi _{1}\left( \xi \right) \right] ^{k}-f_{1}^{+}\left[ \phi _{1}\left( \xi \right) \right] \right) }{\xi -z}\mathrm{d}\xi } \\&\qquad { -f_{1}^{-}\left[ \phi _{1}\left( z\right) \right] }(\phi _{1}^{\prime }(z))^{\frac{1}{p-\varepsilon }}(\phi _{1}(z))^{-\frac{2}{ p-\varepsilon }} \end{aligned}$$

Taking the limit as \(z\rightarrow z^{*}\in \Gamma _{2}\) along all non-tangential paths inside \(\Gamma _{2}\), we reach

$$\begin{aligned}&{f\left( z^{*}\right) -\!\!\sum \limits _{k=0}^{n}a_{k}\Phi _{k,p}\left( z^{*}\right) -\!\!\sum \limits _{k=1}^{n}b_{k}F_{k,p}\left( \frac{1}{z^{*}}\right) } \nonumber \\&\quad =f_{1}^{+}\left[ \phi _{1}\left( z^{*}\right) \right] -\frac{1}{2} (\phi _{1}^{\prime }(z^{*}))^{\frac{1}{p-\varepsilon }}(\phi _{1}(z^{*}))^{-\frac{2}{p-\varepsilon }}\left[ \sum \limits _{k=1}^{n}b_{k} \left[ \phi _{1}\left( z^{*}\right) \right] ^{k}-f_{1}^{+}\left[ \phi _{1}\left( z^{*}\right) \right] \right] \nonumber \\&\qquad -S_{\Gamma _{2}}\left[ (\phi _{1}^{\prime })^{\frac{1}{p}}(\phi _{1})^{- \frac{2}{p-\varepsilon }}\left( \sum \nolimits _{k=1}^{n}b_{k}\phi _{1}^{k}-\left( f_{1}^{+}\circ \phi _{1}\right) \right) \right] (z^{*}) \nonumber \\&\qquad { -\frac{1}{2\pi i}\int \limits _{\Gamma _{1}}\frac{(\phi ^{\prime }(\zeta ))^{\frac{1}{p-\varepsilon }}\left( \sum \nolimits _{k=0}^{n}a_{k}\left[ \phi \left( \zeta \right) \right] ^{k}-f_{0}^{+}\left[ \phi \left( \zeta \right) \right] \right) }{\zeta -z^{*}}\mathrm{d}\zeta } \end{aligned}$$
(2.8)

a.e. on \(\Gamma _{2} \).

Using (2.8), Minkowski’s inequality and the boundedness of \(S_{\Gamma _{2}}\) in \({ L}^{p),\theta }(\Gamma _{2},\omega )\) [33] we get

$$\begin{aligned}&\left\| \!f\!-\!\!\,R_{n}\!\!\left( f,\!z\right) \!\right\| _{L^{p),\theta }\left( \Gamma _{2}\!,\omega \right) }\!\! \nonumber \\&\quad \le _{7}\!\!\left\| \!f_{1}^{+}\!\!\left( w\!\right) \!-\!\!\!\sum \limits _{k=1}^{n}\!\!b_{k}\!w^{k}\!\right\| _{L^{p),\theta }\left( \mathbb {T}\text {,}\omega _{1}\right) }\!\!\!+\!c_{8}\!\left\| \!f_{0}^{+}\!\!\left( \!w\!\right) \!-\!\!\!\sum \limits _{k=0}^{n}\!\!a_{k}\!w^{k}\!\right\| _{L^{p),\theta }\!\left( \mathbb {T}\text {,}\omega _{0}\right) }. \end{aligned}$$
(2.9)

Use of (2.9), Lemma 1.1 and [23] leads to

$$\begin{aligned} \left\| f-R_{n}\left( .,f\right) \right\| _{L^{p),\theta }\left( \Gamma ,\omega \right) }\le & {} c_{9}\left\{ \Omega _{p),\theta \omega _{1}}^{r}\left( f_{1},{1/n}\right) +\Omega _{p),\theta ,\omega _{0}}^{r}\left( f_{0},{1/n}\right) \right\} . \end{aligned}$$

The proof is complete. \(\square \)