Unified Framework for Continuous and Discrete Relations of Gehring and Muckenhoupt Weights on Time Scales

Abstract

This article contains some relations, which include some embedding and transition properties, between the Muckenhoupt classes ${\mathcal{M}}_{\gamma };$$\gamma >1$ and the Gehring classes ${\mathcal{G}}_{\delta };$$\delta >1$ of bi-Sobolev weights on a time scale $\mathbb{T}$. In addition, we establish the relations between Muckenhoupt and Gehring classes, where we define a new time scale $\tilde{\mathbb{T}}=v\left(\mathbb{T}\right)$, to indicate that if the $\tilde{\Delta }$ derivative of the inverse of a bi-Sobolev weight belongs to the Gehring class, then the $\Delta$ derivative of a bi-Sobolev weight on a time scale $\mathbb{T}$ belongs to the Muckenhoupt class. Furthermore, our results, which will be established by a newly developed technique, show that the study of the properties in the continuous and discrete classes of weights can be unified. As special cases of our results, when $\mathbb{T}=\mathbb{R}$, one can obtain classical continuous results, and when $\mathbb{T}=\mathbb{N}$, one can obtain discrete results which are new and interesting for the reader.

1. Introduction

We choose an interval ${\mathcal{I}}_0\subset {\mathbb{R}}_{+}=[0,\infty )$ and consider its subintervals $\mathcal{I}=[c,d],$ where $c<d<\infty .$$\left|\mathcal{I}\right|$ denotes the Lebesgue measure of $\mathcal{I}.$ A weight $\varpi \ge 0$ is defined as a locally integrable function on a bounded interval $\mathcal{I}\subset {\mathcal{I}}_0$ to $[0,+\infty )$. For an exponent $\gamma >1,$ a weight $\varpi$ meets the ${\mathcal{M}}_{\gamma }$ condition and belongs to the Muckenhoupt classes ${\mathcal{M}}_{\gamma }$ (see [1]). If there is a constant $\mathcal{Q}>1$ subject to $\forall \mathcal{I}\subseteq {\mathcal{I}}_0$, then

$$\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}\varpi \left(\vartheta \right)d\vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}{\varpi }^{-{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)d\vartheta \right)}^{\gamma -1}\le \mathcal{Q}.$$

(1)

The ${\mathcal{M}}_{\gamma }$ constant of $\varpi$ is defined as

$${\mathcal{M}}_{\gamma }\left(\varpi \right)=\underset{\mathcal{I}\subset {\mathcal{I}}_0}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}\varpi \left(\vartheta \right)d\vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)d\vartheta \right)}^{\gamma -1}.$$

When $\gamma =1,$ the condition (1) simplifies to

$${\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}\varpi \left(\vartheta \right)d\vartheta \le \mathcal{Q}.ess\underset{\mathcal{I}\subset {\mathcal{I}}_0}{inf}\varpi \left(\vartheta \right),\forall \vartheta \in \mathcal{I},$$

In this case, the weight $\varpi$ satisfies the ${\mathcal{M}}_1$ condition with a constant

$${\mathcal{M}}_1\left(\varpi \right)=\underset{\mathcal{I}\subset {\mathcal{I}}_0}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}\varpi \left(\vartheta \right)d\vartheta \right){\left(ess\underset{\mathcal{I}\subset {\mathcal{I}}_0}{inf}\varpi \right)}^{-1}.$$

In [2], Gehring proposed a novel class of weights, denoted as ${\mathcal{G}}_{\delta };$$\delta >1,$ designed for reverse Hölder’s inequalities. A function $\varphi$ is classified within the Gehring class ${\mathcal{G}}_{\delta }$ with constant $\mathcal{K}$ if, for $\delta >1$, the following condition is met:

$$\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}{\varphi }^{\delta }\left(\vartheta \right)d\vartheta \right)\le \mathcal{K}{\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}\varphi \left(\vartheta \right)d\vartheta \right)}^{\delta },\forall \mathcal{I}\subset {\mathcal{I}}_0.$$

The ${\mathcal{G}}_{\delta }$ constant of $\varphi$ is defined as

$${\mathcal{G}}_{\delta }\left(\varphi \right)=\underset{\mathcal{I}\subset {\mathcal{I}}_0}{sup}{\left[{\displaystyle \frac{\left|\mathcal{I}\right|}{{\int }_{\mathcal{I}}\varphi \left(s\right)ds}}{\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}{\varphi }^{\delta }\left(s\right)ds\right)}^{{\displaystyle \frac{1}{\delta }}}\right]}^{{\delta }^{\ast }},$$

where ${\delta }^{\ast }=\delta /(\delta -1).$

Muckenhoupt and Gehring, in their seminal papers, demonstrated that the classes ${\mathcal{M}}_{\gamma }$ and ${\mathcal{G}}_{\delta }$ both exhibit a remarkable self-improving property:

• Muckenhoupt’s finding regarding this characteristic indicates that if $\varpi \in {\mathcal{M}}_{\delta }\left(\mathcal{Q}\right)\Rightarrow \exists$$ϵ>0$ subject to $\varpi \in {\mathcal{M}}_{\delta -ϵ}\left({\mathcal{Q}}_1\right),$ then ${\mathcal{M}}_{\delta }\left(\mathcal{Q}\right)\subset {\mathcal{M}}_{\delta -ϵ}\left({\mathcal{Q}}_1\right).$
• Gehring’s finding for the same characteristic indicates that if $\varphi \in {\mathcal{G}}_{\delta }\left(\mathcal{K}\right)\Rightarrow \exists$$ϵ>0$ subject to $\varphi \in {\mathcal{G}}_{\delta +ϵ}\left({\mathcal{K}}_1\right),$ then ${\mathcal{G}}_{\delta }\left(\mathcal{K}\right)\subset {\mathcal{G}}_{\delta +ϵ}\left({\mathcal{K}}_1\right).$

In other words, for given pairs of constants $\gamma ,\mathcal{Q}$ and $\delta ,\mathcal{K}$ as described above, there exist limit exponents ${\gamma }^{\ast }={\gamma }^{\ast }(\gamma ,\mathcal{Q})<\gamma$ and ${\delta }^{\ast }={\delta }^{\ast }(\delta ,\mathcal{K})>\delta$ and corresponding constants ${\mathcal{Q}}_l={\mathcal{Q}}_l(\gamma ,\mathcal{Q})$ and ${\mathcal{K}}_t={\mathcal{K}}_t(\delta ,\mathcal{K})$ such that the following conditions are satisfied:

$$\varpi \in {\mathcal{M}}_{\gamma }\left(\mathcal{Q}\right)\Rightarrow \varpi \in {\mathcal{M}}_l\left({\mathcal{Q}}_l\right),\forall l\in ({\gamma }^{\ast },\gamma ],$$

and

$$\varphi \in {\mathcal{G}}_{\delta }\left(\mathcal{K}\right)\Rightarrow \varphi \in {\mathcal{G}}_t\left({\mathcal{K}}_t\right),\forall t\in [\delta ,{\delta }^{\ast }).$$

In [3], the authors explored the relationship between the classes ${\mathcal{M}}_{\gamma }$ and ${\mathcal{G}}_{\delta }$, demonstrating that every Muckenhoupt weight is contained in a Gehring class and vice versa. Specifically, they established transition properties between these classes by proving the following:

• For any $\varpi \in {\mathcal{M}}_{\gamma }\left(\mathcal{Q}\right),$ there exists an index $t^{\ast }=t^{\ast }(\gamma ,\mathcal{Q})$ and corresponding constant ${\mathcal{K}}_t={\mathcal{K}}_t(\gamma ,\mathcal{Q})$ subject to $\varpi \in {\mathcal{G}}_t\left({\mathcal{K}}_t\right),$$\forall t<t^{\ast }.$
• For any $\varphi \in {\mathcal{G}}_{\delta }\left(\mathcal{K}\right),$ there exists an index $l^{\ast }=l^{\ast }(\delta ,\mathcal{K})$ and corresponding constant ${\mathcal{Q}}_l={\mathcal{Q}}_l(\delta ,\mathcal{K})$ subject to $\varphi \in {\mathcal{M}}_l\left({\mathcal{Q}}_l\right),$$\forall l>l^{\ast }.$

It is important to highlight that the self-improving properties of ${\mathcal{M}}_{\gamma }$ and ${\mathcal{G}}_{\delta }$ have valuable applications beyond the investigation of higher integrability and gradient integrability for quasi-conformal mappings. These properties are also pivotal in analyzing the optimal regularity of solutions to certain elliptic partial differential equations (see [4,5]). The characteristics of Muckenhoupt and Gehring classes have been thoroughly studied, particularly in one dimension, with extensive focus on the following aspects:

(I)

Determining the limit exponents ${\gamma }^{\ast }$ and ${\delta }^{\ast }$ that satisfy the self-improving property;

(II)

Identifying the limit exponents $l^{\ast }$ and $t^{\ast }$ that satisfy the transition property;

(III)

Finding the optimal constants ${\mathcal{Q}}_l$ and ${\mathcal{K}}_t$ that satisfy both the ${\mathcal{G}}_t$ condition and the enhanced ${\mathcal{M}}_l$ condition.

D’Apuzzo and Sbordone [6] made a significant breakthrough in addressing problem (I) by determining the optimal integrability exponent for monotonic ${\mathcal{G}}_{\delta }$ weights.

Korenovskii [7] later provided a complete solution to problem (I) by establishing the corresponding self-improving properties for ${\mathcal{M}}_{\gamma }$ and ${\mathcal{G}}_{\delta }$ weights. His work represented a substantial advancement, eliminating the need for monotonicity assumptions. Korenovskii employed classical Hardy-type inequalities and integral estimates to achieve this, deriving optimal exponents from solutions of algebraic equations.

Problem (II) has been completely solved by Malaksiano (see [8,9]), who determined the precise exponents for the transition between ${\mathcal{M}}_{\gamma }$ and ${\mathcal{G}}_{\delta }$ classes and vice versa.

Regarding problem (III), Vasyunin [10] and Dindoš-Wall [11] determined the precise constants for both the self-improving and transition characteristics of ${\mathcal{M}}_{\gamma }$ and ${\mathcal{G}}_{\delta }$ weights. They utilized Bellman functions, a powerful tool for optimization problems, albeit involving greater technical complexity compared to traditional methods.

In [12], Johnson and Neugebauer established some new transition properties between ${\mathcal{M}}_{\gamma }$ and ${\mathcal{G}}_{\delta }$ classes which partially give a solution of the problem (II) and give the relation between them for a certain type of weights called bi-Sobolev maps (will be defined later). In particular, they demonstrated that, for an increasing homeomorphism $w:\mathbb{R}\to \mathbb{R},$ where $w,w^{-1}$ are locally absolutely continuous and differentiable, then

$$w^{\prime }\in {\mathcal{M}}_{\gamma }\iff {\left(w^{-1}\right)}^{\prime }\in {\mathcal{G}}_{\delta },\mathrm{provided}\mathrm{that}\gamma >1\mathrm{and}{\displaystyle \frac{1}{\gamma }}+{\displaystyle \frac{1}{\delta }}=1.$$

(2)

Here, $w^{\prime }$ symbolises the derivative of $w,$ and $w^{-1}$ represents the inverse function. Moreover, they established that the constants ${\mathcal{M}}_{\gamma }\left(w^{\prime }\right)$ and ${\mathcal{G}}_{\delta }\left({\left(w^{-1}\right)}^{\prime }\right)$ are equal.

For the class ${\mathcal{M}}_1$, problems (II) and (III) were resolved by Bojarsky et al. in [13]. They found both the sharp integrability exponents and the precise constants in explicit forms. Their approach and results were extended to the class ${\mathcal{G}}_{\infty }$ by Basile et al. [14].

In [15], Corporente gave the relation between ${\mathcal{M}}_{\infty }$ and ${\mathcal{G}}_1$ for bi-Sobolev weighted functions and proved that

$$w^{\prime }\in {\mathcal{M}}_{\infty }\iff {\left(w^{-1}\right)}^{\prime }\in {\mathcal{G}}_1,$$

(3)

and ${\mathcal{M}}_{\infty }\left(w^{\prime }\right),$${\mathcal{G}}_1\left({\left(w^{-1}\right)}^{\prime }\right)$ are equal where

$${\mathcal{M}}_{\infty }\left(\varpi \right)=\underset{\mathcal{I}\subset {\mathcal{I}}_0}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}\varpi \left(\vartheta \right)d\vartheta \right)exp\left({\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}log{\displaystyle \frac{1}{\varpi \left(\vartheta \right)}}d\vartheta \right)<\infty ,$$

and

$${\mathcal{G}}_1\left(\varphi \right)=\underset{\mathcal{I}\subset {\mathcal{I}}_0}{sup}exp\left({\displaystyle \frac{\left|\mathcal{I}\right|}{{\int }_{\mathcal{I}}\varphi \left(\vartheta \right)d\vartheta }}{\int }_{\mathcal{I}}{\displaystyle \frac{\varphi \left(\vartheta \right)}{{\varphi }_{\mathcal{I}}}}log{\displaystyle \frac{\varphi \left(\vartheta \right)}{{\varphi }_{\mathcal{I}}}}d\vartheta \right)<\infty ,{\varphi }_{\mathcal{I}}={\displaystyle \frac{1}{\left|\mathcal{I}\right|}}{\int }_{\mathcal{I}}\varphi \left(\vartheta \right)d\vartheta .$$

In [16], Sbordone proved some results on continuation and reciprocal embedding for Muckenhoupt class ${\mathcal{M}}_{\gamma }$ and gave relations between the constants of the classes. Specifically, he demonstrated that if $\varpi$ is a positive weight function, then

$${\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1}\le {\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right),\mathrm{if}\gamma >1$$

(4)

and

$${\mathcal{M}}_{\gamma }\left(\varpi \right)\le {\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1},\mathrm{if}1<\gamma \le 2.$$

(5)

In [17], Neugebauer proved some factorization properties of the ${\mathcal{M}}_{\gamma }$ class with two different weighted functions and established that if $\gamma >1,$$u,$$\varphi$ are positive and $\left(u,\varphi \right)\in {\mathcal{M}}_{\gamma },$ then

$$\left({\varphi }^{1-{\gamma }^{\ast }},u^{1-{\gamma }^{\ast }}\right)\in {\mathcal{M}}_{{\gamma }^{\ast }};{\gamma }^{\ast }=\gamma /(\gamma -1).$$

(6)

Here, we remember that the pair $\left(u,\varphi \right)\in {\mathcal{M}}_{\gamma }$ with $\gamma >1$ if and only if there exists a constant $\mathcal{Q}>1$ subject to

$$\left({\int }_{\mathcal{I}}u\left(\vartheta \right)d\vartheta \right){\left({\int }_{\mathcal{I}}{\left[\varphi \left(\vartheta \right)\right]}^{1-{\gamma }^{\ast }}d\vartheta \right)}^{\gamma -1}\le \mathcal{Q}{\left|\mathcal{I}\right|}^{\gamma }.$$

Also in [17], it was showed that if $\left(u,\varphi \right)\in {\mathcal{M}}_{\gamma },$ then

$$\left(u^{{\displaystyle \frac{\lambda -1}{\gamma -1}}},{\varphi }^{{\displaystyle \frac{\lambda -1}{\gamma -1}}}\right)\in {\mathcal{M}}_{\delta };1<\lambda <\gamma <\infty .$$

(7)

In [18], the authors demonstrated that any weight $\varphi$ in the Muckenhoupt class ${\mathcal{M}}_{\infty }$ can be represented as $\varphi =w^{\prime },$ where $w:\mathbb{R}\to \mathbb{R}$ is a bi-Sobolev map. They utilized these results to improve existing findings on exact continuation and reciprocal embedding for ${\mathcal{G}}_{\delta }$ and ${\mathcal{M}}_{\gamma }$ classes, offering precise bounds in each case. Their results give a partial solution of the problem (III) for some special cases and the method that was applied relied on a duality formula ${\mathcal{M}}_{\gamma }\left({\left(w^{-1}\right)}^{\prime }\right)={\mathcal{G}}_{\delta }\left(w^{\prime }\right)$ due to Johnson and Neugebauer.

In recent years, there has been growing interest among researchers in studying the regularity and boundedness of discrete operators on $l^{\gamma }\left(\mathbb{N}\right)$, analogous to $L^{\gamma }\left(\mathbb{R}\right)$ regularity and boundedness. This trend can be observed in works such as [19,20,21] and the references in which they are cited. The motivation behind this surge in interest stems partly from the observation that discrete operators can exhibit behaviors distinct from their continuous counterparts, as demonstrated by examples like the discrete spherical maximal operator [22]. In the given context, the following apply:

• ${\mathbb{Z}}_{+}==\{0,1,2,\dots \}$ represents the set of nonnegative integers.
• $\mathbb{I}=\{b,b+1,b+2,\dots ,N\}\subseteq {\mathbb{Z}}_{+}$.
• For $1\le \gamma <\infty$, the spaces $l^{\gamma }\left(\mathbb{N}\right)$ consists of all sequences $\xi =\left\{\xi \right(j\left)\right\}\in \mathbb{R}$ such that

  $${∥\xi ∥}_{l^{\gamma }\left(\mathbb{N}\right)}:={\left(\sum _{\mathcal{\Im }=1}^{\infty }{\left|\xi (\mathcal{\Im })\right|}^{\gamma }\right)}^{1/\gamma }<\infty .$$
• The $l^{\infty }\left(\mathbb{N}\right)$ norm is defined as ${∥\xi ∥}_{l^{\infty }\left(\mathbb{N}\right)}={sup}_{\mathcal{\Im }\in \mathbb{N}}\left|\xi (\mathcal{\Im })\right|.$
• The difference operator $\Delta \xi (\mathcal{\Im })$ is defined by $\Delta \xi (\mathcal{\Im })=\xi (\mathcal{\Im }+1)-\xi (\mathcal{\Im }),$ and the higher difference ${\Delta }^k\xi (\mathcal{\Im })$ for $k=1,2,\dots ,$ is recursively defined as ${\Delta }^k\xi (\mathcal{\Im })=\Delta \left({\Delta }^{k-1}\xi (\mathcal{\Im })\right)$.
• The discrete Sobolev space $W^{k,\gamma }\left(\mathbb{N}\right),$ corresponding to the Sobolev space of functions, is defined as the set of discrete sequences with a finite $W^{k,\gamma }\left(\mathbb{N}\right)$ norm:

  $${∥\xi ∥}_{W^{k,\gamma }\left(\mathbb{N}\right)}:=\sum _{\mathcal{\Im }=0}^k{∥{\nabla }^{\mathcal{\Im }}\xi ∥}_{l^{\gamma }\left(\mathbb{N}\right)}<\infty .$$

According to the triangle inequality, for $k\ge 1,$ we have ${∥{\nabla }^k\xi ∥}_{l^{\gamma }\left(\mathbb{N}\right)}\le 2^k{∥\xi ∥}_{l^{\gamma }\left(\mathbb{N}\right)}.$ Therefore, any $l^{\gamma }$ bound automatically provides a $W^{k,\gamma }\left(\mathbb{N}\right)$ bound for $k\ge 1.$ This observation establishes that the discrete $W^{k,\gamma }\left(\mathbb{N}\right)$ spaces are equivalent to the classical $l^{\gamma }$ spaces with an equivalent norm, as previously noted in the literature (see [23] or [24]). For the recent developments in the relevant field, see the papers [25,26,27,28,29,30].

To provide a comprehensive overview, we offer some discrete versions of properties related to the discrete Muckenhoupt and Gehring classes. A discrete weight on ${\mathbb{Z}}_{+}$ is defined as a sequence $\varpi ={\left\{\varpi (\mathcal{\Im })\right\}}_{\mathcal{\Im }=0}^{\infty }$ of positive real values. We designate by $l^{\gamma }\left(\varpi \right)=l^{\gamma }({\mathbb{Z}}_{+},\varpi )$ for $1<\gamma <\infty$ the Banach space of all real-valued sequence $\vartheta ={\left\{\vartheta (\mathcal{\Im })\right\}}_{\mathcal{\Im }=0}^{\infty }$ subject to

$${∥\vartheta ∥}_{l^{\gamma }\left(\varpi \right)}:={\left(\sum _{\mathcal{\Im }=1}^{\infty }{\varpi }^{\gamma }(\mathcal{\Im }){\left|\vartheta (\mathcal{\Im })\right|}^{\gamma }\right)}^{1/\gamma }<\infty .$$

We define $\varpi \in {\mathcal{M}}_{\gamma }$ as the Muckenhoupt weight for $l^{\gamma }\left(\varpi \right)$ if a constant $\mathcal{Q}<\infty$ exists subject to

$${\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}{\left(\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^{\gamma }\left(k\right)\right)}^{1/\gamma }{\left(\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^{-\delta }\left(k\right)\right)}^{1/\delta }\le \mathcal{Q},\mathrm{with}0\le \epsilon \le \mathcal{\Im }\mathrm{and}1/\gamma +1/\delta =1.$$

Hunt, Muckenhoupt and Wheeden [31] showed that the matrix

$$S_{+}=\left(\begin{array}{ccccc}0& -1& -1/2& -1/3& \dots \\ 1& 0& -1& -1/2& \dots \\ 1/2& 1& 0& -1& \dots \\ 1/3& 1/2& 1& 0& \dots \\ \dots & \dots & \dots & \dots & \dots \end{array}\right)$$

generates a bounded operator on $l^{\gamma }\left(\varpi \right)\iff \varpi \in {\mathcal{M}}_{\gamma }.$ Weights of the form ${\left\{{(\epsilon +1)}^{\lambda }\right\}}_{\epsilon =0}^{\infty }$ are called power weights. This power weight belongs to ${\mathcal{M}}_{\gamma }$ if and only if $-1/\gamma <\lambda <1/\gamma .$ Clearly, if $-1/\gamma <\lambda <1/\gamma$, then $-1/\gamma <\lambda r<1/\gamma$ for all r values close enough to $1.$ This observation was generalized by Böttcher and Seybold [32] to general Muckenhoupt weights. They demonstrated that if $\varpi \in {\mathcal{M}}_{\gamma }$, then there exists a $\xi ={\xi }_{\gamma ,\varpi }$ subject to ${\varpi }^r\in {\mathcal{M}}_{\gamma },$$\forall r\in (1-\xi ,1+\xi ).$ This means that

$${\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^{r\gamma }\left(k\right)\right)}^{1/\gamma r}{\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^{-r\delta }\left(k\right)\right)}^{1/\delta r}\le {\mathcal{Q}}^r,$$

and then

$${\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^s\left(k\right)\right)}^{1/s}{\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^{-s^{\ast }}\left(k\right)\right)}^{1/s^{\ast }}\le {\mathcal{Q}}^r,$$

where $1/s+1/s^{\ast }=1.$ This indicates that if $\varpi \in {\mathcal{M}}_{\gamma },$ then ∃$\xi ={\xi }_{\gamma ,\varpi }$ subject to $\varpi \in {\mathcal{M}}_s$ where $s\in \left[\gamma \right(1-\xi ),\gamma (1+\xi \left)\right).$ In other words, Böttcher and Seybold’s finding for the self-improving property for discrete weights asserts that if $\varpi \in {\mathcal{M}}_{\gamma }\Rightarrow \exists$$\xi >0$ subject to $\varpi \in {\mathcal{M}}_{\gamma (1-\xi )}$, then

$${\mathcal{M}}_{\gamma }\subset {\mathcal{M}}_{\gamma (1-\xi )}.$$

(8)

A discrete nonnegative sequence $\varphi \in {\mathcal{M}}_1$ (the discrete Muckenhoupt class) for $\gamma >1$ if

$${\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}\varphi \left(k\right)\le \mathcal{Q}·\underset{\mathcal{\Im }\ge \epsilon }{inf}\varphi \left(\epsilon \right)$$

(9)

holds for every $\mathcal{Q}>1.$ We define $\varpi \in {\mathcal{G}}_{\gamma }$ as a discrete Gehring weight for $l^{\gamma }\left(\varpi \right)$ if there exists a constant ${\mathcal{K}}_{\gamma }<\infty$ subject to

$${\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}\varphi \left(k\right)\right)}^{1/\gamma }\le {\mathcal{K}}_{\gamma }{\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}\varphi \left(k\right).$$

In [32], the authors established that if $\varpi \in {\mathcal{M}}_{\gamma }\left(\mathcal{Q}\right)$, then there exists a constant $\beta >0$ and ${\mathcal{Q}}_1<\infty$ depending only on $\gamma$ and $\varpi$ subject to

$${\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^{\gamma (1+\xi )}\left(k\right)\le {\mathcal{Q}}_1{\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varpi }^{\gamma }\left(k\right)\right)}^{1+\xi },$$

(10)

$\forall \xi \in [0,\beta ]$ and all $\left|\mathcal{I}\right|=2^r$ with $r\in \mathbb{N}$. In an inclusion form, this means that if $\varpi \in {\mathcal{M}}_{\gamma }\Rightarrow \exists$$\xi >0$ subject to $\varpi \in {\mathcal{G}}_{\gamma (1+\xi )}$, then

$${\mathcal{M}}_{\gamma }\subset {\mathcal{G}}_{\gamma (1+\xi )}.$$

In [33], the authors established the discrete version of Muckenhoupt results (Lemma 4 in [1]) for higher integrability for nonincreasing sequences in a discrete Muckenhoupt class. They demonstrated that if $\varphi :\mathbb{N}\to {\mathbb{R}}^{+}$ is a nonincreasing sequence and fulfills (9) for $\mathcal{Q}>1,$ and there exists a constant $z>1$ subject to $\left(\mathcal{\Im }+1-b\right)\le z(\mathcal{\Im }-b)$for $\mathcal{\Im }\in \mathbb{I}$, then

$${\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varphi }^{\gamma }\left(k\right)\le \mathcal{N}{\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}\varphi \left(k\right)\right)}^{\gamma },\mathrm{for}\gamma \in [1,\mathcal{Q}/(\mathcal{Q}-1)),\mathcal{\Im }\ge \epsilon ,$$

(11)

where $\mathcal{N}=z^{\gamma }\mathcal{Q}/(\mathcal{Q}-\gamma (\mathcal{Q}-1))$.

Additionally, in [33], the authors demonstrated the discrete form of the self-improving property in the discrete Gehring class which can be considered as the discrete result due to Korenovskii (Theorum 2 in [7]). In particular, they showed that if $\delta >1,$${\mathcal{K}}_{\delta }>1$ and $\varphi (\mathcal{\Im })$ is a nonincreasing sequence and fulfills

$${\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varphi }^{\delta }\left(k\right)\le {\mathcal{K}}_{\delta }{\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}\varphi \left(k\right)\right)}^{\delta },$$

(12)

then there is a constant ${\mathcal{K}}_{\gamma }>1$ subject to

$${\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}{\varphi }^{\gamma }\left(k\right)\le {\mathcal{K}}_{\gamma }{\left({\displaystyle \frac{1}{\mathcal{\Im }+1-\epsilon }}\sum _{k=\epsilon }^{\mathcal{\Im }}\varphi \left(k\right)\right)}^{\gamma },\forall \gamma \in [1,{\delta }^{\ast }),$$

(13)

where ${\delta }^{\ast }$ is obtained using the equation

$${\left({\displaystyle \frac{\vartheta }{\vartheta -1}}\right)}^{-1}{\left({\displaystyle \frac{\vartheta }{\vartheta -\delta }}\right)}^{{\displaystyle \frac{1}{\vartheta }}}=z{\mathcal{K}}_{\delta }.$$

(14)

This establishes that if $\varphi \in {\mathcal{G}}_{\delta }\left(\mathcal{K}\right)$ (the discrete Gehring class), then $\varphi \in {\mathcal{G}}_s\left({\mathcal{K}}_s\right),$$\forall s\in [\delta ,{\delta }^{\ast })$.

Recently, dynamic equations and inequalities on time scales have become an important focus in both pure and applied mathematics. This field addresses the integration of discrete and continuous calculus concepts, bridging gaps between these traditionally separate domains. References such as [34,35,36,37,38,39,40,41,42] provide comprehensive coverage of these developments.

In [43], the authors presented the time scale version of Muckenhoupt’s result for higher integrability. They proved that if $\chi$ is nonincreasing and there exists a constant $\lambda >1$ subject to $\sigma \left(\upsilon \right)\le \lambda \upsilon$ and ${\mathcal{M}}^{\sigma }{\chi }^{\sigma }\le A{\chi }^{\sigma }$ and $A>1$ where

$${\mathcal{M}}^{\sigma }{\chi }^{\sigma }\left(\upsilon \right)={\displaystyle \frac{1}{\sigma \left(\upsilon \right)}}{\int }_0^{\sigma \left(\upsilon \right)}{\chi }^{\sigma }\left(s\right)\Delta s,$$

then

$${\mathcal{M}}^{\sigma }{\left({\chi }^{\sigma }\right)}^{\gamma }\le {\displaystyle \frac{\Re }{\Re -\gamma (\Re -1)}}{\left({\mathcal{M}}^{\sigma }\chi \right)}^{\gamma },$$

for $\gamma \in [1,$$\Re /(\Re -1)],$ where $\Re :=A\lambda >1$ and $\mathbb{T}$ is a time scale that can be any arbitrary closed subset of the real numbers $\mathbb{R}.$ The definition of $\sigma \left(\upsilon \right)$ and related results and terminology on $\mathbb{T}$ will be given later in the subsequent section.

In [43], the authors presented the time scale version for the self-improving property of the Gehring class of monotone functions. They demonstrated that if $\chi \ge 0$ is a nonincreasing integrable function and belongs to the time scale class $G_{\delta }^{\ast }\left({\mathcal{K}}_{\delta }\right),$ the class of nonnegative nonincreasing integrable functions which satisfies

$${\mathcal{M}}^{\sigma }{\chi }^{\delta }\left(\upsilon \right)\le {\mathcal{K}}_{\delta }{\left({\chi }^{\sigma }\left(\upsilon \right)\right)}^{\delta -1}{\mathcal{M}}^{\sigma }\chi \left(\upsilon \right),\forall \upsilon \in {(0,b]}_{\mathbb{T}},\mathrm{and}{\mathcal{K}}_{\delta }>1,$$

(15)

then ${\left({\mathcal{M}}^{\sigma }{\left({\chi }^{\sigma }\right)}^{\gamma }\right)}^{1/\gamma }\le {\mathcal{K}}_{\gamma }{\left({\mathcal{M}}^{\sigma }{\chi }^{\delta }\right)}^{1/\delta },$$\forall \gamma \in [\delta ,\delta +ϵ],$ where $ϵ=\delta /(\beta -1)$,

$${\mathcal{K}}_{\gamma }:={\left[{\displaystyle \frac{{\left(\lambda \beta \right)}^{\frac{\gamma }{\delta }+1}}{\lambda \beta -\frac{\gamma }{\delta }(\lambda \beta -1)}}\right]}^{1/\gamma };\beta ={\left[{\mathcal{Q}}^{\delta }{\left({\displaystyle \frac{\delta }{\delta -1}}\right)}^{\delta }-{\displaystyle \frac{\delta {\mathcal{Q}}^{\delta -1}}{\delta -1}}\right]}^{1/\delta }.$$

We aim to establish the relation between weights in the Muckenhoupt and Gehring classes on $\mathbb{T}$. Also, to obtain, the relation of the derivative of the bi-Sobolev map for the Muckenhoupt and Gehring classes, we must define a new time scale $\tilde{\mathbb{T}}=\nu \left(\mathbb{T}\right)$, where its derivative is $\tilde{\Delta }.$ The result on time scales is a unified version of the result in the continuous case (when $\mathbb{T}=\mathbb{R}$) and discrete case (when $\mathbb{T}=\mathbb{N}$).

Now, the obvious question to ask is the following:

Question. 

Is it feasible to demonstrate the properties (2), (3), (4) and (5) of embedding and transition between Muckenhoupt and Gehring weights on $\mathbb{T}$?

Our goal with this study is to provide a positive response to this question. The article is organized in the following manner: In Section 2, we present preliminary information on time scale theory and the fundamental lemmas that will be utilized in the proofs. In Section 3, first we offer the definitions of all the spaces that we consider on $\mathbb{T}$ and also give the definitions of the Sobolev space and the bi-Sobolev maps on $\mathbb{T}$. Next, we give the answer to the question and prove the properties (2), (3), (4) and (5) proved by Johnson and Neugebauer and Corporente on $\mathbb{T}$. As special instances, one can derive some discrete versions of the properties (2), (3), (4) and (5) of embedding and transition properties in the discrete space $l^{\gamma }\left(\varpi \right)$. To the best of the authors’ information, the discrete properties when $\mathbb{T}=\mathbb{N}$ have not been considered before and are thus essential.

2. Preliminaries and Basic Lemmas

Instead of reiterating the fundamental concepts of time scales and their notation here, we direct the reader to the monograph by Bohner and Peterson [44,45], which provides an organized summary of the theory on time scales.

In what follows, we suppose that the time scale interval $\mathcal{J}\subset \mathbb{T}$ has the form ${[c,d)}_{\mathbb{T}}=[c,d)\cap \mathbb{T}.$ We now present the main theorems on $\mathbb{T}$ that will be utilized to support our conclusions.

Theorem 1

(Hölder’s inequality [45]). Let $\chi ,\phi \in C_{rd}\left(\mathcal{J},\mathbb{R}\right),$ $ћ>1$ and $\delta =ћ/(ћ-1).$ Then,

$${\int }_{\mathcal{J}}\left|\chi \left(\vartheta \right)\phi \left(\vartheta \right)\right|\Delta \vartheta \le {\left({\int }_{\mathcal{J}}{\left|\chi \left(\vartheta \right)\right|}^{ћ}\Delta \vartheta \right)}^{{\displaystyle \frac{1}{ћ}}}{\left({\int }_{\mathcal{J}}{\left|\phi \left(\vartheta \right)\right|}^{\delta }\Delta \vartheta \right)}^{{\displaystyle \frac{1}{\delta }}}.$$

(16)

We now give the concept of $\Delta$-measurable functions and describe the definition of the Lebesgue $\Delta$ measure on $\mathbb{T}$, as performed by Bohner and Guseinov (Chapter 5 in [45]). We define the measure $m_1,$ which allocates the length of each interval ${[c,d)}_{\mathbb{T}}$, that is, $m_1\left(\mathcal{J}\right)=c-d.$ A function $\chi :\mathbb{T}\to \mathbb{R}$ is $\Delta$ measurable if

$${\chi }^{-1}\left([-\infty ,\beta )\right)=\{\vartheta \in \mathbb{T}:\chi \left(\vartheta \right)<\beta \},\forall \beta \in \mathbb{R},$$

is $\Delta$ measurable. Suppose $\mathcal{J}\subset \mathbb{T}$ is a $\Delta$-measurable set, $\gamma \ge 1$ and $\chi :\mathcal{J}\to \mathbb{R}$ is a $\Delta$-measurable function. We denote $\left|\mathcal{J}\right|$ as the delta measure of $\mathcal{J}$. We state that $\chi$ belongs to $L_{\Delta }^{\gamma }\left(\mathcal{J}\right)$ if either

$${∥\chi ∥}_{L_{\Delta }^{\gamma }\left(\mathcal{J}\right)}={\left({\int }_{\mathcal{J}}{\left|\chi \right|}^{\gamma }\Delta s\right)}^{1/\gamma }<\infty ,\mathrm{if}1\le \gamma <\infty ,$$

or there exists a constant $B\in \mathbb{R}$ subject to $\left|\chi \right|\le B$ if $\gamma =+\infty .$ It should be noted that each rd-continuous function defined on a bounded interval of $\mathbb{T}$ is a $\Delta$-measurable function. Throughout the rest of the paper, we suppose that the functions (unless otherwise stated) are nonnegative rd continuous defined on $\mathbb{T}$ and $\Delta$ integrable. We also assume that $u,$ $\varphi$ and $\varpi$ are used for nonnegative rd-continuous weighted functions defined on $\mathbb{T}$.

Theorem 2

([45]). Assuming $\chi :\mathbb{R}\to \mathbb{R}$ is continuously differentiable, and $\psi :\mathbb{T}\to \mathbb{R}$ is Δ differentiable, then

$${\left(\chi \circ \psi \right)}^{\Delta }\left(z\right)=\left\{{\int }_0^1{\chi }^{\prime }\left(\psi \left(z\right)+h\mu \left(z\right){\psi }^{\Delta }\left(z\right)\right)dh\right\}{\psi }^{\Delta }\left(z\right)$$

(17)

holds.

Theorem 3

([45]). Assume $\eta :\mathbb{T}\to \mathbb{R}$ is strictly increasing and $\tilde{\mathbb{T}}:=\eta \left(\mathbb{T}\right)$ is a time scale and let $\varpi :\tilde{\mathbb{T}}\to \mathbb{R}$. If there are ${\eta }^{\Delta }\left(z\right)$ and ${\varpi }^{\tilde{\Delta }}\left(\eta \left(z\right)\right)$ for $z\in \mathbb{T},$ then

$${\left(\varpi \circ \eta \right)}^{\Delta }=\left({\varpi }^{\tilde{\Delta }}\circ \eta \right){\eta }^{\Delta }.$$

Theorem 4.

Let $\eta :\mathbb{T}\to \mathbb{R}$ be strictly increasing and $\tilde{\mathbb{T}}:=\eta \left(\mathbb{T}\right)$ be a time scale. Then,

$${\displaystyle \frac{1}{{\eta }^{\Delta }}}={\left({\eta }^{-1}\right)}^{\tilde{\Delta }}\circ \eta ,$$

provided that ${\eta }^{\Delta }\ne 0$ and ${\left({\eta }^{-1}\right)}^{\tilde{\Delta }}$ exists.

The following two theorems are adapted from [45].

Theorem 5.

Assume $\eta :\mathbb{T}\to \mathbb{R}$ is strictly increasing, $\mathcal{J}\subset \mathbb{T}$ and $\mathcal{L}\subset \tilde{\mathbb{T}}$ where $\tilde{\mathbb{T}}=\eta \left(\mathbb{T}\right)$ is a time scale and $\mathcal{L}=\eta \left(\mathcal{J}\right)$. If $\chi \in C_{rd}\left(\mathbb{T},\mathbb{R}\right)$ and η is differentiable with an rd-continuous derivative, then

$${\int }_{\mathcal{J}}\chi \left(\vartheta \right){\eta }^{\Delta }\left(\vartheta \right)\Delta \vartheta ={\int }_{\mathcal{L}}\left(\chi \circ {\eta }^{-1}\right)\left(s\right)\tilde{\Delta }s.$$

Theorem 6.

Let $\mathbb{T}$ be a time scale and $\beta ,\gamma \in \mathbb{R}$. If $\varkappa \in C_{rd}\left(\mathcal{J},(\beta ,\gamma )\right)$ and $\Phi :(\beta ,\gamma )\to \mathbb{R}$ is continuous and convex, then

$$\Phi \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varkappa \left(\vartheta \right)\Delta \vartheta \right)\le {\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\Phi \left(\varkappa \left(\vartheta \right)\right)\Delta \vartheta .$$

(18)

If Φ is a concave function, then the inequality (18) will have the opposite direction.

Corollary 1.

If $\beta ,\gamma \in \mathbb{R}$ and $\varkappa \in C_{rd}\left(\mathcal{J},(\beta ,\gamma )\right)$ subject to $\varkappa \left(\vartheta \right)>0$ on $\mathcal{J},$ then

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varkappa \left(\vartheta \right)\Delta \vartheta \right)}^{\alpha }\le {\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varkappa }^{\alpha }\left(\vartheta \right)\Delta \vartheta ,\mathrm{for}\alpha <0,\mathrm{or}\alpha >1.$$

(19)

If $0<\alpha <1,$ this inequality will have the opposite direction.

3. Main Results

Before stating and proving the key findings, we present the basic definitions regarding Muckenhoupt and Gehring classes on $\mathbb{T}$, which are adapted from [46].

Definition 1.

A weight $\varpi \in {\mathcal{M}}_{\gamma }$ (the Muckenhoupt class) if it satisfies the condition

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}\le \mathcal{Q},\forall \mathcal{J}\subset \mathbb{T},$$

(20)

where $1<\gamma <\infty$ and $\mathcal{Q}$ is a positive constant. The ${\mathcal{M}}_{\gamma }$ norm of ϖ on $\mathcal{J}$ is

$${\mathcal{M}}_{\gamma }\left(\varpi \right)=\underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}.$$

(21)

It is easily observed from Hölder’s inequality that ${\mathcal{M}}_{\gamma }\left(\varpi \right)\ge 1.$ For a constant $\mathcal{Q}>1,$ if the weight $\varpi$ is in ${\mathcal{M}}_{\gamma }\left(\mathcal{Q}\right),$ then ${\mathcal{M}}_{\gamma }\left(\varpi \right)\le \mathcal{Q}$. These definitions apply for $1<\gamma <\infty$. The next step is to define ${\mathcal{M}}_{\gamma }$ specifically for $\gamma =1$ and $\gamma =\infty .$

Definition 2.

The class ${\mathcal{M}}_1$ consists of weights ϖ for which the ${\mathcal{M}}_1\left(\varpi \right)$ norm is finite. It is defined as follows:

$${\mathcal{M}}_1\left(\varpi \right)=\underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right){\left(ess\underset{\mathcal{J}}{inf}\varpi \right)}^{-1}.$$

Definition 3.

The class ${\mathcal{M}}_{\infty }$ includes all weights ϖ for which the ${\mathcal{M}}_{\infty }\left(\varpi \right)$ norm is finite. It is defined as follows:

$${\mathcal{M}}_{\infty }\left(\varpi \right)=\underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right)\left(exp{\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}log{\displaystyle \frac{1}{\varpi \left(\vartheta \right)}}\Delta \vartheta \right).$$

Definition 4.

A weight $\varphi \in {\mathcal{G}}_{\delta }$ (the Gehring class) if it satisfies the ${\mathcal{G}}_{\delta }$ condition

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\delta }\left(\vartheta \right)\Delta \vartheta \right)}^{{\displaystyle \frac{1}{\delta }}}\le \mathcal{K}{\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varphi \left(\vartheta \right)\Delta \vartheta ,\forall \mathcal{J}\subset \mathbb{T},$$

(22)

where $1<\delta <\infty$ and $\mathcal{K}>0$ is constant. This condition is referred to as a reverse Hölder inequality. The ${\mathcal{G}}_{\delta }$ norm of ϕ on $\mathcal{J}$ is

$${\mathcal{G}}_{\delta }\left(\varphi \right)=\underset{\mathcal{J}}{sup}{\left[{\displaystyle \frac{\left|\mathcal{J}\right|}{{\int }_{\mathcal{J}}\varphi \left(\vartheta \right)\Delta \vartheta }}{\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\delta }\left(\vartheta \right)\Delta \vartheta \right)}^{{\displaystyle \frac{1}{\delta }}}\right]}^{{\delta }^{\ast }},$$

where ${\delta }^{\ast }=\delta /(\delta -1).$

By Hölder’s inequality, it is clear that ${\mathcal{G}}_{\delta }\left(\varphi \right)\ge 1.$ For a fixed constant $\mathcal{K}>1,$ if $\varpi \in {\mathcal{G}}_{\delta }\left(\mathcal{K}\right),$ then ${\mathcal{G}}_{\delta }\left(\varphi \right)\le \mathcal{K}$. Next, we give the specific definition of ${\mathcal{G}}_{\delta }$ when $\delta =1$ and $\delta =\infty .$

Definition 5.

The class ${\mathcal{G}}_1$ includes all weights ϕ for which the ${\mathcal{G}}_1\left(\varphi \right)$ norm is finite. It is defined as follows:

$${\mathcal{G}}_1\left(\varphi \right)=\underset{\mathcal{J}}{sup}exp\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\displaystyle \frac{\varphi \left(\vartheta \right)}{{\varphi }_{\mathcal{J}}}}log{\displaystyle \frac{\varphi \left(\vartheta \right)}{{\varphi }_{\mathcal{J}}}}\Delta \vartheta \right),$$

where ${\varphi }_{\mathcal{J}}=\left(1/\left|\mathcal{J}\right|\right){\int }_{\mathcal{J}}\varphi \left(z\right)\Delta z.$

Definition 6.

The class ${\mathcal{G}}_{\infty }$ includes all weights ϕ for which the ${\mathcal{G}}_{\infty }\left(\varphi \right)$ norm is finite. It is defined as follows:

$${\mathcal{G}}_{\infty }\left(\varphi \right)=\underset{\mathcal{J}}{sup}{\displaystyle \frac{ess{sup}_{\mathcal{J}}\varphi }{\frac{1}{\left|\mathcal{J}\right|}{\int }_{\mathcal{J}}\varphi \left(z\right)\Delta z}}.$$

Definition 7.

Given weights $\psi ,$ ϕ and a constant $\gamma >1,$ we have $\left(\psi ,\varphi \right)\in {\mathcal{M}}_{\gamma }$ if and only if there exists a constant $\mathcal{Q}>1$ such that

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\psi \left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{1-{\gamma }^{\ast }}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}\le \mathcal{Q},\forall \mathcal{J}\subset \mathbb{T},$$

where ${\gamma }^{\ast }=\gamma /\left(\gamma -1\right).$

In the following, we will give a definition of a Sobolev space on a time scale $\mathbb{T}$.

Definition 8.

The Sobolev space on $\mathbb{T}$ is the space ${\mathcal{W}}_{loc}^{k,\gamma }\left(\mathbb{T}\right)$ of rd-continuous Δ-differentiable functions ξ in $L_{\Delta }^{\gamma }\left(\mathbb{T}\right)$ such that ξ and its Δ-derivatives up to an order k have a finite $L_{\Delta }^{\gamma }$ norm for a given γ ($1\le \gamma <\infty ).$ More precisely, we define ${\mathcal{W}}_{loc}^{k,\gamma }\left(\mathbb{T}\right)$ by

$${\mathcal{W}}_{loc}^{k,\gamma }\left(\mathbb{T}\right):=\{\phi \in L_{\Delta }^{\gamma }\left(\mathbb{T}\right):{\phi }^{{\Delta }^j}\in L_{\Delta }^{\gamma }\left(\mathbb{T}\right),\forall j\le k\},$$

where ${\phi }^{{\Delta }^j}={\left({\phi }^{{\Delta }^{j-1}}\right)}^{\Delta }$ with a finite ${\mathcal{W}}^{k,\gamma }\left(\mathbb{T}\right)$ norm

$${∥\phi ∥}_{{\mathcal{W}}^{k,\gamma }\left(\mathbb{T}\right)}:=\sum _{j=0}^k{∥{\phi }^{{\Delta }^j}∥}_{L_{\Delta }^{\gamma }\left(\mathbb{T}\right)}<\infty ,$$

where ${∥\xi ∥}_{L_{\Delta }^{\gamma }\left(\mathbb{T}\right)}={\left({\int \left|\xi \right|}^{\gamma }\Delta \vartheta \right)}^{1/\gamma }<\infty .$

By the triangle inequality, for $k\ge 1,$ it follows that

$${∥\phi ∥}_{{\mathcal{W}}^{k,\gamma }\left(\mathbb{T}\right)}\le 2^k{∥\phi ∥}_{L_{\Delta }^{\gamma }\left(\mathbb{T}\right)}.$$

This implies that, in the time scale setting, any $L_{\Delta }^{\gamma }\left(\mathbb{T}\right)$ bound automatically provides a ${\mathcal{W}}^{k,\gamma }\left(\mathbb{T}\right)$ bound for any $k\ge 1.$

In fact, the ${\mathcal{W}}^{k,\gamma }\left(\mathbb{T}\right)$ spaces can be seen as extensions of the classical $L_{\Delta }^{\gamma }\left(\mathbb{T}\right)$ spaces.

Definition 9.

A nonnegative $rd$-continuous function $\varphi :\mathbb{T}\to \tilde{\mathbb{T}}$, which is increasing and has homomorphism, is a bi-Sobolev map if $\varphi \in {\mathcal{W}}_{loc}^{1,1}\left(\mathbb{T}\right)$ and its inverse ${\varphi }^{-1}\in {\mathcal{W}}_{loc}^{1,1}\left(\tilde{\mathbb{T}}\right)$ exists, where $\tilde{\mathbb{T}}=\varphi \left(\mathbb{T}\right)$ is a time scale. We can easily see that if ϕ is increasing, then ${\varphi }^{-1}$ is also increasing.

Let us now state and demonstrate the important outcomes in this part.

Theorem 7.

Assume $\mathbb{T}$ is a time scale and ϕ is a bi-Sobolev map. If $\delta >1$ and ${\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\in {\mathcal{G}}_{\delta },$ then ${\varphi }^{\Delta }\in {\mathcal{M}}_{\gamma }$ and

$${\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right)\le {\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right),$$

(23)

where $1/\gamma +1/\delta =1$.

Proof. 

Since ${\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\in {\mathcal{G}}_{\delta },$ then for every $\mathcal{L}\subset \tilde{\mathbb{T}},$ we see that

$${\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z\right)}^{{\displaystyle \frac{1}{\delta }}}\le {\left({\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right)\right)}^{{\displaystyle \frac{1}{{\delta }^{\ast }}}}\left[{\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z\right],$$

(24)

where ${\delta }^{\ast }=\delta /(\delta -1)$ and the constant of the class is given by ${\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right).$ Since $\gamma$ and $\delta$ are conjugate, we have $1/\gamma =(\delta -1)/\delta =1/{\delta }^{\ast }$, and then (24) becomes

$${\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z\right)}^{{\displaystyle \frac{1}{\delta }}}\le {\left({\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right)\right)}^{{\displaystyle \frac{1}{\gamma }}}\left[{\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z\right].$$

(25)

Assume there is $\mathcal{J}\subset \mathbb{T}$ such that $\varphi \left(\mathcal{J}\right)=\mathcal{L}$ and define

$$H=\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{{\displaystyle \frac{-1}{\gamma -1}}}\Delta \vartheta \right)}^{\gamma -1}.$$

(26)

Then, by using the relation between $\gamma$ and $\delta ,$ we obtain

$$H=\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{1-\delta }\Delta \vartheta \right)}^{\gamma -1}.$$

(27)

Now, since $\varphi \left(\mathcal{J}\right)=\mathcal{L},$ we have

$${\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(\vartheta \right)\Delta \vartheta =\left|\mathcal{L}\right|.$$

(28)

Applying Theorem 5 with $\chi ={\left[1/{\varphi }^{\Delta }\right]}^{\delta }$ and $\eta =\varphi ,$ we obtain

$${\int }_{\mathcal{L}}{\left[{\displaystyle \frac{1}{{\varphi }^{\Delta }\left({\varphi }^{-1}\left(z\right)\right)}}\right]}^{\delta }\tilde{\Delta }z={\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{1-\delta }\Delta \vartheta .$$

(29)

Now, by applying Theorem 4, we have

$${\displaystyle \frac{1}{{\varphi }^{\Delta }\left({\varphi }^{-1}\left(z\right)\right)}}=\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\circ \varphi \right){\varphi }^{-1}\left(z\right)={\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\left[\varphi \left({\varphi }^{-1}\left(z\right)\right)\right]={\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}.$$

(30)

Substituting (30) into (29), we obtain

$${\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{1-\delta }\Delta \vartheta ={\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z,$$

(31)

and then

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{1-\delta }\Delta \vartheta \right)}^{\gamma -1}={\left({\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}\right)}^{\gamma -1}{\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z\right)}^{\gamma -1}.$$

(32)

Substituting (28) and (32) into (27), we obtain

$$H={\left({\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}\right)}^{\gamma }{\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z\right)}^{\gamma -1}.$$

(33)

By using (25) in (33), we have

$$H\le {\left({\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}\right)}^{\gamma }{\left({\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right)\right)}^{{\displaystyle \frac{\delta (\gamma -1)}{\gamma }}}{\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z\right)}^{\delta (\gamma -1)}.$$

(34)

Now, since ${\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z=\left|\mathcal{J}\right|,$ and $\delta (\gamma -1)=\gamma ,$ we have from (34) that

$$H\le {\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right).$$

(35)

Therefore, we have from (26) and (35) that

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{{\displaystyle \frac{-1}{\gamma -1}}}\Delta \vartheta \right)}^{\gamma -1}\le {\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right).$$

This implies that ${\varphi }^{\Delta }\in {\mathcal{M}}_{\gamma }$ with a constant

$${\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right)=\underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{{\displaystyle \frac{-1}{\gamma -1}}}\Delta \vartheta \right)}^{\gamma -1}\le {\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right),$$

which proves (23). □

Theorem 8.

Suppose $\mathbb{T}$ is a time scale and ϕ is a bi-Sobolev map. If ${\varphi }^{\Delta }\in {\mathcal{M}}_{\gamma }$ for $\gamma >1,$ then ${\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\in {\mathcal{G}}_{\delta }$ and

$${\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right)\le {\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right)$$

(36)

where $1/\gamma +1/\delta =1.$

Proof. 

Since ${\varphi }^{\Delta }\in {\mathcal{M}}_{\gamma },$ then for every $\mathcal{J}\subset \mathbb{T}$, we obtain

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{{\displaystyle \frac{-1}{\gamma -1}}}\Delta \vartheta \right)}^{\gamma -1}\le {\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right).$$

(37)

Since ${\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(\vartheta \right)\Delta \vartheta =\left|\mathcal{L}\right|$ and $-1/(\gamma -1)=1-\delta ,$ we have from (37) that

$${\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{1-\delta }\Delta \vartheta \le {\left|\mathcal{J}\right|}^{\delta }{\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right)\right)}^{{\displaystyle \frac{1}{\gamma -1}}}.$$

(38)

From (31) and the fact that ${\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z=\left|z\right|$, we obtain

$${\displaystyle \frac{{\left|\mathcal{L}\right|}^{\frac{1}{\gamma }}}{{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z}}{\left({\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z\right)}^{{\displaystyle \frac{1}{\delta }}}={\displaystyle \frac{{\left|\mathcal{L}\right|}^{\frac{1}{\gamma }}}{\left|\mathcal{J}\right|}}{\left({\int }_{\mathcal{J}}{\left[{\varphi }^{\Delta }\left(\vartheta \right)\right]}^{1-\delta }\Delta \vartheta \right)}^{{\displaystyle \frac{1}{\delta }}}.$$

(39)

Substituting (38) into (39), we have (note $\delta (\gamma -1)=\gamma$ and $\gamma =\delta /(\delta -1)={\delta }^{\ast }$)

$$\begin{array}{ccc}& & {\displaystyle \frac{{\left|\mathcal{L}\right|}^{\frac{1}{\gamma }}}{{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z}}{\left({\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z\right)}^{{\displaystyle \frac{1}{\delta }}}\hfill \\ & \le & {\left|\mathcal{L}\right|}^{{\displaystyle \frac{1}{\gamma }}}{\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right)\right)}^{{\displaystyle \frac{1}{\delta \left(\gamma -1\right)}}}={\left|\mathcal{L}\right|}^{{\displaystyle \frac{1}{\gamma }}}{\left({\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right)\right)}^{{\displaystyle \frac{1}{\gamma }}}={\left[{\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right)\right]}^{{\displaystyle \frac{1}{{\delta }^{\ast }}}},\hfill \end{array}$$

and then

$${\left[{\displaystyle \frac{{\left|\mathcal{L}\right|}^{\frac{1}{\gamma }}}{{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }z}}{\left({\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\right]}^{\delta }\tilde{\Delta }z\right)}^{{\displaystyle \frac{1}{\delta }}}\right]}^{{\delta }^{\ast }}\le {\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right).$$

This implies that ${\left({\varphi }^{-1}\left(z\right)\right)}^{\tilde{\Delta }}\in {\mathcal{G}}_{\delta }$ with a constant

$${\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right)=\underset{\mathcal{L}}{sup}{\left[{\displaystyle \frac{{\left|\mathcal{L}\right|}^{\frac{1}{\gamma }}}{{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\left(z\right)\tilde{\Delta }z}}{\left({\int }_{\mathcal{L}}{\left[{\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\left(z\right)\right]}^{\delta }\tilde{\Delta }z\right)}^{{\displaystyle \frac{1}{\delta }}}\right]}^{{\delta }^{\ast }}\le {\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right),$$

which proves (36). □

From Theorems 7 and 8, we obtain the following:

Corollary 2.

Assume $\mathbb{T}$ is a time scale and ϕ is a bi-Sobolev map, then

$${\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\in {\mathcal{G}}_{\delta }⟺{\varphi }^{\Delta }\in {\mathcal{M}}_{\gamma },$$

where $\gamma >1$ and $\gamma ,\delta$ are conjugate. Moreover, from (23) and (36), it follows that

$${\mathcal{G}}_{\delta }\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right)={\mathcal{M}}_{\gamma }\left({\varphi }^{\Delta }\right).$$

Remark 1.

As a specific instance of Corollary 2, when $\mathbb{T}=\mathbb{R}$, we derive the result (2) proved by Johnson and Neugebauer (see [12]).

When $\mathbb{T}=\mathbb{N},$ Corollary 2 yields the following result, which gives a transition property between the discrete Muckenhoupt and Gehring classes where the sequences and their inverses are defined on the discrete Sobolev space $W^{1,1}\left(\mathbb{N}\right)$.

Corollary 3.

Let $\varphi :\mathbb{N}\to \mathbb{N}$ be a bi-Sobolev map. Then,

$$\Delta \left({\varphi }^{-1}\right)\in {\mathcal{G}}_{\delta }\iff \Delta \varphi \in {\mathcal{M}}_{\gamma },$$

where ${\mathcal{G}}_{\delta }$ and ${\mathcal{M}}_{\gamma }$ are the Gehring and Muckenhoupt discrete spaces, $\gamma >1$ and $\gamma ,\delta$ are conjugate. Moreover, we have

$${\mathcal{G}}_{\delta }\left(\Delta \left({\varphi }^{-1}\right)\right)={\mathcal{M}}_{\gamma }\left(\Delta \varphi \right).$$

Theorem 9.

Assume $\mathbb{T}$ is a time scale and ϕ is a bi-Sobolev map. If ${\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\in {\mathcal{G}}_1$ and ${\varphi }^{\Delta }\in {\mathcal{M}}_{\infty }$, then

$${\mathcal{M}}_{\infty }\left({\varphi }^{\Delta }\right)={\mathcal{G}}_1\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right).$$

(40)

Proof. 

Define

$$A_{\mathcal{J}}:=\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(z\right)\Delta z\right)exp\left[{\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}log{\displaystyle \frac{1}{{\varphi }^{\Delta }\left(z\right)}}\Delta z\right]$$

(41)

and

$$B_{\mathcal{L}}:=exp\left[{\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\displaystyle \frac{{\left({\varphi }^{-1}\left(s\right)\right)}^{\tilde{\Delta }}}{{\varphi }_{\mathcal{L}}}}log{\displaystyle \frac{{\left({\varphi }^{-1}\left(s\right)\right)}^{\tilde{\Delta }}}{{\varphi }_{\mathcal{L}}}}\tilde{\Delta }s\right].$$

(42)

where ${\varphi }_{\mathcal{L}}={\displaystyle \frac{1}{\left|\mathcal{L}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(y\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }y.$ Using Theorem 5 with $\chi \left(z\right)={\displaystyle \frac{1}{{\varphi }^{\Delta }\left(z\right)}}log{\displaystyle \frac{1}{{\varphi }^{\Delta }\left(z\right)}},$ we obtain

$${\int }_{\mathcal{J}}log{\displaystyle \frac{1}{{\varphi }^{\Delta }\left(z\right)}}\Delta z={\int }_{\mathcal{L}}{\displaystyle \frac{1}{{\varphi }^{\Delta }\left({\varphi }^{-1}\left(\vartheta \right)\right)}}log{\displaystyle \frac{1}{{\varphi }^{\Delta }\left({\varphi }^{-1}\left(\vartheta \right)\right)}}\tilde{\Delta }\vartheta .$$

(43)

Substituting (30) into (43) yields

$${\int }_{\mathcal{J}}log{\displaystyle \frac{1}{{\varphi }^{\Delta }\left(z\right)}}\Delta z={\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}log{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\tilde{\Delta }\vartheta .$$

(44)

Since ${\int }_{\mathcal{J}}{\varphi }^{\Delta }\left(z\right)\Delta z=\left|\mathcal{L}\right|,$ then we have from (41) and (44) that

$$A_{\mathcal{J}}={\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}exp\left[{\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}log{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\tilde{\Delta }\vartheta \right].$$

(45)

Since ${\int }_{\mathcal{L}}{\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\left(y\right)\tilde{\Delta }y=\left|\mathcal{J}\right|,$ we obtain

$$\begin{array}{ccc}& & {\int }_{\mathcal{L}}{\displaystyle \frac{\left|\mathcal{L}\right|{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}}{{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(y\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }y}}log{\displaystyle \frac{\left|\mathcal{L}\right|{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}}{{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(y\right)\right)}^{\tilde{\Delta }}\tilde{\Delta }y}}\tilde{\Delta }\vartheta \hfill \\ & =& {\int }_{\mathcal{L}}{\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}log\left[{\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\right]\tilde{\Delta }\vartheta \hfill \\ & =& {\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\left(log{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}-log{\displaystyle \frac{\left|\mathcal{J}\right|}{\left|\mathcal{L}\right|}}\right)\tilde{\Delta }\vartheta \hfill \\ & =& {\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\left(\vartheta \right)log{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\tilde{\Delta }\vartheta \hfill \\ & & -{\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}\left(log{\displaystyle \frac{\left|\mathcal{J}\right|}{\left|\mathcal{L}\right|}}\right){\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\tilde{\Delta }\vartheta \hfill \\ & =& {\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}log{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\tilde{\Delta }\vartheta -\left|\mathcal{L}\right|log{\displaystyle \frac{\left|\mathcal{J}\right|}{\left|\mathcal{L}\right|}}.\hfill \end{array}$$

(46)

Substituting (46) into (42), we have

$$\begin{array}{ccc}\hfill B_{\mathcal{L}}& =& exp\left[{\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}log{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\tilde{\Delta }\vartheta -log{\displaystyle \frac{\left|\mathcal{J}\right|}{\left|\mathcal{L}\right|}}\right]\hfill \\ & =& {\displaystyle \frac{\left|\mathcal{L}\right|}{\left|\mathcal{J}\right|}}exp\left[{\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{L}}{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}log{\left({\varphi }^{-1}\left(\vartheta \right)\right)}^{\tilde{\Delta }}\tilde{\Delta }\vartheta \right].\hfill \end{array}$$

(47)

From (45) and (47), we obtain

$$A_{\mathcal{J}}=B_{\mathcal{L}}.$$

(48)

From (48) and ${\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\in {\mathcal{G}}_1$, we observe

$$A_{\mathcal{J}}=B_{\mathcal{L}}\le \underset{\mathcal{L}}{sup}{B}_{\mathcal{L}}={\mathcal{G}}_1\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right),$$

and then

$${\mathcal{M}}_{\infty }\left({\varphi }^{\Delta }\right)=\underset{\mathcal{J}}{sup}{A}_{\mathcal{J}}\le {\mathcal{G}}_1\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right).$$

(49)

From (48) and the fact that ${\varphi }^{\Delta }\in {\mathcal{M}}_{\infty },$ we have $B_{\mathcal{L}}=A_{\mathcal{J}}\le {sup}_{\mathcal{J}}{A}_{\mathcal{J}}={\mathcal{M}}_{\infty }\left({\varphi }^{\Delta }\right),$ and then

$${\mathcal{G}}_1\left({\left({\varphi }^{-1}\right)}^{\tilde{\Delta }}\right)=\underset{\mathcal{L}}{sup}{B}_{\mathcal{L}}\le {\mathcal{M}}_{\infty }\left({\varphi }^{\Delta }\right).$$

(50)

From (49) and (50), we obtain (40). □

Remark 2.

As a specific instance of Theorem 9, when $\mathbb{T}=\mathbb{R}$, we obtain (3), proved by Corporente.

When $\mathbb{T}=\mathbb{N},$ Theorem 9 yields the following result where the sequences and their inverses are defined on the discrete Sobolev space $W^{1,1}\left(\mathbb{N}\right)$.

Corollary 4.

Let $\varphi :\mathbb{N}\to \mathbb{N}$ be a bi-Sobolev map. Suppose $\Delta \left({\varphi }^{-1}\right)\in {\mathcal{G}}_1$ and $\Delta \varphi \in {\mathcal{M}}_{\infty }$, then

$${\mathcal{M}}_{\infty }(\Delta \varphi )={\mathcal{G}}_1\left(\Delta \left({\varphi }^{-1}\right)\right),$$

where ${\mathcal{G}}_1$ and ${\mathcal{M}}_{\infty }$ are Gehring and Muckenhoupt discrete spaces.

Theorem 10.

Let ϖ be a positive rd-continuous function defined on ${\mathcal{J}}_0\subset \mathbb{T}$.

(I).

If $\gamma >1$ and $\varpi ,$$1/\varpi \in {\mathcal{M}}_{\gamma },$ then ${\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\in {\mathcal{M}}_2$ and

$${\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1}\le {\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right).$$

(51)

(II).

If $1<\gamma \le 2$ and ${\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\in {\mathcal{M}}_2,$ then $\varpi \in {\mathcal{M}}_{\gamma }$and

$${\mathcal{M}}_{\gamma }\left(\varpi \right)\le {\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1}.$$

(52)

Proof. 

First, we prove (51). By applying the Cauchy–Schwarz inequality for every $\mathcal{J}\subset {\mathcal{J}}_0$, we obtain

$$\left|\mathcal{J}\right|={\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{2}}}\left(\vartheta \right){\varpi }^{{\displaystyle \frac{-1}{2}}}\left(\vartheta \right)\Delta \vartheta \le {\left({\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{\mathcal{J}}{\displaystyle \frac{1}{\varpi \left(\vartheta \right)}}\Delta \vartheta \right)}^{{\displaystyle \frac{1}{2}}},$$

and then

$${\left|\mathcal{J}\right|}^2\le \left({\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right)\left({\int }_{\mathcal{J}}{\displaystyle \frac{1}{\varpi \left(\vartheta \right)}}\Delta \vartheta \right).$$

(53)

From (53), we see that

$$\begin{array}{ccc}& & {\left[\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\right]}^{\gamma -1}\hfill \\ & \le & \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}\hfill \\ & & \times \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\displaystyle \frac{1}{\varpi \left(\vartheta \right)}}\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}.\hfill \end{array}$$

(54)

Since $\varpi$ and $1/\varpi \in {\mathcal{M}}_{\gamma },$ we have from (54) that

$$\begin{array}{ccc}& & {\left[\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\right]}^{\gamma -1}\hfill \\ & \le & \underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}\hfill \\ & & \times \underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\displaystyle \frac{1}{\varpi \left(\vartheta \right)}}\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}\hfill \\ & =& {\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right),\hfill \end{array}$$

and hence

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\le {\left[{\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right)\right]}^{{\displaystyle \frac{1}{\gamma -1}}}.$$

Then, we obtain that

$$\underset{\mathcal{J}}{sup}\left[\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\right]\le {\left[{\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right)\right]}^{{\displaystyle \frac{1}{\gamma -1}}},$$

which implies that ${\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\in {\mathcal{M}}_2$ with a constant

$${\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\le {\left[{\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right)\right]}^{{\displaystyle \frac{1}{\gamma -1}}},$$

and then

$${\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1}\le {\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right),$$

which is (51). Now, we prove (52) for $1<\gamma \le 2.$ By applying (19), we obtain

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right)}^{{\displaystyle \frac{1}{\gamma -1}}}\le {\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta ,$$

and then

$$\begin{array}{ccc}& & \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}\hfill \\ & \le & {\left[\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)\right]}^{\gamma -1}\hfill \\ & \le & {\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1},\hfill \end{array}$$

which implies that $\varpi \in {\mathcal{M}}_{\gamma }$ with a constant

$${\mathcal{M}}_{\gamma }\left(\varpi \right)=\underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(\vartheta \right)\Delta \vartheta \right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\Delta \vartheta \right)}^{\gamma -1}\le {\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1},$$

which is (52). □

Remark 3.

As a specific instance of Theorem 10, where $\mathbb{T}=\mathbb{R}$, we derive the results demonstrated by Sbordone [16].

Corollary 5.

If $\mathbb{T}=\mathbb{N}$, $\gamma >1,$ $\varpi ,$ $1/\varpi \in {\mathcal{M}}_{\gamma },$ then ${\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\in {\mathcal{M}}_2$ and the relation

$${\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1}\le {\mathcal{M}}_{\gamma }\left(\varpi \right){\mathcal{M}}_{\gamma }\left({\displaystyle \frac{1}{\varpi }}\right),$$

holds, and if $1<\gamma \le 2$and ${\varpi }^{{\displaystyle \frac{1}{\gamma -1}}}\in {\mathcal{M}}_2$, then $\varpi \in {\mathcal{M}}_{\gamma }$ and the following relation holds:

$${\mathcal{M}}_{\gamma }\left(\varpi \right)\le {\left[{\mathcal{M}}_2({\varpi }^{{\displaystyle \frac{1}{\gamma -1}}})\right]}^{\gamma -1},$$

where

$${\mathcal{M}}_{\gamma }\left(\varpi \right)=\underset{\mathcal{J}\subset \mathbb{N}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}\sum _{\vartheta \in \mathcal{J}}\varpi \left(\vartheta \right)\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\vartheta \in \mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(\vartheta \right)\right)}^{\gamma -1},$$

and

$${\mathcal{M}}_2\left(\varpi \right)=\underset{\mathcal{J}\subset \mathbb{N}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}\sum _{\vartheta \in \mathcal{J}}\varpi \left(\vartheta \right)\right)\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\vartheta \in \mathcal{J}}{\displaystyle \frac{1}{\varpi \left(\vartheta \right)}}\right).$$

Theorem 11.

Let ϖ be a positive rd-continuous function defined on ${\mathcal{J}}_0\subset \mathbb{T}$ and $1<\kappa \le \gamma <\infty .$

(I).

If $\varpi \in {\mathcal{M}}_{\kappa },$ then $\varpi \in {\mathcal{M}}_{\gamma }$ and

$$1\le {\mathcal{M}}_{\gamma }\left(\varpi \right)\le {\mathcal{M}}_{\kappa }\left(\varpi \right).$$

(55)

(II).

If $\varpi \in {\mathcal{G}}_{\gamma },$ then $\varpi \in {\mathcal{G}}_{\kappa }$and

$$1\le {\left[{\mathcal{G}}_{\kappa }\left(\varpi \right)\right]}^{{\displaystyle \frac{1}{{\kappa }^{\ast }}}}\le {\left[{\mathcal{G}}_{\gamma }\left(\varpi \right)\right]}^{{\displaystyle \frac{1}{{\gamma }^{\ast }}}},$$

(56)

where ${\kappa }^{\ast }=\kappa /(\kappa -1)$ and ${\gamma }^{\ast }=\gamma /(\gamma -1).$

Proof. 

For $1<\kappa <\gamma <\infty ,$ we have $0<\kappa -1<\gamma -1$ and $0<(\kappa -1)/(\gamma -1)<1.$ Applying the reverse of (19) with $\varkappa ={\varpi }^{{\displaystyle \frac{-1}{\kappa -1}}}$ and $\alpha =(\kappa -1)/(\gamma -1)<1,$ we obtain for every subinterval $\mathcal{J}\subset {\mathcal{J}}_0$ that

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\Delta z\right)}^{{\displaystyle \frac{\kappa -1}{\gamma -1}}}\ge {\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varpi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\right]}^{{\displaystyle \frac{\kappa -1}{\gamma -1}}}\Delta z={\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\Delta z,$$

and then

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\Delta z\right)}^{\gamma -1}\le {\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\Delta z\right)}^{\kappa -1}.$$

(57)

From (57) and since $\varpi \in {\mathcal{M}}_{\kappa },$ we obtain

$$\begin{array}{ccc}& & \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\Delta z\right)}^{\gamma -1}\hfill \\ & \le & \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\Delta z\right)}^{\kappa -1}\le {\mathcal{M}}_{\kappa }\left(\varpi \right).\hfill \end{array}$$

(58)

This implies that $\varpi \in {\mathcal{M}}_{\gamma }$ with a constant

$${\mathcal{M}}_{\gamma }\left(\varpi \right)=\underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\Delta z\right)}^{\gamma -1}\le {\mathcal{M}}_{\kappa }\left(\varpi \right).$$

(59)

By applying (19) with $\alpha =1-\gamma <0$ and $\varkappa ={\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}},$ we have

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\Delta z\right)}^{1-\gamma }\le {\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\right]}^{1-\gamma }\Delta z={\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z.$$

From this, we deduce that

$$\begin{array}{ccc}\hfill 1& \le & \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\Delta z\right)}^{\gamma -1}\hfill \\ & \le & \underset{\mathcal{J}}{sup}\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{{\displaystyle \frac{-1}{\gamma -1}}}\left(z\right)\Delta z\right)}^{\gamma -1}={\mathcal{M}}_{\gamma }\left(\varpi \right).\hfill \end{array}$$

(60)

Based on (59) and (60), we may conclude that $1\le {\mathcal{M}}_{\gamma }\left(\varpi \right)\le {\mathcal{M}}_{\kappa }\left(\varpi \right)$, which is (55).

Now, we prove (56). Applying (19) with $\alpha =\gamma /\kappa >1$ and $\varkappa ={\varpi }^{\kappa },$ we obtain

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{\kappa }\left(z\right)\Delta z\right)}^{{\displaystyle \frac{\gamma }{\kappa }}}\le {\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{\gamma }\left(z\right)\Delta z,$$

and then

$$\begin{array}{ccc}\hfill {\left[{\displaystyle \frac{\left|\mathcal{J}\right|}{{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z}}{\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{\kappa }\left(z\right)\Delta z\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]}^{{\kappa }^{\ast }}& \le & {\left[{\displaystyle \frac{\left|\mathcal{J}\right|}{{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z}}{\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{\gamma }\left(z\right)\Delta z\right)}^{{\displaystyle \frac{1}{\gamma }}}\right]}^{{\kappa }^{\ast }}\hfill \\ & \le & {\left[{\mathcal{G}}_{\gamma }\left(\varpi \right)\right]}^{{\displaystyle \frac{{\kappa }^{\ast }}{{\gamma }^{\ast }}}}.\hfill \end{array}$$

Therefore, $\varpi \in {\mathcal{G}}_{\gamma }$ with a constant

$${\mathcal{G}}_{\kappa }\left(\varpi \right)=\underset{\mathcal{J}}{sup}{\left[{\displaystyle \frac{\left|\mathcal{J}\right|}{{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z}}{\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{\kappa }\left(z\right)\Delta z\right)}^{{\displaystyle \frac{1}{\kappa }}}\right]}^{{\kappa }^{\ast }}\le {\left[{\mathcal{G}}_{\gamma }\left(\varpi \right)\right]}^{{\displaystyle \frac{{\kappa }^{\ast }}{{\gamma }^{\ast }}}},$$

and hence

$${\left[{\mathcal{G}}_{\kappa }\left(\varpi \right)\right]}^{{\displaystyle \frac{1}{{\kappa }^{\ast }}}}\le {\left[{\mathcal{G}}_{\gamma }\left(\varpi \right)\right]}^{{\displaystyle \frac{1}{{\gamma }^{\ast }}}}.$$

(61)

By applying (19) with $\alpha =\kappa >1$ and $\varkappa =\varpi ,$ we observe

$${\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{\kappa }\left(z\right)\Delta z\ge {\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z\right)}^{\kappa },$$

and then we have for $\varpi \in {\mathcal{G}}_{\kappa }$ that

$$1\le {\displaystyle \frac{\left|\mathcal{J}\right|}{{\int }_{\mathcal{J}}\varpi \left(z\right)\Delta z}}{\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varpi }^{\kappa }\left(z\right)\Delta z\right)}^{{\displaystyle \frac{1}{\kappa }}}\le {\left[{\mathcal{G}}_{\kappa }\left(\varpi \right)\right]}^{{\displaystyle \frac{1}{{\kappa }^{\ast }}}}.$$

(62)

From (61) and (62), we obtain

$$1\le {\left[{\mathcal{G}}_{\kappa }\left(\varpi \right)\right]}^{{\displaystyle \frac{1}{{\kappa }^{\ast }}}}\le {\left[{\mathcal{G}}_{\gamma }\left(\varpi \right)\right]}^{{\displaystyle \frac{1}{{\gamma }^{\ast }}}},$$

which is (56). □

Theorem 12.

Suppose u and ϕ are defined on ${\mathcal{J}}_0\subset \mathbb{T}$. If $\left(u,\varphi \right)\in {\mathcal{M}}_{\kappa }$, then $\left(u,\varphi \right)\in {\mathcal{M}}_{\vartheta }$, where $1<\kappa <\vartheta <\infty .$

Proof. 

For $1<\kappa <\vartheta <\infty ,$ we have $0<\kappa -1<\vartheta -1$ and $0<(\kappa -1)/(\vartheta -1)<1.$ By applying the reverse of (19) with $\varkappa ={\varphi }^{{\displaystyle \frac{-1}{\kappa -1}}}$ and $\alpha =(\kappa -1)/(\vartheta -1)<1,$ we obtain for every subinterval $\mathcal{J}\subset {\mathcal{J}}_0$ that

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\Delta z\right)}^{{\displaystyle \frac{\kappa -1}{\vartheta -1}}}\ge {\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\varphi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\right]}^{{\displaystyle \frac{\kappa -1}{\vartheta -1}}}\Delta z={\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{{\displaystyle \frac{-1}{\vartheta -1}}}\left(z\right)\Delta z,$$

and then

$${\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{{\displaystyle \frac{-1}{\vartheta -1}}}\left(z\right)\Delta z\right)}^{\vartheta -1}\le {\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\Delta z\right)}^{\kappa -1}.$$

(63)

From (63), we see (note u is positive on $\mathcal{J}$) that

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}u\left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{{\displaystyle \frac{-1}{\vartheta -1}}}\left(z\right)\Delta z\right)}^{\vartheta -1}\le \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}u\left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{{\displaystyle \frac{-1}{\kappa -1}}}\left(z\right)\Delta z\right)}^{\kappa -1}.$$

(64)

Since $-1/\left(\vartheta -1\right)=1-{\vartheta }^{\ast }$ and ${\vartheta }^{\ast }=\vartheta /\left(\vartheta -1\right),$ the inequality (64) becomes

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}u\left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{1-{\vartheta }^{\ast }}\left(z\right)\Delta z\right)}^{\vartheta -1}\le \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}u\left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\varphi }^{1-{\kappa }^{\ast }}\left(z\right)\Delta z\right)}^{\kappa -1}.$$

(65)

Since $\left(u,\varphi \right)\in {\mathcal{M}}_{\kappa },$ we have from (65) that $\left(u,\varphi \right)\in {\mathcal{M}}_{\vartheta }.$ □

Theorem 13.

Assume $\mathbb{T}$ is a time scale and ${\mathcal{J}}_0\subset \mathbb{T}$ is a finite interval. Furthermore, suppose that ψ and ω are defined on ${\mathcal{J}}_0$. If $\left(\psi ,\omega \right)\in {\mathcal{M}}_{\mathcal{\ell }},$ then $\left({\omega }^{1-{\mathcal{\ell }}^{\ast }},{\psi }^{1-{\mathcal{\ell }}^{\ast }}\right)\in {\mathcal{M}}_{{\mathcal{\ell }}^{\ast }},$ where $\mathcal{\ell }>1$ and ${\mathcal{\ell }}^{\ast }=\mathcal{\ell }/\left(\mathcal{\ell }-1\right).$

Proof. 

Since ${\mathcal{\ell }}^{\ast }=\mathcal{\ell }/\left(\mathcal{\ell }-1\right),$ we notice that

$${\left({\mathcal{\ell }}^{\ast }\right)}^{\ast }={\displaystyle \frac{{\mathcal{\ell }}^{\ast }}{{\mathcal{\ell }}^{\ast }-1}}=\mathcal{\ell },\mathrm{and}{\mathcal{\ell }}^{\ast }-1={\displaystyle \frac{1}{\mathcal{\ell }-1}},$$

(66)

and then

$$\left(1-{\mathcal{\ell }}^{\ast }\right)\left(1-{\left({\mathcal{\ell }}^{\ast }\right)}^{\ast }\right)=\left(1-{\mathcal{\ell }}^{\ast }\right)\left(1-\mathcal{\ell }\right)=1.$$

(67)

By noting that

$$\begin{array}{ccc}& & \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\psi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\omega }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\Delta z\right)}^{\mathcal{\ell }-1}\hfill \\ & =& {\left[\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\omega }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\psi \left(z\right)\Delta z\right)}^{{\displaystyle \frac{1}{\mathcal{\ell }-1}}}\right]}^{\mathcal{\ell }-1},\hfill \end{array}$$

(68)

we have from (66) and (67) that

$$\begin{array}{ccc}& & \left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\psi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\omega }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\Delta z\right)}^{\mathcal{\ell }-1}\hfill \\ & =& {\left[\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\omega }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\psi }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\right]}^{1-{\left({\mathcal{\ell }}^{\ast }\right)}^{\ast }}\Delta z\right)}^{{\mathcal{\ell }}^{\ast }-1}\right]}^{\mathcal{\ell }-1}.\hfill \end{array}$$

(69)

Since $\left(\psi ,\omega \right)\in {\mathcal{M}}_{\mathcal{\ell }}$, then there exists a constant $\mathcal{Q}>1$ such that

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}\psi \left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\omega }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\Delta z\right)}^{\mathcal{\ell }-1}\le \mathcal{Q}.$$

(70)

From (69) and (70), we see that

$$\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\omega }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\Delta z\right){\left({\displaystyle \frac{1}{\left|\mathcal{J}\right|}}{\int }_{\mathcal{J}}{\left[{\psi }^{1-{\mathcal{\ell }}^{\ast }}\left(z\right)\right]}^{1-{\left({\mathcal{\ell }}^{\ast }\right)}^{\ast }}\Delta z\right)}^{{\mathcal{\ell }}^{\ast }-1}\le {\mathcal{Q}}^{{\displaystyle \frac{1}{\mathcal{\ell }-1}}},$$

which gives us

$$\left({\omega }^{1-{\mathcal{\ell }}^{\ast }},{\psi }^{1-{\mathcal{\ell }}^{\ast }}\right)\in {\mathcal{M}}_{{\mathcal{\ell }}^{\ast }}.$$

□

Remark 4.

As a particular instance of Theorem 13, when $\mathbb{T}=\mathbb{R}$, we obtain (6), proved by Neugebauer.

Theorem 14.

Suppose $\mathbb{T}$ is a time scale and u and ϕ are defined on ${\mathcal{J}}_0\subset \mathbb{T}$. If $\left(u,\varphi \right)\in {\mathcal{M}}_{\gamma }$, then $\left(u^{{\displaystyle \frac{\delta -1}{\gamma -1}}},{\varphi }^{{\displaystyle \frac{\delta -1}{\gamma -1}}}\right)\in {\mathcal{M}}_{\delta },$ where $1<\delta <\gamma <\infty .$

Proof. 

By using the assumption $\left(u,\varphi \right)\in {\mathcal{M}}_{\gamma },$ and applying Theorem 13 with $\psi =u,$ $\omega =\varphi$ and $\mathcal{\ell }=\gamma ,$ we have

$$\left({\varphi }^{1-{\gamma }^{\ast }},u^{1-{\gamma }^{\ast }}\right)\in {\mathcal{M}}_{{\gamma }^{\ast }},$$

(71)

where ${\gamma }^{\ast }=\gamma /\left(\gamma -1\right).$ Since $\delta <\gamma ,$ we see that $1-(1/\delta )<1-(1/\gamma ),$ and then ${\delta }^{\ast }>{\gamma }^{\ast },$ where ${\delta }^{\ast }=\delta /\left(\delta -1\right).$ Since $\gamma >1$, we observe that ${\gamma }^{\ast }>1,$and then

$$1<{\gamma }^{\ast }<{\delta }^{\ast }.$$

(72)

By using (71) and (72) and by applying Theorem 12 with $\kappa ={\gamma }^{\ast },$ $\vartheta ={\delta }^{\ast },$ $\lambda ={\varphi }^{1-{\gamma }^{\ast }}$ and $\phi =u^{1-{\gamma }^{\ast }},$ we see that

$$\left({\varphi }^{1-{\gamma }^{\ast }},u^{1-{\gamma }^{\ast }}\right)\in {\mathcal{M}}_{{\delta }^{\ast }}.$$

(73)

By using (73) and applying Theorem 13 with $\mathcal{\ell }={\delta }^{\ast },$$\psi ={\varphi }^{1-{\gamma }^{\ast }}$ and $\omega =u^{1-{\gamma }^{\ast }},$ we obtain

$$\left({\left[u^{1-{\gamma }^{\ast }}\right]}^{1-{\left({\delta }^{\ast }\right)}^{\ast }},{\left[{\varphi }^{1-{\gamma }^{\ast }}\right]}^{1-{\left({\delta }^{\ast }\right)}^{\ast }}\right)\in {\mathcal{M}}_{{\left({\delta }^{\ast }\right)}^{\ast }}.$$

By using the fact that ${\left({\delta }^{\ast }\right)}^{\ast }={\displaystyle \frac{{\delta }^{\ast }}{{\delta }^{\ast }-1}}=\delta$ and $1-{\gamma }^{\ast }=-1/(\gamma -1),$ we obtain $\left(u^{{\displaystyle \frac{\delta -1}{\gamma -1}}},{\varphi }^{{\displaystyle \frac{\delta -1}{\gamma -1}}}\right)\in {\mathcal{M}}_{\delta }.$ □

Remark 5.

As a particular instance of Theorem 14, when $\mathbb{T}=\mathbb{R}$, we obtain (7), proved by Neugebauer.

Remark 6.

We mention here that other discrete results can be obtained from the above theorems as special cases. Due to limited space, the details are left to the interested reader.

4. Conclusions

This paper has explored the relationships between Muckenhoupt and Gehring classes, focusing on their inclusion and transition properties across time scales. Overall, this study has made significant strides in understanding the behavior of Muckenhoupt and Gehring classes, providing new insights into their properties and applications. Future research can build on these results to explore more scenarios and further refine the constants and conditions identified here. This work lays a robust foundation for the continued exploration of weighted inequalities and their applications in mathematical analysis and beyond. In addition, we proved the relations between Muckenhoupt and Gehring weights on time scales where any result on the time scales is a unified version of the continuous analog and discrete analog. As special cases, when $\mathbb{T}=\mathbb{R}$, one can obtain the continuous version, and when $\mathbb{T}=\mathbb{N}$, one can obtain the discrete version of any result. Also, we obtained the relation of a bi-Sobolev weight between Muckenhoupt and Gehring classes, and we used two time scales $\mathbb{T}$ and $\tilde{\mathbb{T}}=v\left(\mathbb{T}\right),$ where this idea is new.