In this paper, we construct a new integral operator \(T^{(r)}_{k}\) which generalizes the classical Hardy-Knopp type integral operator \(A_{k}\) by considering the power mean of the non-negative measurable functions. We state and prove a new refined Hardy-Knopp type inequality related to the weighted Lebesgue spaces. As a special case of our results, the refinements of multidimensional Hardy-Knopp type inequalities are obtained. Finally, we also apply a similar idea to prove some new norm inequalities in Orlicz spaces in which the properties of N-functions and superquadratic functions are involved.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.
1 Introduction
It is well known that the classical Hardy inequality in [1] reads
for any \(f\in L^{p}(\mathbb{R}_{+})\) with \(1< p<\infty\); the constant \((\frac{p}{p-1} )^{p}\) is the best possible. After that, the inequality has been tremendously studied and applied in an almost unbelievable way. By replacing f with \(f^{\frac{1}{p}}\) and letting \(p\rightarrow\infty\) in (1.1), we obtain the limiting case which is referred to as Knopp’s inequality:
As we know, since the above inequalities (1.1), (1.2), and (1.3) were established, they have been developed and generalized in different directions; see [1‐4]. It should be particularly emphasized that the above inequalities are various special cases of the following Hardy-Knopp type inequality, which was pointed out by Oguntuase et al. in [5] and Kaijser et al. in [6]:
where Φ is a convex function on \((0,\infty)\). Note that the above inequality (1.4) can be proved by using Jensen’s inequality and Fubini’s theorem, whose idea comes from those papers [7‐9].
Anzeige
Recently, Krulić et al. [10] unified the all above results in an abstract way by introducing the Hardy-Knopp type integral operator \(A_{k}\) in the measure space. Let \((\Omega_{1},\Sigma_{1},\mu _{1})\) and \((\Omega_{2},\Sigma_{2},\mu_{2})\) be measure spaces with positive σ-finite measures, respectively. Suppose that \(u:\Omega _{1}\rightarrow\mathbb{R}\) and \(k:\Omega_{1}\times\Omega_{2}\rightarrow \mathbb{R}\) are two non-negative measurable functions with
Then we have the following modular Hardy type inequality in [10]: for \(0< p\leq q<\infty\) and any measurable functions \(f:\Omega _{2}\rightarrow\mathbb{R}\) such that \(f(\Omega_{2})\subseteq I\) we have
In addition, Čižmešija et al. [11] obtained a class of new sufficient conditions for a weighted modular inequality involving the above operator \(A_{k}\), so that they refined the classical Godunova inequality. Adeleke et al. [3] generalized the classical Hardy-Knopp type inequality to the class of arbitrary non-negative functions bounded from below and above with a convex function multiplied by positive real constants.
Motivated by the idea from [3, 6, 10‐12], in this paper we will establish a generalized Hardy-Knopp type inequality by introducing a new integral operator \(T^{(r)}_{k}\) as follows: for a non-negative measurable function f defined on \(\Omega_{2}\) and a real number \(r>0\), let
Then we will attain a strengthened Hardy-Knopp type inequality which includes all the above results, so as to give a refined version with multidimensional form as corollary.
Moreover, some new norm inequalities in Orlicz spaces are established. The assertion that the Orlicz norm \(\|A_{k}f\|_{\Phi(u)}\) is bounded by a constant K if the N-function Φ satisfies the \(\Delta _{2}\)-condition is proved. Additionally, under the assumption that the composition of two N-functions \(\Phi_{1}\circ\Phi^{-1}_{2}\) is also an N-function, we prove a new norm inequality \(\|A_{k}f\|_{\Phi _{2}(u)}\leq C\|f\|_{\Phi_{1}(u)}\) which may characterize the Hardy-Knopp operators in abstract spaces. Further, we obtain the upper bound of the operator norm \(\|A_{k}\|_{*}\) which implies the continuity of the Hardy-Knopp operator between two different Orlicz spaces. This conclusion is also applied to some useful examples.
Anzeige
The paper is organized as follows. To make the proofs as self-contained as possible, some notations of Orlicz spaces and superquadratic functions are stated in Section 2 and we also present some preliminaries. In Section 3, we prove the generalized Hardy-Knopp type inequalities as regards the operator \(T^{r}_{k}\) and derive the corresponding conclusions in a multidimensional form. The norm inequalities in Orlicz spaces are formulated and discussed in Section 4.
2 Preliminaries
Throughout this paper, all measures are assumed to be positive and all functions are assumed to be measurable. For a real parameter \(p>1\), we denote its conjugate exponent by \(p^{\prime}\) and \(p^{\prime}=\frac {p}{p-1}\), that is, \(\frac{1}{p} + \frac{1}{p^{\prime}} =1\). Moreover, by a weight function we mean a non-negative measurable function on the actual interval or more general set.
Before stating and proving the related norm inequality on the integral operator \(A_{k}\) in Orlicz spaces, let us first describe some properties of the \(\Delta_{2}\)-condition and superquadratic functions involved later. We know that the seminal textbook by Krasnosel’skii et al. [13] contains all the fundamental properties about Orlicz spaces. More recently, the textbooks by Rao and Ren [14] or by Adams and Fournier [15] were concerned with very general situations including the possible pathologies of Young’s functions and the concept of the Orlicz-Sobolev space. Following the notations in [13, 16], we use the class of ‘N-functions’ as defining functions Φ for Orlicz spaces. This class is not as wide as the class of Young’s functions used in [17]. However, N-functions are simpler to deal with and are adequate for our purpose. First, we recall the concepts of an N-function and its complement (see [13, 14] for details).
Definition 2.1
A real-valued function \(\Phi (x)=\int^{x}_{0}\phi(t)\,dt\) is called an N-function if ϕ is a real-valued function defined on \([0,\infty)\) and satisfies the following conditions:
Given any ϕ with the assumptions (a)-(c) above, we let \(\phi^{-1}(s):=\sup\{t>0 : \phi(t)\leq s\} \) be a right continuous inverse function of ϕ. Denoting
$$ \Psi(x)=\int^{x}_{0} \phi^{-1}(s)\,ds, $$
(2.1)
then \(\Psi(x)\) is called the complementary function of \(\Phi(x)\). Note that it is an N-function itself.
Definition 2.3
An N-function Φ is said to satisfy the \(\Delta_{2}\)-condition (globally) if there is a positive constant C such that
$$ \Phi(2t)\leq C\Phi(t) \quad\mbox{for all } t\in\mathbb{R}_{+}. $$
(2.2)
Next, we recall the concept of the Orlicz space \(L_{\Phi(u)}\); see [13] for details.
Definition 2.4
Let \(u(x)\) be a weight function and \(\Phi(x)\) be an N-function on a σ-finite measure space \((\Omega,\Sigma,\mu)\). The Orlicz space \(L_{\Phi(u)}\) consists of all non-negative measurable functions f (module equivalent almost everywhere) with
If anN-function satisfies the\(\triangle_{2}\)-condition, then there are constantsαandβwith\(1\leq\beta\leq\alpha <\infty\)such that\(s^{\beta}\Phi(t)\leq\Phi(st)\leq s^{\alpha}\Phi(t)\)when\(s\geq1\)and\(t\geq0\), and\(s^{\alpha}\Phi(t)\leq\Phi(st)\leq s^{\beta}\Phi(t)\)when\(0\leq s\leq1\)and\(t\geq0\).
In fact, the verification of propositions above can be found in pp.23-26 in [13] and pp.59-62 in [14]. Another main tool in the proofs is to use superquadratic functions and a generalization of Jensen’s inequality given by Abramovich et al. in [18].
Definition 2.7
A function \(f:[0,\infty)\rightarrow \mathbb{R}\) is superquadratic provided that for each \(x\geq0\) there exists a constant \(C_{x}\in\mathbb{R}\) such that
holds for all probability measuresμand all non-negativeμ-integrable functionsfif and only ifϕis superquadratic.
For convenience later, we also recall the following convexity concepts and Jensen’s inequalities in n dimensional variables; see [19] for details.
Definition 2.10
A function \(\Phi: D\rightarrow\mathbb{R}\) for which D is a convex set of \(\mathbb{R}^{n}\) is said to be convex on D if for all \(\mathbf {x}\in\mathbb{R}^{n}\), \(\mathbf{y}\in\mathbb{R}^{n}\) and \(\lambda\in [0,1]\) we have
Let\(D\subseteq\mathbb{R}^{n}\)be convex and open, \(\phi:D\rightarrow \mathbb{R}\)be twice differentiable. Thenϕis convex onDif and only if its Hessian matrix\((\frac{\partial^{2}f}{\partial x_{i}\,\partial y_{j}}(\mathbf{x}))_{n\times n}\)is positive semi-definite for all\(\mathbf{x}\in D\subset\mathbb{R}^{n}\).
Lemma 2.12
(n-variable Jensen’s inequality)
Let\(p(x)\)be a non-negative continuous function on\(I=[a, b]\subseteq\mathbb{R}\)such that\(\int_{I}p(t)\,dt>0\). If\(f_{i}:I\rightarrow[m_{i},M_{i}]\)is a real-valued continuous function for each\(i\in{1,2,\ldots,n}\)on\([a,b]\)and Φ is convex on\(\Delta_{n}=\prod^{n}_{i=1}[m_{i},M_{i}]\subseteq\mathbb {R}^{n}\), then we have
The Orlicz spaces really extend the usual \(L_{p}\) spaces. In fact, the function \(\Phi(x)=x^{p}\) entering the definition of \(L_{p}\) is replaced by a more general convex N-function \(\Phi(x)\). The Propositions 2.5 and 2.6 are crucial for the proofs of the norm inequalities in Orlicz space. The concepts of a superquadratic function and Jensen’s inequality in n variables are used to prove the generalized Hardy-Knopp type inequalities.
3 Generalizations for Hardy-Knopp type operators in weighted Lebesgue spaces
Our analysis starts with a powerful sufficient condition for a new inequality related to the operator \(T^{(r)}_{k}\). As its conclusion, a new norm inequality in weighted Lebesgue space is obtained. Now, we point out the monotonicity of \(T^{(r)}_{k}(x)\) on r when x is fixed.
Lemma 3.1
Fix\(x\in\Omega_{1}\)and define\(M(r)=T^{(r)}_{k}f(x)\)for each\(r>0\), then the function\(M:\mathbb{R}^{+}\rightarrow[0,\infty)\)is non-decreasing.
Proof
(I) First the case of \(0< r<1\). Let \(p=\frac{1}{r}\). By Hölder’s inequality we have
Therefore, for any \(0< s_{1}< s_{2}\leq1\) let \(r=\frac{s_{1}}{s_{2}}<1\). Then, by replacing f with \(f^{s_{2}}\) one deduces \(M(s_{1})\leq M( s_{2})\leq M(1)\).
similar to the case above, one gets \(M(1)\leq M(r_{1})\leq M(r_{2})\) for any \(1\leq r_{1}< r_{2}\). This completes the proof. □
Theorem 3.2
For\(1<\beta\leq q\), \(0< p\leq\beta\), and\(0< r\leq1\), let\((\Omega_{1},\Sigma_{1},\mu_{1})\)and\((\Omega _{2},\Sigma_{2},\mu_{2})\)be measure spaces with positiveσ-finite measures, ube a positive weight function on\(\Omega_{1}\), vbe a positive weight function on\(\Omega_{2}\)and\(k:\Omega _{1}\times\Omega_{2}\rightarrow\mathbb{R}\)be a non-negative measurable function. Suppose that\(K:\Omega_{1}\rightarrow\mathbb{R}\)is as in (1.5) so that the function\(x\rightarrow u(x) (\frac {k(x,y)}{K(x)} )^{q}\)is integrable on\(\Omega_{1}\)for each fixed\(y\in\Omega_{2}\). Assume Φ is a non-negative increasing convex function on an interval\(I\subseteq[0,\infty)\)and there is a positive measurable function\(w: \Omega_{2}\rightarrow\mathbb{R}\)such that
is valid for all measurable functions\(f:\Omega_{2}\rightarrow I\subseteq\mathbb{R}\)and\(T^{(r)}_{k}\)is defined by (1.9).
Proof
Denote \(g(y)=v(y)\Phi^{p}(f(y))\), then \(\Phi(f(y))=g^{\frac{1}{p}}(y)v^{-\frac{1}{p}}(y)\). First, by Hölder’s inequality, we have the following estimate:
Notice that \(T^{(r)}_{k}f(x)\in I\), \(x\in\Omega_{1}\) and inequality (3.2). Applying Jensen’s inequality, Minkowski’s inequality as well as monotonicity of the convex function \(\Phi(x)\) on I and \(M(r)\) on \((0,\infty)\), by Lemma 3.1 for any measurable function \(f:\Omega_{2}\rightarrow I \) we get a series of inequalities:
Theorem 3.2 is our main result in the first part of Section 3. Enlightened by the work of Lour [20], we will derive a series of examples based on it, including several averaging operators and integral transforms in the weighted Lebesgue spaces. Before stating their descriptions we need to give some notations.
First, for \(\mathbf{x}=(x_{1},\ldots,x_{n})\in\mathbb{R}^{n}_{+}\) and \(\mathbf{y}=(y_{1},\ldots,y_{n})\in\mathbb{R}^{n}_{+}\) we denote \(\frac{\mathbf{y}}{\mathbf{x}}= (\frac{y_{1}}{x_{1}},\ldots,\frac {y_{n}}{x_{n}} ) \), \(\mathbf{x}^{\mathbf{y}}={x_{1}}^{y_{1}}\cdots {x_{n}}^{y_{n}}\); in particular, \(\mathbf{x}^{\mathbf{1}}=\prod_{i=1}^{n}x_{i}\). Additionally, let \(S=\{\mathbf{x}\in\mathbb{R}^{n}:|\mathbf{x}|=1\}\) be the unit sphere in \(\mathbb{R}^{n}\) with the standard Euclidean norm \(|\mathbf{x}|\) of x, and \(E\subseteq\mathbb{R}^{n}\) be a spherical cone with \(E=\{\mathbf{x}\in\mathbb{R}^{n}:\mathbf{x}=r \mathbf{b},0< r<\infty,\mathbf{b}\in A \}\) for any measurable subset A of S. Suppose that \(\Omega_{1}=\Omega_{2}=E\) in Theorem 3.2, \(d\mu _{1}(\mathbf{x})=d\mathbf{x} \), and \(d\mu_{2}(\mathbf{y})=d\mathbf{y}\). For all non-negative functions f on E, we list the following examples with the averaging integral operators.
Example 3.5
(Averaging operator of Laplace type) Consider the case that \(k(\mathbf {x},\mathbf{y})=|\mathbf{x}|^{n}e^{-|\mathbf{x}||\mathbf{y}|}\), \(1<\beta =p\leq q\), \(r=1\), and \(w(\mathbf{y})\equiv1\). Then we have \(K(\mathbf {x})=\int_{E}k(\mathbf{x},\mathbf{y})\,d\mathbf{y}=|A|(n-1)!\), and consequently
Consider the case that \(k(\mathbf{x},\mathbf{y})=(|\mathbf{x}|+|\mathbf{y}|)^{-\rho}\) (\(\rho >0\)), \(1<\beta=p\leq q\), \(r=1\), and \(w(\mathbf{y})\equiv1\). Then we have \(K(\mathbf{x})=\int_{E}k(\mathbf{x},\mathbf{y})\,d\mathbf {y}=|A|B(\rho-n,n)|\mathbf{x}|^{-\rho+n}\) with a Beta function \(B(\cdot ,\cdot)\), and consequently
Finally, consider the case that \(k(\mathbf{x},\mathbf{y})=|\mathbf{y}|(e^{|\mathbf{x}||\mathbf {y}|}-1)^{-1}\), \(1<\beta=p\leq q\), \(r=1\), and \(w(\mathbf{y})\equiv1\). Then we attain \(K(\mathbf{x})=\int_{E}k(\mathbf{x},\mathbf{y})\,d\mathbf {y}=|A|l_{1}|\mathbf{x}|^{-n-1}\) with \(l_{r}=\int^{\infty }_{0}t^{r+n-1}(e^{t}-1)^{-r}\,dt\), \(r>0\), and consequently
Indeed, the above conclusions can be reformulated with particular convex functions such as power or exponential functions, especially with the N-function \(\Phi=\int^{x}_{0}\phi(t)\,dt\). This leads to multidimensional analogs of corollaries and examples by way of the previous theorems.
Now, we are in the position to consider the superquadratic function Φ. On the basis of a refinement of Jensen’s inequality (2.6), we can refine the inequality (3.4) above with respect to the operator \(A_{k}\). Therefore, we get the following theorem as the second part of this section.
Theorem 3.8
Let\(t\in[1,\infty)\), \((\Omega_{1},\Sigma_{1},\mu_{1})\), and\((\Omega _{2},\Sigma_{2},\mu_{2})\)be measure spaces with positiveσ-finite measures, ube a weight function on\(\Omega_{1}\), and\(k:\Omega_{1}\times\Omega_{2}\rightarrow\mathbb{R}\)be a non-negative measurable function. Suppose that\(K:\Omega_{1}\rightarrow\mathbb{R}\)is as in (1.5), that the function\(x\rightarrow u(x) (\frac{k(x,y)}{K(x)} )^{t}\)is integrable on\(\Omega_{1}\)for each fixed\(y\in\Omega_{2}\), and that the weight functionvis defined by
Observe that for \(t=1\) the inequality (3.5) may result from Theorem 5.1 in [3]. Moreover, the above conclusions can be rewritten by a special convex functions such as a power function, an exponential function, and an N-function \(\Phi=\int^{x}_{0}\phi(t)\,dt\) with a continuous function ϕ such that \(\frac {\phi(t)}{t}\) is non-decreasing or \(\phi(t)\) is superadditive on \([0,\infty)\), since the N-function Φ is a superquadratic function by Lemma 2.8.
Let \(\Omega_{1}=\Omega_{2}=\mathbb{R}^{n}_{+}\), \(d\mu_{1}(\mathbf {x})=d\mathbf{x}\), \(d\mu_{2}(\mathbf{y})=d\mathbf{y}\), and the kernel k in (1.5) be as the form \(k(\mathbf{x},\mathbf{y})=h(\frac {\mathbf{y}}{\mathbf{x}})\), where \(h:\mathbb{R}^{n}_{+}\rightarrow \mathbb{R}\) is a non-negative measurable function. If \(u(\mathbf{x})\) and \(v(\mathbf{y})\) are substituted by \(\frac{u(\mathbf{x})}{\mathbf {x}^{\mathbf{1}}}\) and \(\frac{w(\mathbf{y})}{\mathbf{y}^{\mathbf {1}}}\), recall that \(\mathbf{x}^{\mathbf{1}} =\prod_{1}^{n}x_{i}\) above, and by Theorem 3.8 we have the following corollary.
Corollary 3.10
Let\(t\in[1,\infty)\), andube a weight function on\(\mathbb {R}^{n}_{+}\)such that\(H(\mathbf{x})=\mathbf{x}^{\mathbf{1}} \int_{\mathbb{R}^{n}_{+}}h(\mathbf{y})\,d\mathbf{y}\)satisfies\(0< H(\mathbf {x})<\infty\)for all\(\mathbf{x}\in\mathbb{R}^{n}_{+}\)and that the function\(\mathbf{x}\rightarrow u(\mathbf{x}) (\frac{\frac{\mathbf {y}}{\mathbf{x}}}{H(\mathbf{x})} )^{t}\)is integrable on\(\mathbb {R}^{n}_{+}\)for each fixed\(\mathbf{y}\in\mathbb{R}^{n}_{+}\). The weight functionwis defined by
with any non-negative measurable functions\(f:\mathbb {R}^{n}_{+}\rightarrow\mathbb{R}\)with values inIand\(A_{k}f\)as in (1.9).
In virtue of the above corollary, one can deduce a generalization of Godunova’s inequality in [21]. The following result is based on Lemma 2.12 and its proof is similar to the proof of Theorem 3.2 above.
Theorem 3.11
Suppose that\(t\in[1,\infty)\), \(I_{1}=[a,b]\subseteq\mathbb{R}\), \(I_{2}=[c,d]\subseteq\mathbb{R}\), and\(p(x)\)is as in Lemma 2.12. Letube a weight function on\(I_{1}\), \(k:I_{1}\times I_{2}\rightarrow\mathbb{R}\)be a non-negative measurable function, \(K(x)=\int_{I_{2}}k(x,y)d(y)>0\), \(x\in I_{1}\), the function\(x\rightarrow u(x) (\frac{k(x,y)}{K(x)} )^{t}\)be integrable on\(I_{1}\)for each fixed\(y\in I_{2}\), and the weight functionvbe defined by
with any non-negative measurable functions\(f_{i}:I_{2}\rightarrow [m_{i},M_{i}]\). Further, inequality (3.9) holds in the reversed direction if Φ is a non-negative concave function and\(t\in(0,1]\).
Proof
By using Lemma 2.12, Minkowski’s inequality, and Fubini’s theorem, we observe that
Note that if Φ is a non-negative concave function and \(t\in(0,1]\), it is completed by reversing the inequality sign in (3.10). □
In the process of proving Theorem 3.9, assume that \(\Phi:H\rightarrow\mathbb{R}\) is a twice differentiable function on an open convex set H which contains the compact set \(\Delta_{n}= \prod^{n}_{i=1}[m_{i},M_{i}]\) such that its Hessian matrix \((\frac{\partial ^{2}f}{\partial x_{i}\,\partial y_{j}}(\mathbf{x}))_{n\times n}\) is positive semi-definite for all \(\mathbf{x}\in\mathbb{R}^{n}\). Then according to Lemma 2.11 Φ is a convex function on H and inequality (3.9) holds for any non-negative measurable functions \(f_{i}:I_{2}\rightarrow [m_{i},M_{i}]\). As a special case, the following corollary is derived.
Corollary 3.12
Under the same conditions as Theorem 3.11, let\(\Phi (x_{1},x_{2},\ldots,x_{n})=\mathbf{x}^{T}\mathbf{A}\mathbf{x}\)be a quadratic form innindependent variables, with associated symmetric matrixAwhich is positive semi-definite. Then we have the following inequality:
Reversely, the inequality (3.11) holds in the reversed direction ifAis a negative semi-definite and\(t\in(0,1]\).
4 The norm inequalities in Orlicz spaces
In this section, by combining some basic properties of Orlicz spaces and the arguments of the preceding sections, we establish some new norm inequalities which may characterize the Hardy-Knopp type operators in abstract spaces.
Theorem 4.1
Suppose that\((\Omega_{1},\Sigma_{1},\mu_{1})\)and\((\Omega_{2},\Sigma _{2},\mu_{2})\), \(u(x)\), \(k(x,y)\), and\(K(x)\)are as in Theorem 3.2. Let the function\(x\rightarrow\frac{u(x)k(x,y)}{K(x)} \)be an integrable function on\(\Omega_{1}\)for each fixed\(y\in\Omega _{2}\), and the weight functionωbe defined by
If Φ is aN-function satisfying the\(\Delta_{2}\)-condition, then there are constantsαandβwith\(1\leq\beta\leq\alpha <\infty\)such that the following norm inequality holds:
Moreover, if\(\phi(t)\)is a continuous function such that\(\frac{\phi (t)}{t}\)is non-decreasing or\(\phi(t)\)is superadditive on\([0,\infty )\), then the following refined normal inequality holds:
By Proposition 2.6, there are constants α and β with \(1\leq\beta\leq\alpha<\infty\) such that \(s^{\beta}\Phi (t)\leq\Phi(st)\leq s^{\alpha}\Phi(t)\) when \(s\geq1\) and \(t\geq0\), and \(s^{\alpha}\Phi(t)\leq\Phi(st)\leq s^{\beta}\Phi(t)\) when \(0\leq s\leq 1\) and \(t\geq0\).
Case I. If \(\lambda\leq1\), let \(s=\frac{1}{\lambda}\). Then it follows that
First, we consider the case of \(M_{f}<1\), by letting \(\lambda=1\) in (4.1) then we have \(\|A_{k}f\|_{\Phi(u)}\leq1\). Hence, it is sufficient to consider the case that \(\lambda\leq1\). If \(\lambda\geq M_{f}^{\frac{1}{\alpha}}\) then
due to inequality (4.1). Consequently, \(\|A_{k}f\|_{\Phi(u)}\leq M_{f}^{\frac{1}{\alpha}}<1\) by the definition of the Luxemburg norm. Now, we are in a position to consider another case of \(M_{f}\geq1\). If \(\| A_{k}f\|_{\Phi(u)}\geq1\), we have the norm inequality \(\|A_{k}f\|_{\Phi (u)}\leq M_{f}^{\frac{1}{\beta}}\) due to inequality (4.2). Therefore, \(\|A_{k}f\|_{\Phi(u)}\leq\max (1,M_{f}^{\frac{1}{\beta}})= M_{f}^{\frac{1}{\beta}}\), which completes the first part of this theorem.
Finally, let \(\phi(t)\) be a continuous function such that \(\frac{\phi (t)}{t}\) is a non-decreasing or \(\phi(t)\) is superadditive on \([0,\infty )\). Then \(\Phi(x)\) is superquadratic by Lemma 2.8, and consequently we employ the refinement Jensen’s inequality as follows (cf. Lemma 2.6):
which completes the rest of the proof by way of repeating the above discussion (4.1) and (4.2). □
Theorem 4.2
Let\((\Omega,\Sigma,\mu)\)be a measure space with positiveσ-finite measure, ube a weight function on Ω, and\(k:\Omega \times\Omega\rightarrow\mathbb{R}\)be a non-negative measurable function. Let the weight functionvbe defined by
Suppose that\(\Phi_{1}\)and\(\Phi_{2}\)areN-functions, where\(\Phi _{2}\)satisfies the\(\Delta_{2}\)-condition, so that\(\Phi_{1}\circ\Phi ^{-1}_{2}\)is anN-function. The complementary function of\(\Phi_{1}\circ\Phi^{-1}_{2}\)is denoted by Ψ. If\(\|\frac{v}{u}\|_{\Psi (u)}<\infty\), then there exists a constantCsuch that the following norm inequality:
holds for any non-negative functionf, then we have\(\|\frac{u}{v}\| _{\Psi(v)}<\infty\).
Proof
Without loss of generality, to prove the first statement we may assume that \(\|f\|_{\Phi_{1}(u)}=1\), which implies \(\|\Phi_{2}(f)\| _{\Phi_{1}\circ\Phi^{-1}_{2}(u)}\leq1\). By Hölder’s inequality in Orlicz spaces (2.6) it yields
Now we take \(C=\max(1,2\|\frac{v}{u}\|_{\Psi(u)})\), then one deduces \(\int_{\Omega}u(t)\Phi_{2}(\frac{A_{k}f(t)}{C})\,d\mu(t)\leq1\). This proves (4.3).
Conversely, since the Luxemburg norm is dominated by the Orlicz norm itself, it suffices to show that
Note that \(\Phi_{2}\) satisfies the \(\Delta_{2}\)-condition, Proposition 2.5, and hence the inequality \(\int_{\Omega}f(x)u(x)\,d\mu (x)\leq C_{1}\) holds for some constant \(C_{1}\). Then we have \(\|\frac{u}{v}\|_{\Psi(v)}\leq C_{1}<\infty\). □
Corollary 4.3
Suppose that\((\Omega,\Sigma,\mu)\), u, k, andvare as in Theorem 4.2. Let\(\Phi_{1}\)and\(\Phi_{2}\)beN-functions such that\(\Phi_{2}\)satisfies the\(\Delta_{2}\)-condition and\(\Phi_{1}\circ\Phi ^{-1}_{2}\)is anN-function. Denote by Ψ the complementary function of\(\Phi_{1}\circ\Phi^{-1}_{2}\). If the inequality\(\|\frac{v}{u}\|_{\Psi(u)}<\infty\)holds, then the linear operator\(A_{k}:L_{\Phi _{1}(u)}\rightarrow L_{\Phi_{2}(u)}\)is continuous and we have the following estimate:
According to the proof of inequality (4.3), we conclude that \(\frac{\|A_{k}f\|_{\Phi_{2}(u)}}{\|f\|_{\Phi _{1}(u)}}\leq\max(1, 2\|\frac{v}{u}\|_{\Psi(u)})\) holds for any non-negative function \(f(x)\). Then we have \(\|A_{k}\|_{*}\leq\max(1, 2\| \frac{v}{u}\|_{\Psi(u)})\) and hence \(A_{k}\) is continuous. □
Let \(\Phi_{1}(x)=\frac{1}{p} x^{p}\) and \(\Phi_{2}(x)=\frac{1}{q} x^{q}\) in Theorem 4.2, where \(1< q< p<\infty\). It is clear that \(\Phi _{1}\), \(\Phi_{2}\) are N-functions satisfying the \(\Delta_{2}\)-condition, and \(\Phi_{1}\circ\Phi^{-1}_{2}=\int^{x}_{0} q^{\frac{p}{q} -1}t^{\frac{p}{q} -1}\,dt\) is also an N-function. Furthermore, the complementary N-function of Φ is calculated by \(\Psi(x)=\frac{p-q}{pq}x^{\frac{p}{p-q}}\). Then we have the following conclusion.
Corollary 4.4
Let\((\Omega,\Sigma,\mu)\)be a measure space with positiveσ-finite measure, \(u(x)\), \(k(x,y)\), \(v(x)\)be as in Theorem 4.2. Suppose that\(\Phi_{1}(x)=\frac{1}{p} x^{p}\)and\(\Phi_{2}(x)=\frac{1}{q} x^{q}\)where\(1< q< p<\infty\). Then there exists a constantCsuch that the norm inequality holds:
for any non-negative functionfand\(\|\frac{v}{u}\|_{\Psi(u)}<\infty\)with\(\Psi(x)=\frac{p-q}{pq}x^{\frac{p}{p-q}}\). Moreover, if there exists a constantCsuch that the following inequality is valid:
for any non-negative functionf, then\(\|\frac{u}{v}\|_{\Psi(v)}<\infty\)holds.
Proposition 4.5
Suppose that\((\Omega_{1},\Sigma_{1},\mu_{1})\)and\((\Omega_{2},\Sigma _{2},\mu_{2})\)areσ-finite measure spaces and thatTis a linear operator which maps any non-negative measurable functions on\(\Omega_{2}\)to some non-negative measurable functions on\(\Omega_{1}\). Let\(\Phi(x)\)be anN-function, then
holds for all\(\epsilon>0\)withCindependent ofϵ (see Bloom’s paper in [22]).
Corollary 4.6
Assume that the assumptions in Proposition 4.5are satisfied. Let\(T^{(r)}_{k}\)be the linear operator defined in (1.9) and\(\Phi(x)\)be anN-function, then
It is clear that \(\Phi(x)=\int^{x}_{0}\phi(t)\,dt\) in which \(\phi (t)=e^{t}-1\) is an N-function. Then, by applying Proposition 4.5 to the linear operator \(T^{(r)}_{k}\) and replacing \(f(x)\) by \(\ln f(x)\), we obtain the following important example.
Example 4.7
Assume that the assumptions in Proposition 4.5 are satisfied and that \(f(x)\) is a measurable function such that \(f(x)\geq 1\) for all \(x\in\Omega_{2}\). Then the following inequality:
holds for all \(\epsilon>0\) with C independent of ϵ.
Acknowledgements
The authors would like to thank the referees for their valuable comments and suggestions which essentially improved the quality of this paper. This work is partially supported by the National Science Foundation of China grant 11371050 and Project No. 14017002 supported by National Training Program of Innovation and Entrepreneurship for Undergraduates.
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed to each part of this work equally, and they all read and approved the final manuscript.