On refined Hardy-Knopp type inequalities in Orlicz spaces and some related results
- Jinjie Zhang^{1} and
- Shenzhou Zheng^{1}Email author
https://doi.org/10.1186/s13660-015-0686-4
© Zhang and Zheng 2015
Received: 15 January 2015
Accepted: 8 May 2015
Published: 28 May 2015
Abstract
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.
Keywords
Hardy-Knopp type inequalities Orlicz spaces norm inequalities convex functions superquadratic functionsMSC
26D10 26D15 46E301 Introduction
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.
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.
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)
\(\phi(0)=0\), \(\phi(t)>0\) whenever \(t>0\), \(\lim_{t\rightarrow \infty}\phi(t)= \infty\);
- (b)
\(\phi(t)\) is non-decreasing;
- (c)
\(\phi(t)\) is right continuous.
Definition 2.2
Definition 2.3
Next, we recall the concept of the Orlicz space \(L_{\Phi(u)}\); see [13] for details.
Definition 2.4
- (a)the Luxemburg norm$$\|f\|_{\Phi(u)}:=\inf \biggl\{ \lambda>0 : \int_{\Omega}\Phi \biggl(\frac {f(x)}{\lambda} \biggr)u(x)\,d\mu(x)\leq1 \biggr\} < +\infty. $$
- (b)The Orlicz norm$$\|f\|^{\prime}_{\Phi(u)}:=\sup \biggl\{ \int_{\Omega}f(x)g(x)u(x)\,d\mu(x) : \int_{\Omega}\Psi\bigl(g(x)\bigr)u(x)\,d\mu(x)\leq1 \biggr\} < +\infty, $$
Now, we give some basic properties of the Orlicz space \(L_{\Phi(u)}\) (cf. [13]), which will be used to prove our main results.
Proposition 2.5
- (a)
\(\Phi(\alpha x)\leq\alpha\Phi(x)\) for \(0\leq\alpha\leq1\).
- (b)
\(\alpha\Phi(x)\leq\Phi(\alpha x)\) for \(1\leq\alpha<\infty\).
- (c)
For any measurable function \(f\ge0\), \(\|f\|_{\Phi(u)}\leq1\) if and only if \(\int_{\Omega}\Phi(f(x))u(x)\,d\mu(x)\leq1\).
Proposition 2.6
- (a)\(L_{\Phi}\) is a Banach space such that the Luxemburg and Orlicz norms are equivalent; indeed,$$ \|f\|_{\Phi(u)}\leq\|f\|^{\prime}_{\Phi(u)}\leq2 \|f\|_{\Phi(u)}. $$(2.3)
- (b)Hölder’s inequality:$$ \int_{\Omega}f(x)g(x)u(x)\,d\mu(x)\leq\|f \|_{\Phi(u)}\|g\|^{\prime}_{\Psi(u)}. $$(2.4)
- (c)
If an N-function satisfies the \(\triangle_{2}\)-condition, then 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 \(0\leq s\leq1\) 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
Lemma 2.8
A function \(\phi:[0,\infty]\rightarrow\mathbb{R}\) is continuously differentiable and \(\phi(0)\leq0\). If \(\phi^{\prime}\) is superadditive or \(\frac{\phi^{\prime}(x)}{x}\) is non-decreasing, then ϕ is superquadratic.
Lemma 2.9
(Refinement of Jensen’s inequality)
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
Lemma 2.11
Let \(D\subseteq\mathbb{R}^{n}\) be convex and open, \(\phi:D\rightarrow \mathbb{R}\) be twice differentiable. Then ϕ is convex on D if 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)
Remark 2.13
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
Theorem 3.2
Proof
Remark 3.3
Remark 3.4
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
Example 3.6
(Averaging operator of Stieltjes type)
Example 3.7
(Averaging operator of Lambert type)
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
Proof
Remark 3.9
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
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
Proof
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
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
Proof
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 \(t\geq0\).
Theorem 4.2
Proof
Corollary 4.3
Proof
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
Proposition 4.5
Corollary 4.6
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
Declarations
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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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