1 Introduction
It is well known that the classical Hardy inequality in [
1] reads
$$ \int^{\infty}_{0} \biggl( \frac{1}{x}\int^{x}_{0}f(t)\,dt \biggr)^{p}\,dx \leq \biggl(\frac{p}{p-1} \biggr)^{p}\int ^{\infty}_{0}f^{p}(x)\,dx, $$
(1.1)
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:
$$ \int^{\infty}_{0}\exp \biggl( \frac{1}{x} \int^{x}_{0}\ln f(t)\,dt \biggr)\,dx \leq e\int^{\infty}_{0}f(x)\,dx, $$
(1.2)
for all positive functions
\(f\in L^{1}(\mathbb{R}_{+})\). Another important classical Hardy-Hilbert inequality is closely associated with (
1.1):
$$ \int^{\infty}_{0} \biggl(\int ^{\infty}_{0}\frac{f(x)}{x+y}\,dx \biggr)^{p}\,dy \leq \biggl(\frac{\pi}{\sin(\frac{\pi}{p})} \biggr)^{p}\int^{\infty}_{0}f^{p}(x)\,dx. $$
(1.3)
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]:
$$ \int^{\infty}_{0}\Phi \biggl( \frac{1}{x}\int^{x}_{0}f(t)\,dt \biggr) \frac{dx}{x} \leq\int^{\infty}_{0}\Phi\bigl(f(x) \bigr)\frac{dx}{x}, $$
(1.4)
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].
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
$$ K(x):=\int_{\Omega_{2}}k(x,y)\,d\mu_{2}(y)< \infty, \quad x\in\Omega_{1}. $$
(1.5)
If
f is a real-valued measurable function defined on
\(\Omega_{2}\), the general Hardy-Knopp type operator
\(A_{k}\) is defined by
$$ A_{k}f(x):=\frac{1}{K(x)}\int_{\Omega_{2}}k(x,y)f(y)\,d\mu_{2}(y), \quad x\in\Omega_{1}. $$
(1.6)
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
$$ \biggl(\int_{\Omega_{1}}u(x)\Phi^{\frac{q}{p}} \bigl(A_{k}f(x)\bigr)\,d\mu_{1}(x) \biggr)^{\frac{1}{q}} \leq \biggl(\int_{\Omega_{2}}v(y)\Phi\bigl(f(y)\bigr)\,d\mu_{2}(y) \biggr)^{\frac{1}{p}}, $$
(1.7)
where Φ is a non-negative convex function defined on a convex set
\(I\subseteq\mathbb{R}\) and
$$ v(y)= \biggl[\int_{\Omega_{1}}u(x) \biggl( \frac{k(x,y)}{K(x)} \biggr)^{\frac{q}{p}}\,d\mu_{1}(y) \biggr]^{\frac{p}{q}},\quad y\in\Omega_{2}. $$
(1.8)
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
$$ T^{(r)}_{k}f(x):= \biggl(\frac{1}{K(x)}\int _{\Omega_{2}}k(x,y)f^{r}(y)\,d\mu _{2}(y) \biggr)^{\frac{1}{r}}, \quad x\in\Omega_{1}. $$
(1.9)
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.
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).
Next, we recall the concept of the Orlicz space
\(L_{\Phi(u)}\); see [
13] for details.
Now, we give some basic properties of the Orlicz space
\(L_{\Phi(u)}\) (
cf. [
13]), which will be used to prove our main results.
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].
For convenience later, we also recall the following convexity concepts and Jensen’s inequalities in
n dimensional variables; see [
19] for details.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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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.