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On refined HardyKnopp type inequalities in Orlicz spaces and some related results
Journal of Inequalities and Applications volume 2015, Article number: 169 (2015)
Abstract
In this paper, we construct a new integral operator \(T^{(r)}_{k}\) which generalizes the classical HardyKnopp type integral operator \(A_{k}\) by considering the power mean of the nonnegative measurable functions. We state and prove a new refined HardyKnopp type inequality related to the weighted Lebesgue spaces. As a special case of our results, the refinements of multidimensional HardyKnopp 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 Nfunctions and superquadratic functions are involved.
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 \((\frac{p}{p1} )^{p}\) 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:
for all positive functions \(f\in L^{1}(\mathbb{R}_{+})\). Another important classical HardyHilbert inequality is closely associated with (1.1):
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 HardyKnopp 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].
Recently, Krulić et al. [10] unified the all above results in an abstract way by introducing the HardyKnopp 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 σfinite measures, respectively. Suppose that \(u:\Omega _{1}\rightarrow\mathbb{R}\) and \(k:\Omega_{1}\times\Omega_{2}\rightarrow \mathbb{R}\) are two nonnegative measurable functions with
If f is a realvalued measurable function defined on \(\Omega_{2}\), the general HardyKnopp type operator \(A_{k}\) is defined by
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
where Φ is a nonnegative convex function defined on a convex set \(I\subseteq\mathbb{R}\) and
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 HardyKnopp type inequality to the class of arbitrary nonnegative 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 HardyKnopp type inequality by introducing a new integral operator \(T^{(r)}_{k}\) as follows: for a nonnegative measurable function f defined on \(\Omega_{2}\) and a real number \(r>0\), let
Then we will attain a strengthened HardyKnopp 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 Nfunction Φ satisfies the \(\Delta _{2}\)condition is proved. Additionally, under the assumption that the composition of two Nfunctions \(\Phi_{1}\circ\Phi^{1}_{2}\) is also an Nfunction, we prove a new norm inequality \(\A_{k}f\_{\Phi _{2}(u)}\leq C\f\_{\Phi_{1}(u)}\) which may characterize the HardyKnopp operators in abstract spaces. Further, we obtain the upper bound of the operator norm \(\A_{k}\_{*}\) which implies the continuity of the HardyKnopp 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 selfcontained 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 HardyKnopp 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 \(p^{\prime}=\frac {p}{p1}\), that is, \(\frac{1}{p} + \frac{1}{p^{\prime}} =1\). Moreover, by a weight function we mean a nonnegative 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 OrliczSobolev space. Following the notations in [13, 16], we use the class of ‘Nfunctions’ as defining functions Φ for Orlicz spaces. This class is not as wide as the class of Young’s functions used in [17]. However, Nfunctions are simpler to deal with and are adequate for our purpose. First, we recall the concepts of an Nfunction and its complement (see [13, 14] for details).
Definition 2.1
A realvalued function \(\Phi (x)=\int^{x}_{0}\phi(t)\,dt\) is called an Nfunction if ϕ is a realvalued function defined on \([0,\infty)\) and satisfies the following conditions:

(a)
\(\phi(0)=0\), \(\phi(t)>0\) whenever \(t>0\), \(\lim_{t\rightarrow \infty}\phi(t)= \infty\);

(b)
\(\phi(t)\) is nondecreasing;

(c)
\(\phi(t)\) is right continuous.
Definition 2.2
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
then \(\Psi(x)\) is called the complementary function of \(\Phi(x)\). Note that it is an Nfunction itself.
Definition 2.3
An Nfunction Φ is said to satisfy the \(\Delta_{2}\)condition (globally) if there is a positive constant C such that
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 Nfunction on a σfinite measure space \((\Omega,\Sigma,\mu)\). The Orlicz space \(L_{\Phi(u)}\) consists of all nonnegative measurable functions f (module equivalent almost everywhere) with

(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
$$\f\^{\prime}_{\Phi(u)}:=\sup \biggl\{ \int_{\Omega}f(x)g(x)u(x)\,d\mu(x) : \int_{\Omega}\Psi\bigl(g(x)\bigr)u(x)\,d\mu(x)\leq1 \biggr\} < +\infty, $$
where Ψ is the complementary function of Φ.
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
Let Φ be an Nfunction with \(\Phi(0)=0\). Then

(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
Let Φ be an Nfunction and Ψ be the complementary of Φ. Then we have

(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 Nfunction 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.2326 in [13] and pp.5962 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
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 nondecreasing, then ϕ is superquadratic.
Lemma 2.9
(Refinement of Jensen’s inequality)
Let \((\Omega,\mu)\) be a probability measure space. The inequality
holds for all probability measures μ and all nonnegative μintegrable functions f if and only if ϕ is 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
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 semidefinite for all \(\mathbf{x}\in D\subset\mathbb{R}^{n}\).
Lemma 2.12
(nvariable Jensen’s inequality)
Let \(p(x)\) be a nonnegative 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 realvalued 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
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 Nfunction \(\Phi(x)\). 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 HardyKnopp type inequalities.
3 Generalizations for HardyKnopp 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 nondecreasing.
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 \(M(s_{1})\leq M( s_{2})\leq M(1)\).
(II) Next the case of \(r>1\). Since
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 positive σfinite measures, u be a positive weight function on \(\Omega_{1}\), v be a positive weight function on \(\Omega_{2}\) and \(k:\Omega _{1}\times\Omega_{2}\rightarrow\mathbb{R}\) be a nonnegative 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 nonnegative 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
where
Then the following inequality:
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:
Immediately, it yields (3.1) from (3.3). □
Remark 3.3
Suppose that the weight function v is defined by
in Theorem 3.2. Then we can proceed to the inequalities (3.3) in the following way:
where \(q\geq1\). Let \(r=1\) and replace q with \(\frac{p}{q}\) in (3.4); we also get the modular Hardy type inequality (1.7).
Remark 3.4
If K is the best possible constant in (3.1), then
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 nonnegative 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 \(K(\mathbf {x})=\int_{E}k(\mathbf{x},\mathbf{y})\,d\mathbf{y}=A(n1)!\), and consequently
as an averaging operator of Laplace type. According to Theorem 3.2 it follows that
for any nonnegative measurable functions \(f:E\rightarrow I\), where
Example 3.6
(Averaging operator of Stieltjes type)
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 \(K(\mathbf{x})=\int_{E}k(\mathbf{x},\mathbf{y})\,d\mathbf {y}=AB(\rhon,n)\mathbf{x}^{\rho+n}\) with a Beta function \(B(\cdot ,\cdot)\), and consequently
as an averaging operator of Stieltjes type. By Theorem 3.2 we obtain
with any nonnegative measurable functions \(f:E\rightarrow I\), where
Example 3.7
(Averaging operator of Lambert type)
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 \(K(\mathbf{x})=\int_{E}k(\mathbf{x},\mathbf{y})\,d\mathbf {y}=Al_{1}\mathbf{x}^{n1}\) with \(l_{r}=\int^{\infty }_{0}t^{r+n1}(e^{t}1)^{r}\,dt\), \(r>0\), and consequently
as an averaging operator of Stieltjes type. In terms of Theorem 3.2 we deduce
where \(f: E\rightarrow I\) is a nonnegative measurable function, and
Indeed, the above conclusions can be reformulated with particular convex functions such as power or exponential functions, especially with the Nfunction \(\Phi=\int^{x}_{0}\phi(t)\,dt\). 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 positive σfinite measures, u be a weight function on \(\Omega_{1}\), and \(k:\Omega_{1}\times\Omega_{2}\rightarrow\mathbb{R}\) be a nonnegative 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 function v is defined by
If Φ is a nonnegative superquadratic function on an interval \(I\subseteq[0,\infty)\), then we have
for any nonnegative measurable functions \(f:\Omega_{2}\rightarrow I\subseteq\mathbb{R}\), where \(A_{k}f\) defined on \(\Omega_{1}\) by (1.6).
Proof
According to Lemma 2.9 it yields
As a consequence of Bernoulli’s inequality, we derive
Multiplying (3.6) by \(u(x)\) and integrating it over \(\Omega_{1}\), by Minkowski’s inequality, it follows that
So, the inequality (3.5) follows from (3.7). □
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 Nfunction \(\Phi=\int^{x}_{0}\phi(t)\,dt\) with a continuous function ϕ such that \(\frac {\phi(t)}{t}\) is nondecreasing or \(\phi(t)\) is superadditive on \([0,\infty)\), since the Nfunction 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 nonnegative 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 \(t\in[1,\infty)\), and u be 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 function w is defined by
If Φ is a nonnegative increasing superquadratic function on an interval \(I\subseteq[0,\infty)\), then we have the following inequality:
with any nonnegative measurable functions \(f:\mathbb {R}^{n}_{+}\rightarrow\mathbb{R}\) with values in I and \(A_{k}f\) 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. Let u be a weight function on \(I_{1}\), \(k:I_{1}\times I_{2}\rightarrow\mathbb{R}\) be a nonnegative 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 function v be defined by
If Φ is a nonnegative convex function on \(\Delta_{n}=\prod^{n}_{i=1}[m_{i},M_{i}]\subseteq\mathbb{R}^{n}\), then we have
with any nonnegative measurable functions \(f_{i}:I_{2}\rightarrow [m_{i},M_{i}]\). Further, inequality (3.9) holds in the reversed direction if Φ is a nonnegative 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 nonnegative 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 semidefinite 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 nonnegative 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 in n independent variables, with associated symmetric matrix A which is positive semidefinite. Then we have the following inequality:
for any nonnegative measurable functions \(f_{i}:I_{2}\rightarrow\mathbb {R}\), where
Reversely, the inequality (3.11) holds in the reversed direction if A is a negative semidefinite 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 HardyKnopp 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 function ω be defined by
If Φ is a Nfunction satisfying the \(\Delta_{2}\)condition, then there are constants α and β with \(1\leq\beta\leq\alpha <\infty\) such that the following norm inequality holds:
for any nonnegative measurable functions \(f:\Omega_{2}\rightarrow [0,\infty)\), where
and \(A_{k}f\) is defined on \(\Omega_{1}\) by (1.6).
Moreover, if \(\phi(t)\) is a continuous function such that \(\frac{\phi (t)}{t}\) is nondecreasing or \(\phi(t)\) is superadditive on \([0,\infty )\), then the following refined normal inequality holds:
for all nonnegative measurable functions \(f:\Omega_{2}\rightarrow [0,\infty)\), where
and \(A_{k}f\) is defined on \(\Omega_{1}\) by (1.6).
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 \(t\geq0\).
Case I. If \(\lambda\leq1\), let \(s=\frac{1}{\lambda}\). Then it follows that
Case II. If \(\lambda>1\), let \(s=\frac{1}{\lambda}\). Similarly to (4.1), we can get
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 nondecreasing 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 positive σfinite measure, u be a weight function on Ω, and \(k:\Omega \times\Omega\rightarrow\mathbb{R}\) be a nonnegative measurable function. Let the weight function v be defined by
Suppose that \(\Phi_{1}\) and \(\Phi_{2}\) are Nfunctions, where \(\Phi _{2}\) satisfies the \(\Delta_{2}\)condition, so that \(\Phi_{1}\circ\Phi ^{1}_{2}\) is an Nfunction. 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 constant C such that the following norm inequality:
holds for any nonnegative measurable function f. Moreover, if there exists a constant C such that the inequality
holds for any nonnegative function f, 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
Let \(\int_{\Omega}\Phi_{1}\circ\Phi^{1}_{2}(f(x))v(x)\,d\mu(x)\leq1\), then \(\\Phi^{1}_{2}(f)\_{\Phi_{1}(v)}\leq1\). By (4.4) we have
According to the definition of the Luxemburg norm this shows 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, and v are as in Theorem 4.2. Let \(\Phi_{1}\) and \(\Phi_{2}\) be Nfunctions such that \(\Phi_{2}\) satisfies the \(\Delta_{2}\)condition and \(\Phi_{1}\circ\Phi ^{1}_{2}\) is an Nfunction. 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:
Here \(\\cdot\_{*}\) is the operator norm.
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 nonnegative 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 Nfunctions 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 Nfunction. Furthermore, the complementary Nfunction of Φ is calculated by \(\Psi(x)=\frac{pq}{pq}x^{\frac{p}{pq}}\). Then we have the following conclusion.
Corollary 4.4
Let \((\Omega,\Sigma,\mu)\) be a measure space with positive σfinite 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 constant C such that the norm inequality holds:
for any nonnegative function f and \(\\frac{v}{u}\_{\Psi(u)}<\infty\) with \(\Psi(x)=\frac{pq}{pq}x^{\frac{p}{pq}}\). Moreover, if there exists a constant C such that the following inequality is valid:
for any nonnegative function f, then \(\\frac{u}{v}\_{\Psi(v)}<\infty\) holds.
Proposition 4.5
Suppose that \((\Omega_{1},\Sigma_{1},\mu_{1})\) and \((\Omega_{2},\Sigma _{2},\mu_{2})\) are σfinite measure spaces and that T is a linear operator which maps any nonnegative measurable functions on \(\Omega_{2}\) to some nonnegative measurable functions on \(\Omega_{1}\). Let \(\Phi(x)\) be an Nfunction, then
if and only if
holds for all \(\epsilon>0\) with C independent of ϵ (see Bloom’s paper in [22]).
Corollary 4.6
Assume that the assumptions in Proposition 4.5 are satisfied. Let \(T^{(r)}_{k}\) be the linear operator defined in (1.9) and \(\Phi(x)\) be an Nfunction, then
if and only if
holds for all \(\epsilon>0\) with C independent of ϵ.
It is clear that \(\Phi(x)=\int^{x}_{0}\phi(t)\,dt\) in which \(\phi (t)=e^{t}1\) is an Nfunction. 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:
where \(I=\int_{\Omega_{2}}(\ln f(y)+1)\,d\mu_{2}(y)\int_{\Omega _{1}}(T^{(r)}_{k}\ln f(x)+1)\,d\mu_{1}(x)\) holds, if and only if
holds for all \(\epsilon>0\) with C independent of ϵ.
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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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Zhang, J., Zheng, S. On refined HardyKnopp type inequalities in Orlicz spaces and some related results. J Inequal Appl 2015, 169 (2015). https://doi.org/10.1186/s1366001506864
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DOI: https://doi.org/10.1186/s1366001506864