Some sharp inequalities for integral operators with homogeneous kernel
 Dag Lukkassen^{1, 2},
 LarsErik Persson^{1, 3}Email author and
 Stefan G Samko^{4}
https://doi.org/10.1186/s1366001610379
© Lukkassen et al. 2016
Received: 25 September 2015
Accepted: 10 February 2016
Published: 9 April 2016
Abstract
One goal of this paper is to show that a big number of inequalities for functions in \(L^{p}(R_{+})\), \(p\geq1\), proved from time to time in journal publications are particular cases of some known general results for integral operators with homogeneous kernels including, in particular, the statements on sharp constants. Some new results are also included, e.g. the similar general equivalence result is proved and applied for \(0< p<1\). Some useful new variants of these results are pointed out and a number of known and new HardyHilbert type inequalities are derived. Moreover, a new PólyaKnopp (geometric mean) inequality is derived and applied. The constants in all inequalities in this paper are sharp.
Keywords
inequalities Hardy type inequalities HardyHilbert type inequalities homogeneous kernels sharp constantsMSC
26D10 26D151 Introduction
Let \(p>0\) and denote by \(p^{\prime}\) the conjugate parameter defined by \(\frac{1}{p}+\frac{1}{p^{\prime}}=1\) (\(p^{\prime}=\infty\) when \(p=1\)). We also let f and g denote arbitrary measurable positive functions on \((0,\infty)\). The constants in all inequalities below and in all of this paper are sharp.
Remark 1
Hilbert himself considered only the case \(p=2\) and the corresponding discrete form of (1) (see his paper [1] from 1906 and also [2, 3] and the historical description in [4]). \(L^{p}\)spaces with \(p\neq2\) appeared only later (around 1920). Concerning the equivalence of (1) and (2) see our Lemma 15 for a more general statement.
Remark 2
For the case \(\alpha=0\) (3) is the classical Hardy inequality. The almost 10 years of research until Hardy finally proved this inequality in 1925 (see [5]) is described in detail in [4]. In particular, it is completely clear that Hardy’s motivation was to find an elementary proof of Hilbert’s inequality for the discrete case. Also the weighted variant (3) was first proved by Hardy (see [6]). The further development of inequalities (3) and (4) to what today is called Hardytype inequalities is very extensive and still a very active area of research (see e.g. the monographs [7] and [8] and [9]) and the references given there.
Remark 3
There are a huge number of papers devoted to the proof of (5) and (6) for concrete kernels \(k(x,y)\) other than the classical Hilbert kernel \(k(x,y)=1/(x+y)\). In this connection we refer to the monograph [10] and the references there. Moreover, we announce that by using a standard dilation argument in (5)(6) we see that such kernels must be homogeneous of degree −1. One weakness with many of these results is that the authors do not refer to the fact that already in 1999 (see [11] and also [12]) it was given necessary and sufficient conditions for (5) to hold and with sharp constant and general kernel of degree −1. See Theorem 5.
 (a)
A general reversed version of the inequalities described in Remark 3 yielded for \(0< p<1\). See Theorem 7.
 (b)
A corresponding equivalence theorem for homogeneous kernels of any order λ but with the righthand sides in Theorem 5 and Remark 6 replaced by some corresponding weighted \(L^{p}\)spaces so that our main results can be used. See Theorem 10 and Remark 11.
 (c)
In order to be able to cover also some other results in the literature we derive a version for ‘skew symmetric’ kernels of order −1 (for the definition see (21)). See Theorem 13 and Remark 14.
 (d)
A completely new geometric mean (PólyaKnopp) type inequality is derived (see Theorem 30). Moreover, we present a number of applications of this, which seems to be new too.
 (e)
As applications a number of new (and also wellknown) sharp inequalities are presented.
Remark 4
The paper is organized as follows: Some main results are presented and commented in Section 2. The detailed proofs are given in Section 3. Some applications concerning Hardy and HardyHilbert type inequalities are presented in Section 4. Finally, Section 5 is reserved for another main result, namely the announced new PólyaKnopp type inequality. Some applications of this result are also given. All inequalities in this paper have sharp constants.
2 Main results
Our first main results reads as follows.
Theorem 5
 (i):

The constant \(\kappa_{p}<\infty\).
 (ii):

The inequalityholds for some finite constant C for all \(f\in L_{p}\) and \(g\in L_{p^{\prime}}\).$$ \int_{0}^{\infty} \int_{0}^{\infty}k(x,y)f(x)g(y)\,dx\,dy\leq C\Vert f \Vert _{p}\Vert g\Vert _{p^{\prime}} $$(12)
 (iii):

The inequalityholds for the same finite constant C as in (12) and all \(f\in L_{p}\).$$ \int_{0}^{\infty} \biggl( \int_{0}^{\infty}k(x,y)f(x)\,dx \biggr) ^{p} \,dy\leq C^{p} \int_{0}^{\infty}f^{p}(x)\,dx $$(13)
Moreover, the constant \(C=\kappa_{p}\) is sharp in both (12) and (13).
Remark 6
The proof of (12) under the condition \(\kappa _{p}<\infty\) was given already in the book [14], Theorem 3.19. Apart from the original proof in [14], this sufficiency part may be derived, via a change of variables, from the Young theorem for convolutions in R, for details see [12] and [11]. In this way the sharpness of the constant is derived from the fact that the Young inequality \(\Vert h\ast f\Vert _{p}\leq \Vert h\Vert _{1}\Vert f\Vert _{p}\) holds with the sharp constants \(\Vert h \Vert _{1}\) when h is nonnegative. Hence, by using the results in [12] and [11] and the equivalence result in Lemma 15, Theorem 5 is essentially known even if it has not been formulated in this way before. However, to make our paper selfcontained we include a proof which also guides us how to prove the other results in this section.
Theorem 7
Let \(0< p<1\) and the kernel \(k(x,y)\) satisfy (10). Moreover, assume that (14)(16) hold. Then all the statements in Theorem 5 hold with inequalities (12) and (13) holding in reversed direction.
Since \(p^{\prime}<0\) in this case we have \(\Vert g\Vert _{p^{\prime}}= ( \int_{0}^{\infty} \vert g ( y ) \vert ^{p/(p1)}\,dy ) ^{\frac{p1}{p}}\) and we assume that \(0<\Vert g\Vert _{p^{\prime}}<\infty\) here and in the sequel.
Remark 8
For the proof of the fact that \(\kappa_{p}<\infty\) implies the equivalent reversed conditions (12) and (13) we do not need the restriction (16).
Remark 9
By using a standard dilation argument it is seen that the inequalities considered in Theorem 5 can hold if and only if \(\lambda=1\). However, by changing the norms in the lefthand sides in (12) and (13) to powerweighted norms we can from our result obtain a similar result for homogeneous kernels of any degree λ. In order to be able to compare with a result in [15] we formulate this result as follows.
Theorem 10
 (i^{∗}):

The constant \(\kappa_{p,\beta}<\infty\).
 (ii^{∗}):

The inequalityholds for some finite constant C for all \(f\in L_{p,x^{\alpha}}\) and \(g\in L_{p^{\prime},x^{\beta}}\).$$ \int_{0}^{\infty} \int_{0}^{\infty}k_{\lambda _{0}}(x,y)f(x)g(y)\,dx\,dy \leq C\Vert f\Vert _{p,x^{\alpha}} \Vert g\Vert _{p^{\prime },x^{\beta}} $$(19)
 (iii^{∗}):

The inequalityholds for the same finite constant C as in (19) and all \(f\in L_{p,x^{\alpha}}\).$$ \int_{0}^{\infty} \biggl( y^{\beta} \int_{0}^{\infty}k(x,y)f(x)\,dx \biggr) ^{p} \,dy\leq C^{p} \int_{0}^{\infty}f^{p}(x)x^{\alpha p}\,dx $$(20)
Remark 11
By choosing \(\lambda=\lambda_{0}\), \(\alpha=1\frac{ \lambda}{r}\frac{1}{p}\), \(\beta=1\frac{\lambda}{s}\frac {1}{p^{\prime}}\) (\(=\frac{\lambda}{s}+\frac{1}{p}\)) with \(s>1\), \(\frac{1}{r}+\frac{1}{s}=1\) we can compare with Theorem 2.1 in [15]. For the case \(p>1\), \(\lambda _{0}>0\) the equivalence in (ii^{∗}) and (iii^{∗}) were established already in this Theorem and also the sharpness in (iv^{∗}) for these cases. However, the necessity pointed out in (i^{∗}) was not explicitly pointed out in this paper.
Remark 12
By using our Theorem 7 and making similar calculations as in the proof of Theorem 10 we can obtain a similar complement and strengthening of Theorem 2.2 in [15] yielding for \(0< p<1\) and kernels of any homogeneity \(\lambda_{0}\in R\).
Theorem 13
 (i):

The constant \(\kappa_{p}(a,b)<\infty\).
 (ii):

The inequalityholds for some finite constant C for all \(f\in L_{p}\) and \(g\in L_{p^{\prime}}\).$$ \int_{0}^{\infty} \int_{0}^{\infty}k(x,y)f(x)g(y)\,dx\,dy\leq C\Vert f \Vert _{p}\Vert g\Vert _{p^{\prime}} $$(22)
 (iii):

The inequalityholds for the same finite constant C as in (22) and all \(f\in L_{p}\).$$ \int_{0}^{\infty} \biggl( \int_{0}^{\infty}k(x,y)f(x)\,dy \biggr) ^{p} \,dx\leq C^{p} \int_{0}^{\infty}f^{p}(x)\,dx $$(23)
Remark 14
By using a similar proof to that of Theorem 13 we can obtain a similar consequence (and formal extension) also of our Theorem 7.
For the proof of these Theorems we need a lemma of independent interest, which we state and prove in a little more general form. Let \(k(x,y)\) denote a positive kernel on \(R_{+}\times R_{+}\).
Lemma 15
 (a)Let \(p\geq1\). The following statements are equivalent:
 (i)The inequalityholds for some finite constant C and all \(f\in L_{p}\) and \(g\in L_{p^{\prime}}\).$$ \int_{0}^{\infty} \int_{0}^{\infty}k(x,y)f(x)g(y)\,dx\,dy\leq C\Vert f \Vert _{p}\Vert g\Vert _{p^{\prime}} $$(24)
 (ii)The inequalityholds for the same finite constant C as in (24) and all \(f\in L_{p}\).$$ \int_{0}^{\infty} \biggl( \int_{0}^{\infty}k(x,y)f(x)\,dx \biggr) ^{p} \,dy\leq C^{p} \int_{0}^{\infty}f^{p}(x)\,dx $$(25)
 (i)
 (b)
Let \(0< p<1\). A similar equivalence to that in (a) holds also in this case but with the inequalities in (24) and (25) reversed (here we use the same convention concerning \(\Vert g\Vert _{p^{\prime}}\) as before, see the sentence after Theorem 7).
Remark 16
The statement in (a) is well known and follows from a more general statement in functional analysis. However, we give here another simple direct proof which works also to prove that part (b) holds, which seems not to have been explicitly stated before.
3 Proofs
Proof of Lemma 15
Let \(p=1\) so \(p^{\prime}=\infty\). By applying (24) with \(g(y)\equiv1\) we see that (24) implies (25). Moreover, by using that \(g(y)\leq \Vert g\Vert _{\infty} \), \(y\in (0,\infty) \), we find that (25) implies (24).
(b) Hölder’s inequality holds in the reversed direction in this case. Hence, the proof of (b) only consists of obvious modifications of the proof of (a). □
Proof of Theorem 5
Moreover, by using Lemma 15, we see that statements (i) and (ii) are equivalent including the fact that the constant \(C= {\kappa}_{p}\) is sharp also in (12). We have thus also proved that statement (iv) is correct.
Proof of Theorem 7
Proof of Theorem 10
Proof of Theorem 13
4 Examples of inequalities covered by the results in Section 3
First we present two simple standard examples.
Example 17
Remark 18
Inequalities of the type (31) and (32) are in several papers called HardyHilbert or Hilbert type inequalities. As we have pointed out they are a consequence of Theorem 5 and can be obtained if and only if the kernel \(k(x,y)\) is homogeneous of type −1. A great number of examples have been presented in the literature but most such results can also be derived from Theorem 5 for \(1\leq p<\infty\) and from the reversed forms from Theorem 7 for \(0< p<1\).
Example 19
Remark 20
The inequality (33) is the first weighted form of Hardy’s original inequality proved by Hardy himself in 1928 (see [6]). Equation (34) is sometimes called the dual form of (33), in fact these inequalities are in a sense equivalent.
In our next example we unify and generalize the inequalities in Examples 17 and 19 by presenting a scale of inequalities between these inequalities (a genuine HardyHilbert inequality).
Example 21
Remark 22
 (∗∗):

if \(a=\infty\), \(\beta=1\), \(\alpha=0\) we get the Hilbert inequality in Example 17,
 (∗∗∗):

in all (Hardy like) cases \(\beta=0\) we have the sharp constant$$ \frac{a^{\frac{1}{p^{\prime}}\alpha}}{\frac{1}{p^{\prime}}\alpha },\quad \alpha< \frac{1}{p^{\prime}}. $$
Remark 23
The following example is a dual counterpart to Example 21.
Example 24
Example 25
(HardyLittlewood inequality [17])
The following example is also a particular case of Theorem 5.
Example 26
As a simple generalization of Example 17, the next example also easily follows from Theorem 5.
Example 27
(Hilbert type inequality)
We finish this section by also giving the following application of our Theorem 13.
Example 28
 (i)
\(\int_{0}^{\infty}\int_{0}^{\infty} ( \frac{1}{x^{\lambda }+y^{\mu}} ) ^{\alpha}f ( x ) g ( y ) \,dx\,dy\leq C\Vert f\Vert _{p}\Vert g\Vert _{p^{\prime}}\) for all \(f\in L_{p} \)and \(g\in L_{p^{\prime}}\).
 (ii)
\(\int_{0}^{\infty} ( \int_{0}^{\infty} ( \frac{1}{x^{\lambda}+y^{\mu}} ) ^{\alpha}f ( x ) \,dy ) ^{p}\,dx\leq C^{p}\int_{0}^{\infty}f^{p} ( x ) \,dx\) for all \(f\in L_{p}\).
In fact, the proof follows by just using Theorem 13 with \(a=\frac{1}{\lambda\alpha}\), \(b=\frac{1}{\mu\alpha}\) and making some straightforward calculations.
Remark 29
5 A new general geometric mean type inequality
Our new general geometric mean inequality reads as follows.
Theorem 30
Proof
Example 31
(Generated by a weighted Hardy inequality)
Example 32
(Generated by weighted Hilbert inequality)
Example 33
(Generated by the HardyLittlewood inequality)
Declarations
Acknowledgements
For the third author the research was supported by grant No. 150102732 of the Russian Fund of Basic Research.
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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