Abstract
We present necessary and sufficient conditions for boundedness and compactness of Hardy operator (1.4) with kernel (1.3) for .
MSC:26D10, 26D15, 47B07, 47B34, 47B38.
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1 Introduction
First, let us recall that the weighted Lebesgue space , with and w a weight function on , is defined as a set of all functions such that
We will investigate the Hardy-type inequality
with and u, v weight functions on .
This inequality was investigated by many authors. For , we obtain the ‘classical’ Hardy inequality
the case can be easily reduced to the classical case with modified weights and instead of u and v, respectively. Stepanov [1] has investigated ‘convolutionary’ kernels like , and probably the most general approach is connected with the name of Oinarov who investigated positive kernels k such that
(see [2]) and also several more general kernels (see [3, 4]). Also, inequalities with modified or generalized kernels have been investigated; let us mention the recent book [5] where multiple Hardy-type inequalities with the so-called product kernels are considered. See also [6] and [7] for further details.
Here we consider kernels of the type
and we want to find conditions on the weight functions u, v and on the functions , , for which the integral operator
maps the space continuously into .
Remark 1.1 The case of kernel (1.3) with special functions was investigated by Rychkov [8] for . Such kernels appear for general p, q by the investigation of higher-order Hardy inequalities; see [9].
Now, let us denote, for given , , u and v,
Then we can rewrite inequality (1.1) [for functions f] as the unweighted inequality
[for functions ] with
Remark 1.2 If , we have , and we can rewrite (1.6) as
which is in fact the classical (weighted) Hardy inequality
[for the function ] with the weight functions and .
It is well known (see, e.g., [6] or [7]) that for the case , inequality (1.8) holds for all functions if and only if the so-called Muckenhoupt-Bradley condition
is satisfied, where
with .
It follows from (1.9) that we need the integrability of on and of on for all , i.e.,
[where we denote by the classical Lebesgue spaces of functions defined on ].
In accordance with this remark, we suppose throughout the paper that the functions , from (1.7) satisfy
for all and .
2 Sufficient conditions
For and u, v weight functions on , let us define
Remark 2.1 Let us recall that the condition
is necessary and sufficient for the validity of the classical Hardy inequality (1.2) for .
It is easy to find a sufficient condition for (1.1) to hold, if we consider the ‘partial’ operators
as operators from into .
Theorem 2.2 Let . For defined by (2.1), denote
where , are the functions in (1.3). Then the Hardy-type inequality (1.1) with kernel from (1.3) holds if the weight functions u, v satisfy for the conditions
Proof Conditions (2.4) guarantee the validity of the Hardy inequality
for functions g; if we take , we can rewrite the foregoing Hardy inequality as
Now, using (1.3), the Minkowski and the last inequality, we obtain
i.e., we have derived inequality (1.1). □
Remark 2.3 Let us mention that the expression in (2.3) is nothing else than the expression from (1.9), where we replace A, B by , , respectively, with , from (1.5).
3 Necessary conditions
First let us introduce some auxiliary notions.
Definition 3.1 We will say that the system (matrix) of real numbers satisfies the ellipticity condition if there exists a constant such that
Remark 3.2 (i) The ellipticity condition is equivalent to the positive definiteness of the quadratic form
where
with some .
-
(ii)
A sufficient condition for the ellipticity to be satisfied is
Indeed, if we denote , then
To find necessary conditions for (1.1) to hold, we consider three cases.
-
(I)
The case , . Denote for this case
Theorem 3.3 Let and . Then the following condition
is necessary for inequality (1.1) to hold.
Proof For and for from (1.5) let be an arbitrary maximal linearly independent subsystem of in , which, for simplicity, we denote by . Using the Gram-Schmidt method of orthogonalization to the system in , we obtain a system such that
for and . Using this we rewrite the kernel in (1.6) in the form
for , where .
Let , then choosing and using the orthogonality of the system , we estimate the left-hand side of (1.6) as
This estimate together with (1.6) implies
Using the Minkowski inequality, we estimate in the form
Since is arbitrary and inequality (1.1) (i.e., inequality (1.6)) holds, we conclude that C is finite and we get (3.1). □
Moreover, we can show that condition (2.4) is also necessary if we add some assumptions. For this purpose, denote
where .
Definition 3.4 We say that the condition is satisfied if there exists a constant such that for every the system satisfies the ellipticity condition.
Theorem 3.5 Let and . Suppose that the system from (3.5) satisfies the condition . Then (sufficient) condition (2.4) is necessary for (1.1) to hold.
Proof Using Theorem 3.3, formula (3.4), the condition and the Minkowski inequality, we obtain
for arbitrary . Consequently, from (3.3) and (3.6), we get (2.4). The theorem is proved. □
-
(II)
The case , . Let us denote
where belongs to since
Definition 3.6 We will say that the condition is satisfied if there exists a constant such that for every the system satisfies the ellipticity condition for .
Theorem 3.7 Let and . If the condition holds, then (2.4) is necessary for inequality (1.1) to hold.
Proof Let and let . Let be a maximal linearly independent subsystem of in , which, for simplicity, we denote by . Thus, using the method of orthogonalization, we get an orthogonal system in such that
If we denote , and choose the following test function in (1.6) as
then the left- and right-hand sides are estimated in the forms:
and
where .
Using (3.7) we rewrite the kernel in the form
where .
Then we have from (3.8), (3.9) and (1.6) that
Now, repeating the proofs of the foregoing theorems with respect to (3.10) with the kernel , we can also get a similar estimate as in (3.2) in the form
Then supposing the condition , we also obtain the following estimate:
i.e.,
which holds for and .
Using this estimate and (3.11), we finally get that
The proof is complete. □
-
(III)
The case , . Another approach how to investigate inequality (1.1) is based on the following lemma (for details, see [7]).
Lemma 3.8 Let . Then inequality (1.6) holds for all if and only if the conjugate inequality
holds for all , where
and .
The lemma enables to replace the investigation of inequality (1.6) by the investigation of inequality (3.13).
Denote
where belongs to since
Definition 3.9 We will say that the condition is satisfied if there exists a constant and for every such that the system satisfies the ellipticity condition for .
Theorem 3.10 Let and . If the condition holds for the system from (3.15), then (2.4) is necessary for inequality (1.6) to hold.
Proof Let and let . Let be a maximal linearly independent subsystem of in , which, for simplicity, we denote by . Thus, using the method of orthogonalization, we get an orthogonal system in such that
If we denote , and choose the test function in the form
then the left- and right-hand sides are similarly estimated in the forms:
and
where .
Using (3.7) we can rewrite the kernel in the form
where .
Then we have from (3.17), (3.18) and (3.13) that
Now repeating the proof of the foregoing theorem with respect to (3.19) with the kernel , we can also get a similar estimate as in (3.11) in the form
Then, supposing that the condition is satisfied, we also obtain the following estimate:
i.e.,
which holds for all and .
Using this estimate and (3.20), we finally get that
The proof is complete. □
4 Criteria of compactness
As far as the compactness of the imbedding is concerned, we have the following.
Theorem 4.1 Let . Let us suppose that the functions from (2.4) satisfy
Then the operator k from (1.4) maps into compactly.
Proof Conditions (4.1) guarantee that the operators of from into are compact (see, e.g., [7]). Since , the compactness of k follows. □
Theorem 4.2 Let and . Let the condition be satisfied. If the operator k from (1.4) is compact from to , then conditions (2.4) and (4.1) are satisfied.
Proof Let us suppose that operator (1.4) is compact. Then it is bounded, and, by using Theorems 3.3 and 3.7, we get (2.4) and also the compactness of the operator
as an operator from to .
Let and . Moreover, we choose the function as in the proof of Theorem 3.7. Let be arbitrary, then using (3.8) we have
from which it follows that
as , i.e., the class of functions weakly converges to zero in as .
This and the compactness of operator (4.2) imply that the class of images strongly converges to zero in as , i.e.,
Analogously as in the proof of Theorem 3.7, (3.12) can also be obtained, i.e.,
which with (4.3) implies that for all .
Now we show that for all . The compactness of operator (4.2) follows from the compactness of the conjugate operator (3.14) from to .
Let and choose . It can be shown as in foregoing cases that the class of functions weakly converges to zero in as . Then the class of images strongly converges to zero in as , i.e.,
Using the dual principle of , we have
Now, choosing as in the proof of Theorem 3.7 instead of f in (4.6), we have
Since the operator K is bounded from to , i.e., (3.10) is satisfied, from which we will have that , . Then, choosing the function , and from (4.7), we have
As in the proof of Theorem 3.7, by estimating the right-hand side, we obtain also that
Consequently, from this and (4.5) we have that for all .
The theorem is proved. □
Theorem 4.3 Let and . Let the condition be satisfied. If operator (1.4) is compact from to , then conditions (2.4) and (4.1) are satisfied.
Proof First we show that , for which we use the compactness of the dual operator and the proof of Theorem 3.10. Then it can be shown that the class of functions also weakly converges to zero in as and
Analogously as in the proof of Theorem 3.10, (3.21) can be obtained, i.e.,
which with (4.8) implies that for all .
To prove , we use the duality principle as in the proof of the foregoing theorem. The formulation of the corresponding proof is left to the reader as an exercise. □
Corollary 4.4 Let and . Let us suppose that the condition is satisfied. Then operator (1.4) from into is bounded and compact if and only if (2.4) and (4.1) are satisfied, respectively.
Corollary 4.5 Let and . Let us suppose that the condition is satisfied. Then operator (1.4) from into is bounded and compact if and only if (2.4) and (4.1) are satisfied, respectively.
Corollary 4.6 Let . Let us suppose that the conditions and are satisfied. Then operator (1.4) from into is bounded and compact if and only if (2.4) and (4.1) are satisfied, respectively.
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Acknowledgements
The first author was supported by RVO: 67985840. The second author was supported by Leverhulme Trust; grant No. RPG-167 and Wales Institute of Mathematical & Computational Sciences. The third author was supported by the Scientific Committee of Ministry of Education and Science of the Republic of Kazakhstan, grant No.1529/GF, on priority area ‘Intellectual potential of the country’.
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Kufner, A., Kuliev, K. & Oinarov, R. Some criteria for boundedness and compactness of the Hardy operator with some special kernels. J Inequal Appl 2013, 310 (2013). https://doi.org/10.1186/1029-242X-2013-310
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DOI: https://doi.org/10.1186/1029-242X-2013-310