Abstract
In the paper, for a certain class of Hardy operators with kernels, we consider the problem of their boundedness from a second order weighted Sobolev space to a weighted Lebesgue space.
1. Introduction
Let and . Let and be positive functions locally integrable on the interval . In addition, suppose that , where .
Let be a set of functions having generalized derivatives up to the second order on with the finite norm where is the standard norm of the space , .
In the paper, we consider the problem of boundedness of the integral operator with a kernel from the weighted space to the weighted space with the norm . This problem is equivalent to the validity of the following inequality
Let be the set of compactly supported functions infinitely time continuously differentiable on . Due to the assumptions on , we have that . Denote by the closure of the set with respect to norm defined by (1). Depending on the behaviour of the function at zero and infinity, the set can be dense or not dense in the space , i.e., or respectively.
In the paper, we study inequality (3) under condition (4) for a certain class of integral operators. Note that in the case when is the identity operator , inequalities of form (3) have been studied in many papers. Some results with proofs and a survey of other results with comments are given in Chapter 4 of the book [3]. Our work is related to the works [5, 6], in which inequality (3) with was studied under various zero boundary conditions for .
The boundedness of integral operators in form (2) from a first order weighted Sobolev space to a weighted Lebesgue space has been investigated in the series of papers (see, e.g., [1, 2] and references given therein).
The paper is organized as follows. In Section 2, we present definitions and statements required to prove the main results. In Section 3, we present and prove the main results, especially we obtain necessary and sufficient conditions for the validity of inequality (3). In Section 4, we present corollaries that follow from the results of Section 3.
2. Axillary Definitions and Statements
Let . In the paper, is the characteristic function of the interval . Moreover, the notation means and means .
From the book [3], we have the following theorem.
Theorem 1. Let . (i)The inequalityholds if and only if In addition, , where is the best constant in (5). (ii)The inequalityholds if and only if
In addition, , where is the best constant in (7).
The following definitions and statements are from the paper [7].
Definition 2. Let be a nonnegative function measurable on the set and nonincreasing in the second argument. We say that the function belongs to the class if there exist nonnegative functions and measurable on such that for ; moreover, the equivalence coefficients in (9) do not depend on , , and .
Definition 3. Let be a nonnegative function measurable on the set and nonincreasing in the second argument. We say that the function belongs to the class if there exist and nonnegative functions , , and measurable on such that for ; moreover, the equivalence coefficients in (10) do not depend on , , and .
Definition 4. Let be a nonnegative function measurable on the set and nonincreasing in the second argument. We say that the function belongs to the class if there exist , , and nonnegative functions , , , and measurable on such that
for ; moreover, the equivalence coefficients in (11) do not depend on , , and .
Let
Theorem 5. Let . Let the kernel of operator (2) belong to the class . Then, the inequality
holds if and only if . In addition, , where is the best constant in (15).
Let
Theorem 6. Let . Let the kernel of operator (2) belong to the class . Then, inequality (15) holds if and only if . In addition, , where is the best constant in (15).
Let
Theorem 7. Let . Let the kernel of operator (2) belong to the class . Then, inequality (15) holds if and only if . In addition, , where is the best constant in (15).
For , we assume that , , , and regardless of whether they are finite or infinite.
The following statement is from the paper [4].
Theorem 8. Let . If the conditions hold; then, for , there exist the finite values , , and such that
3. Main Results
First, we state some necessary lemmas. Some of them are new and of independent interest, and therefore proved in detail.
Lemma 9. Let , where for . Then for ;
Proof. (i)For , we haveTherefore, by (10), we get that .
(ii)For , it easily follows that(iii)Using (28), for , we haveThen, in view of (11), we obtain that . The proof is complete.
Let and . Assume that
By using part (iii) of Lemma 9 and Theorem 7, we have one more lemma.
Lemma 10. Let and . Then, the inequality
holds if and only if . In addition, , where is the best constant in (38).
Let
Using (28) and the inverse Hölder’s inequality, by Theorem 5, we have the following lemma.
Lemma 11. Let and . Then, the inequality
holds if and only if . In addition, , where is the best constant in (42).
Let
From part (i) of Theorem 1, we can state the following lemma.
Lemma 12. Let . Then, the inequality
holds if and only if . In addition, , where is the best constant in (44).
Assume that
By using part (i) of Lemma 9 and Theorem 6, we get the following statement.
Lemma 13. Let and . Then, the inequality
holds if and only if . In addition, , where is the best constant in (49).
Let
Lemma 14. Let and . Then, the inequality holds if and only if . In addition, , where is the best constant in (54).
Proof. Since , by Lemma 9, we have that Hence, inequality (54) is equivalent to simultaneous fulfilment of the following inequalities: In addition, , where and are the best constants in (56) and (57), respectively. By Theorem 5, inequality (56) holds if and only if , and in addition, . By part (i) of Theorem 1, inequality (57) holds if and only if , and in addition, . Then, inequality (54) holds if and only if and . The proof is complete.
Assume that
By using part (ii) of Theorem 1, we have the following lemma.
Lemma 15. Let . Then, the inequality holds if and only if . In addition, , where is the best constant in (59).
Let infinitely differentiable functions and be such that , , for , for , for and for . Moreover, , for and for all .
Assume that and are polynomials such that and , where , . Denote by and the sets of polynomials in the form and , respectively.
Let the conditions of Theorem 8 hold. Then, from (26), we have where means the direct sum.
From Theorem 8, it follows that (1) is equivalent to the norm
Therefore, for we have
First, using (60), we establish inequality (3) on the set , which, due to (62), has the form:
Assume that
Our first main result reads.
Theorem 16. Let and . Let condition (25) hold. Then, inequality (63) holds if and only if . In addition, , where is the best constant in (63).
Proof. Sufficiency. From (25) by Theorem 8, it follows the validity of (26). As in Theorem 2.1 of [5], using (26), for , we get
where . Assuming in (66), we have that . Moreover, from (26), it follows that .
Assume that . Then, in (66), the condition is equivalent to the condition . Replacing (66) into the left-hand side of (63), we find that
Therefore, inequality (63) has the form
In the left-hand side of (68), using the Minkowski’s inequality for sums, then, applying Lemmas 10, 11, 12, 13, and 14 to each term, we get
Since the left-hand side of inequality (63) does not depend on , then, taking in the right-hand side of (69) infimum with respect to , we can conclude that
where is the best constant in (63).
Necessity. By the conditions of Theorem 16, we have that . Then, for any , there exists such that
in addition, increases in , and .
Let us use the ideas in the paper [5]. For , we consider two sets and . For each and , we, respectively, construct the functions and so that for and for belongs to the set .
We define a strictly increasing function from the relation
where is inverse to . From (73), it follows that the functions and are locally absolutely continuous, and .Differentiating both relations in (73), we have
Then, for , we construct
while for , we construct
Changing the variables and using the first equality in (74), we find that
Similarly, using the second equality in (74), we get
From (77) and (78), assuming that for and for , we have
i.e., . For any integrating both sides of (75) from to and (76) from to , we find that
Hence, constructed from the functions and , the function belongs to . Replacing it into (68), we get
where all terms in the left-hand side are nonnegative.
Let the function constructed from the function . Then, from (81) and (79), we have
Due to arbitrariness of , by Lemmas 10, 11, and 12, the latter gives that
Similarly, due to (81) and (79), for the function constructed from the function , we obtain
From (83) and (84), we find that
Therefore, , which, together with (70), yields that , where is the best constant in (63). The proof is complete.
Let
Our main result concerning Hardy-type inequality (3) reads.
Theorem 17. Let and . Let conditions in (25) hold. Then, inequality (3) holds if and only if . In addition, , where is the best constant in (3).
Proof. Due to (60), we consider inequality (3) on the set
The function has the form
Hence, almost everywhere on . Therefore, on the basis of (61), we have
Let be such that
Then, from (89), we obtain
which implies that .
Let . Then, . Replacing the function into the left-hand side of (89), we get
The latter, together with , gives that . Then, by Theorem 16, it follows that , where is the best constant in (3). The proof is complete.
4. Corollaries
Assume that the kernel of operator (2) satisfies the Oinarov condition which is often applied for integral operators. Then, in , the expression turns to the expression in the expression turns to the expression in the expression turns to the expression and in , the expressions , , respectively, turn to the expressions
After these changes, we denote by , by , and by and get the following statement.
Corollary 18. Let and the kernel of (2) satisfy condition (93). Let conditions (25) hold. (i)Inequality (63) holds if and only if . In addition, , where is the best constant in (63)(ii)Inequality (3) holds if and only if . In addition, , where is the best constant in (3)Let . Instead of operator (2), we consider the operator of Riemann-Liouville , defined by The kernel of the operator satisfies condition (93), and therefore, it belongs to the class . In this case, we replace and by and assume that . For the kernel inequality, (38) has the form Then, according to Theorem 5, we have , where For the sum of kernels of the operators in (42) and (44), we deduce that Then instead of inequalities (42) and (44), we, respectively, have By part (i) of Theorem 1, this yields that Assume that . Now, for the sum of kernels of the operators in (49) and (54), we get Then instead of inequalities (49) and (54), we obtain Hence, by part (i) of Theorem 1 we have For the kernel inequality (59) can be written as follows: Therefore, by using part (ii) of Theorem 1, we find Assume that Thus, for inequality (63) with operator (99) we can conclude the following statement.
Corollary 19. Let and conditions (25) hold. Then, inequality (115) holds if and only if . In addition, , where is the best constant in (115).
Assume that in (115). Then, and , . Moreover, and inequality (115) turns to the inequality with conditions
From Corollary 19, we get one more corollary.
Corollary 20. Let and conditions (25) hold. Then, inequality (116) with conditions (117) holds if and only if . In addition, , where is the best constant in (116).
The statement of Corollary 20 gives one of the results of the work [6].
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that she has no conflicts of interest.
Acknowledgments
This work was supported by the Ministry of Education and Science of the Republic of Kazakhstan in the area “Scientific research in the field of natural sciences” (grant number AP09259084).