Weighted Estimates of a Class of Integral Operators with Three Parameters

Abstract

We characterize the validity of a Hardy-type inequality with a kernel and three parameters under some conditions on three weight functions , , and .

1. Introduction

Let , , , and . Let , , and be positive functions locally integrable on , hereinafter referred to as weights. Suppose that for two nonnegative quantities and the expression means with some constant that through the paper depends only on the parameters , , and . The notation means . Moreover, .

We consider the following inequalities:for all , where the kernel satisfies the conditions for all , , and such that .

A class of Volterra type integral operators with kernels satisfying condition (3) was introduced in [1] and independently in [2]. Later such kernels were considered in many works (see, e.g., [3–8]).

The main aim of this paper is to find necessary and sufficient conditions on the weights , , and for the validity of inequalities (1) and (2) in the case . The same problem for was considered in [9, 10].

Assume

Two-sided estimates of the values and with kernels satisfying condition (3) were found in [11]. Moreover, when we get standard Hardy-type estimates that have been extensively investigated by many authors. A complete review of Hardy-type estimates and generalized Hardy-type estimates can be found in books [12, 13] and references given there.

The following theorem will be used for the main results.

Theorem A (see [11]). (1) If , then for all we have (2) If , then for all we have

Remark 1. Since the expressions , , , and are decreasing in and increasing in , then from (5) and (6) we have that are equivalent to a decreasing function in and an increasing function in . This means that there exists a constant depending only on and such that for .

2. Main Results

2.1. Case

Theorem 2. Let and . Inequality (1) holds for all if and only if . Moreover, , where is the best constant in (1).

Theorem 3. Let and . Inequality (2) holds for all if and only if . Moreover, , where is the best constant in (2).

Remark 4. Let us prove only Theorem 2 since the proof of Theorem 3 is similar.

Proof of Theorem 2.  
Sufficiency. Let . For any integer we introduce It is obvious that for any we have . However, when we have . Therefore, Let . Then Suppose that ; then from (8), twice applying Minkowski’s inequality, we get Since , we can use (3) so that the last givesFrom (9) and (10) we haveFrom (12) and (13) it follows thatNext, we separately estimate , , and for and .
Let . From (5) we get To estimate we use Hölder’s inequality:To estimate we again use Hölder’s inequality and get From (14), (15), (18), and (21) it follows that for inequality (1) is correct. Moreover, where is the best constant in (1).
Let us turn to the case . In the same way as above from (6) we get To estimate we work with (17). Since we haveSimilarly, working with (20) we have that yields Combining (14), (23), (25), and (27), we have that for inequality (1) is correct. Moreover, where is the best constant in (1).
Necessity. Let (1) be valid. Let and be an arbitrary function such that . Suppose that If we substitute the function in (1) we have From (30) we have Therefore, Moreover, from (22), (28), and (32) we have , where is the best constant in (1). The proof of Theorem 2 is complete.

2.2. Case

In this section we consider the case , , and and present sufficient conditions for the validity of inequalities (1) and (2).

Let

Theorem 5. Let , , and . Inequality (1) holds if . Moreover, , where is the best constant in (1).

Theorem 6. Let , , and . Inequality (2) holds if . Moreover, , where is the best constant in (2).

Remark 7. Let us prove only Theorem 5 since the proof of Theorem 6 is similar.

Proof of Theorem 5. The first steps of the proof are similar to those in Theorem 2 up to (14), where This means that we need to separately estimate , , and .
Let us start with . Let us notice that we use Hölder’s inequality: Now, we turn to the estimation of . Again Hölder’s inequality is used: The last step is to estimate :Combining (14), (35), (36), and (37), we have that inequality (1) is correct. Moreover, , where is the best constant in (1).

Remark 8. Let us consider the following inequalities: It is obvious that, in view of (3), the validity of inequality (38) is equivalent to the simultaneous validity of inequality (1) and the following inequalities: Inequality (40) can be treated by Theorem A, while inequality (41) is the standard Hardy-type inequality. This means that if we combine Theorem 2 and the known results on Hardy-type inequalities, we can characterize (38) for the case and . Similar splitting can be done for inequality (39). In [14] inequalities (38) and (39) are completely characterized for all relations between , , and , where , , and . The characterization method in [14] is not based on the integral splitting. Thus, due to the splitting, our main inequalities (1) and (2) allow characterizing inequalities (38) and (39). However, inversely, inequalities (38) and (39) do not help to characterize inequalities (1) and (2).
Let us also notice that when inequalities (38) and (39) were considered in [15–17].

3. Applications

(1) Let a function have generalized derivatives up to th order; . Let . Now we consider the inequality where the inside norm is taken with respect to the second argument of the function and the function is the th remainder of Taylor’s formula of ; that is, In the case we have Moreover, stands for By integration by parts it is easy to see that for we have Similarly, for we get

Therefore, inequality (42) holds if and only if the following inequalities simultaneously hold:

Thus, if we denote and when , from Theorems 2, 3, 5, and 6 we have the following.

Theorem 9. Let . Let and . Inequality (42) holds if and only if . Moreover, , where is the best constant in (42).

Theorem 10. Let . Let , , and . Inequality (42) holds if . Moreover, , where is the best constant in (42).

Remark 11. If inequality (42) holds, then the inequalitiesalso hold.

(2) In this part of the paper we investigate the inequality for with the conditions

When inequality (50) turns to the inequality Characterization of inequality (52) with conditions (51) is associated with open problem 2 in book [13, page 297].

First, we consider the inequality for with the conditions Let ; then we have Therefore, for , In the case and we have In the case and for we use the following relation: From (56), (57), and (58) we have where when and when .

From (59) we obtain

Consequently, the validity of inequality (53) is equivalent to the simultaneous validity of the following inequalities: for and , where , .

Inequality (61) can be characterized by Theorems 2 and 5 when .

Inequality (62) can be characterized by Theorem A if we denote and , where we replace by , by , and by . Then by Theorem A inequality (62) is valid if and only if for and for .

Thus, the following hold.

Proposition 12. Let and . Let and . Suppose that satisfies condition (54). Then inequality (53) holds if and only if . Moreover, , where is the best constant in (53).

Proposition 13. Let and . Let and . Suppose that satisfies condition (54). Then inequality (53) holds if . Moreover, , where is the best constant in (53).

A similar result can be written for the inequality with the conditions Here we need the following notations: and , where we replace by , by for and , where , , and by .

Proposition 14. Let and . Let and . Suppose that satisfies condition (64). Then inequality (63) holds if and only if . Moreover, , where is the best constant in (63).

Proposition 15. Let and . Let and . Suppose that satisfies condition (64). Then inequality (63) holds if . Moreover, , where is the best constant in (63).

The validity of inequality (50) with conditions (51) is equivalent to the simultaneous validity of inequalities (53) with (54) and (63) with (64). Therefore, from Propositions 12, 13, 14, and 15 we have the following.

Theorem 16. Let . Let and . Suppose that satisfies conditions (51). Then inequality (50) holds if and only if . Moreover, , where is the best constant in (50).

Theorem 17. Let . Let and . Suppose that satisfies conditions (51). Then inequality (50) holds if . Moreover, , where is the best constant in (50).

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

The paper was written under financial support by the Scientific Committee of the Ministry of Education and Science of Kazakhstan, Grant no. 5499/GF4 on priority area “Intellectual Potential of the Country.”