Journal of Function Spaces and Applications

Volume 2013 (2013), Article ID 435480, 11 pages

http://dx.doi.org/10.1155/2013/435480

Research Article

Estimates of the Fundamental Solution for Higher Order Schrödinger Type Operators and Their Applications

Satoko Sugano

Kobe City College of Technology, 8-3 Gakuen-higashimachi, Nishi-ku, Kobe 651-2194, Japan

Received 3 October 2013; Accepted 21 November 2013

Academic Editor: Hendra Gunawan

Copyright © 2013 Satoko Sugano. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We establish estimates of the fundamental solution and show boundedness of some Schrödinger type operators.

1. Introduction

Let be a nonnegative potential and consider the Schrödinger type operators on , where is a positive integer and . When is a nonnegative polynomial, Zhong proved estimates of the fundamental solution for and and showed some estimates for and (see [1]). More precisely, he showed the boundedness of the operators , , and , where . He also proved that the operators and are Calderón-Zygmund operators. Recently, in [2], the authors showed the boundedness of the operators , , and for nonnegative polynomial potential  .

For the potential which belongs to the reverse Hölder class, which includes nonnegative polynomials, Shen generalized Zhong's results on (see [3]). Actually, he established estimates of the fundamental solution for and showed the estimates for the operators , , , and so on. For the operator with reverse Hölder class potentials, further results have been investigated by many researchers. See [4–7], for example. For the operator with reverse Hölder class potentials, in [8], the author established estimates of the fundamental solution for and showed the boundedness of the operators , where . In [8], the author also showed that the operator is a Calderón-Zygmund operator. Recently, in [9], the authors showed boundedness of the operator and that the operator is a Calderón-Zygmund operator.

In this paper, we study with reverse Hölder class potential on , where is a nonnegative integer and . We establish estimates of the fundamental solution for and show the boundedness of the operators , where is an integer satisfying .

As mentioned above, in [2], the authors proved some results on , where is a nonnegative polynomial and is an integer, . They proved their results by making use of [2, Lemma 3.3 and Corollary 3.1] which have been proved only for nonnegative polynomial . In this paper, our strategy is different from the one in [2], since the question whether the above two results can be proved for reverse Hölder class potentials is yet to be settled. The purpose of this paper is to show some results on with potential which belongs to the reverse Hölder class, which includes nonnegative polynomials. However, our results are only for , where is an integer, .

We recall the definitions of the reverse Hölder class (e.g., [3]). We denote by the ball centered at with radius .

Definition 1 (reverse Hölder class). Let .

For one says that if and there exists a positive constant such that

holds for every and .

One says that if and there exists a positive constant such that

holds for every and .

Remark 2. If is a polynomial and , then belongs to (see [10, page 146]).

For , it is easy to see .

Definition 3 (see [3, Definition 1.3]). (1) Let and . Then it is well known that there exists a positive such that (see [11, Lemma 2]). Then the function is well defined by

and satisfies for every .

One recalls that

(see [3, page 519]).

Remark 4. If , , then there exists a positive constant such that

(cf. [3, Lemma 1.8] and [5, Lemma 2.2 (a)]).

If then there exists a positive constant such that (see [3, Remark 2.9]).

Let denote the multi-index with , , and . Define and for . For any positive integer and a function , denote that and . We denote by the fundamental solution for . The operator is the integral operator with as its kernel.

Now we state our theorems.

Theorem 5. Let , , and be integers, , , and . Suppose that for some . Then there exist positive constants such that

where .

In Theorem 5, the case was shown in [3, Theorems 2.7 and 4.13].

Remark 6. In Theorem 5, for the case , inequality (6) was shown in [3, Remark 2.9] under the assumption . If we take the limit , then class becomes and implies “ and .” See [8, Remark 3] and also [8, Theorem 1].

Remark 7. We assume “ and .” Then inequality (6) is true for . See Section 7 for the details. We also remark that the inequality (6) is true for and under the assumption “ and .” See [3, Remark 2.9] and also [8, Theorem 1 (2)].

Theorem 8. Let , , and be integers, , , and . Suppose that for some , . Then there exist positive constants such that (6) holds, where and .

To prove Theorems 5 and 8 we need the estimates of the fundamental solution (Theorems 9 and 10). The following Theorem 9 generalizes the results in [2, Theorem 3.1] to the operator with potential which belongs to the reverse Hölder class.

Theorem 9. Let and be integers, , and . Suppose that . Then for any positive integer there exists a positive constant such that

In Theorem 9, the cases and were shown in [3, Theorem 2.7] and [8, Theorem 2], respectively. We prove Theorem 9 by induction; that is, we assume that Theorem 9 is true for and show the case . We also prove the following theorem which states derivative estimates of the fundamental solution.

Theorem 10. Let , , and be integers, , , and . Suppose that . Then for any positive integer there exists a positive constant such that

In Theorem 10, the cases and were shown in [3, page 537] and [8, Theorem 6], respectively.

The plan of this paper is as follows. In Section 2, we describe some lemmas needed later. In Section 3, we assume that Theorem 9 is true for and show some estimates for which are needed to prove the case in Theorem 9. In Section 4, we prove the case in Theorem 9. Section 5 is devoted to proof of Theorem 10. In Section 6, we prove Theorems 5 and 8. Finally, in Section 7, we state some remarks. Throughout this paper the letter stands for a constant not necessarily the same at each occurrence.

2. Preliminaries

In this section, we describe some lemmas needed later. First, we remark that , , implies that is a doubling measure; that is, there exists a positive constant such that

(see [3, page 518] and also [5, Remark 1.2 (1)]).

Lemma 11 (see [3, Lemma 1.2]). Assume that for some . Then there exists a positive constant such that, for ,

Lemma 12 (see [3, Lemma 1.4 (b), (c)]). Suppose that . Then there exist positive constants and such that, for ,

Suppose . Then there exist positive constants and such that, for , ,

Lemma 13 (Caccioppoli type inequality). Let , , and be integers, , , and . Assume that in for some . Then there exists a positive constant such that

For readers’ convenience, we give the proof of Lemma 13 at the end of this section. If is an even number, then letting in Lemma 13, we have the following.

Corollary 14 (see [2, Lemma 3.5]). Let be an integer, an even number, , and . Assume that in for some . Then there exists a positive constant such that

Lemma 15 (Caccioppoli type inequality). Let , , and be integers, , , and . Assume that in for some . Then there exists a positive constant such that

If is an odd number, then letting in Lemma 15, we have the following.

Corollary 16 (see [2, Lemma 3.5]). Let be an integer, an odd number, , and . Assume that in for some . Then there exists a positive constant such that

Lemma 17. Let , , and be integers, , , and . Assume that , , in for some . Then there exists a positive constant such that

In Lemma 17, the cases and have been proved (see [3, page 523] and [1, Corollary 5.6]).

Proof of Lemma 17. We proceed following the proof of [1, Corollary 5.6]. We prove Lemma 17 for which satisfies , , in . Let and be nonnegative integers and choose such that on and . Let be the fundamental solution for . Note that

where

and the summation is taken over all integers and satisfying , , and . Then, integrating by parts, for we have

Let

Then we have

It remains to estimate . Since , it follows that . Hence is subharmonic. Then for all ,

Then we have

Then using Lemmas 13 and 15 repeatedly, we have

It follows that

From (26) we have, for all ,

Then we arrive at the desired estimate.

Lemma 18. Let , , and be integers, , , and . Suppose that for some , . Assume also that in for some . Then there exists a positive constant such that

where .

In Lemma 18, the cases and were shown in [3, Lemma 4.6] and [8, Lemma 7], respectively.

Proof of Lemma 18. We show Lemma 18 by a method similar to the one used in the proof of [3, Lemma 4.6]. Let such that on and , where and are nonnegative integers. We denote by the fundamental solution for . Note that

where the summation is taken over all integers and satisfying , , and . Then integrating by parts, for , we have

It then follows from the well-known theorem on fractional integrals that

where and we have used Remark 4 (1).

At the end of this section, we give the following.

Proof of Lemma 13. Let and be nonnegative integers and choose such that on and . Multiplying by and integrating over by integrating by parts, we have

where the summation is taken over all integers and satisfying , , and . Let be a positive real number which will be determined later. Then the right-hand side of (32) is bounded by

Then choosing such that , we arrive at the desired estimate.

3. Estimates for

In this section, we assume that Theorem 9 is true for and show estimates for which is needed to prove the case in Theorem 9.

Lemma 19. Let , , and be integers, , and . Suppose that . Assume also that Theorem 9 is true for . Then there exists a positive constant such that

where .

Proof. We can prove Lemma 19 by the same way as in the proof of [3, Corollary 2.8]. It follows from the inductive assumption that

If , then

Let . Then using Hölder's inequality and (35) we have

It follows that

Finally, note that, by the inductive assumption and Lemma 12 (1),

if we choose . Then Lemma 19 follows.

We also need the following lemma.

Lemma 20. Let , , and be integers, , , and . Suppose that for some , . Assume also that Theorem 9 is true for . Then for there exists a positive constant such that

where .

Proof. We show Lemma 20 by a method similar to the one used in the proof of [3, Theorem 4.13]. Suppose that for some . Then for some , . Let

The adjoint of is given by

By duality, it suffices to show that

where . Let . We choose and such that , . Thus . By Hölder’s inequality we have

Let be a positive integer and . It follows from Lemma 18, Lemma 12 (1), and the inductive assumption that

where is a finite integer not depending on and . Thus

where we choose and is the Hardy-Littlewood maximal operator. Hence it follows that

Then (43) follows since .

For the case in Lemma 20, if , letting , we have the following.

Corollary 21. Let , , and be integers, , , and . Suppose that . Assume also that Theorem 9 is true for . Then there exists a positive constant such that

4. Proof of Theorem 9 (Case)

In this section, we assume that Theorem 9 is true for and prove the case in Theorem 9. It follows easily from the following lemma.

Lemma 22. Let , , and be integers, , and . Suppose that . Assume also that in for some and Theorem 9 is true for . Then for any positive integer there exists a positive constant such that

Assuming Lemma 22 for the moment, we give

Proof of Theorem 9  . Fix , and put . Then is a solution of on . Using the estimate and (49), we arrive at the desired estimate.

To prove Lemma 22, we need some lemmas.

Lemma 23. Let , , and be integers, , , and . Suppose that . Assume also that Theorem 9 is true for . Then there exists a positive constant such that

where .

Proof. By the case in Lemma 19 and Corollary 21, we have

If and , then letting and in Lemma 18 we have the following.

Corollary 24. Let , , and be integers, , , and . Suppose that . Assume also that in for some . Then there exists a positive constant such that

Now we are ready to give the following.

Proof of Lemma 22. Let such that on , , where is an integer satisfying . Applying Lemma 23 to and using Corollary 14 we have

From Lemma 12 (2) it follows for each integer satisfying that

Then we have

Repeating the above argument, for any positive integer we have

Then using Lemma 17, Corollary 24, and estimate (56), we arrive at the desired estimate.

5. Proof of Theorem 10

In this section, we prove Theorem 10 which states derivative estimates of the fundamental solution for . We arrive at Theorem 10 combining Lemmas 25 with 22.

Lemma 25. Let , , and be integers, , , and . Suppose that . Assume also that in for some . Then there exist positive constants and such that

Proof. Let such that on and , where and are nonnegative integers. We use (29); integrating by parts, we have

Since , it follows that for some . We choose such that and . By Hölder’s inequality we have

where we have used Remark 4 (1). From (59) we have, for all ,

Using Lemma 12 (1), we have

Then the proof is complete.

6. Proofs of Theorems 5 and 8

In this section, we prove Theorems 5 and 8.

Proof of Theorem 5. We assume that for some . Then for some . Let

The adjoint of is given by

Let . By duality, it suffices to show that

where . Let . By Theorems 9 and 10 and Hölder's inequality we have

where . Since we have

For the case , by Lemma 11 and (4), we have

For the case , by the doubling condition (9) and (4), we have

where . Hence, choosing , we obtain

It follows that

Then (64) follows since .

Proof of Theorem 8. We assume that for some , . Then for some , . We use (62) and (63) again and it suffices to show (64) for and . We choose and such that and . Thus,

Let . By Hölder’s inequality we have

It follows from Lemma 18 and Theorem 9 that

Thus, as in the proof of Theorem 5, we have

Then (64) follows since .

7. Remarks

We show estimate (57) under another assumption.

Lemma 26. Let , , and be integers, , , and . Suppose that and there exists a positive constant such that . Assume also that in for some . Then there exist constants and such that

Proof. We use (58). From Lemma 12 (1) we have

Then by the same argument as in the proof of Lemma 25, we arrive at the desired estimate.

The following theorem immediately follows from Lemmas 22 and 26.

Theorem 27. Let and be integers, , , and . Suppose and there exists a positive constant such that . Then for any positive integer there exists a constant such that

Using Theorem 27 we can prove the following theorem by the same way as the proof of Lemma 19.

Theorem 28. Let , , and be integers, , , and . Suppose that and there exists a positive constant such that . Then there exist positive constants such that (6) holds, where .

We obtain the following pointwise estimate stated with the Hardy-Littlewood maximal operator.

Theorem 29. Let , , and be integers, , , and . Suppose that and there exists a positive constant such that . Then there exists a constant such that

In Theorem 29, the cases and were shown in [5, Theorem 1.3] and [8, Lemma 10], respectively.

Proof of Theorem 29. Let . Since , it follows from Theorems 9 and 27 that

Then choosing we arrive at the desired estimate.

Remark 30 (see [5, Remark 1.6]). Once we obtain the estimate (78), we can obtain boundedness of the operator on any space on which is bounded, for example, Morrey spaces (see [12]).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The author is grateful to the referee(s) for the helpful comment. The author is also grateful to Professor Kazuhiro Kurata, who read the first version of this paper and gave useful comment. This work was supported by Grant-in-Aid for Scientific Research (C) (no. 25400152), Japan Society for the Promotion of Science.

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