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

Let V(x) be a nonnegative potential and consider the Schrödinger type operatorsHk = (−Δ) k + V k on R, where k is a positive integer and n ≥ 2k + 1. When V is a nonnegative polynomial, Zhong proved estimates of the fundamental solution forH1 andH2 and showed some estimates forH1 and H2 (see [1]). More precisely, he showed the L p boundedness of the operators VH 1 , V∇H 1 , and V∇H 2 , where j = 0, 1, 2, 3, 4. He also proved that the operators ∇H 1 and

As mentioned above, in [2], the authors proved some results on   = (−Δ)  +   , where  is a nonnegative polynomial and  is an integer,  ≥ 3.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   = (−Δ)  +   with potential  which belongs to the reverse Hölder class, which includes nonnegative polynomials.However, our results are only for  = 2  , where  is an integer,  ≥ 2.
(1) For 1 <  < ∞ one says that  ∈ ()  if  ∈   loc (R  ) and there exists a positive constant  such that holds for every  ∈ R  and 0 <  < ∞.
(2) One says that  ∈ () ∞ if  ∈  ∞ loc (R  ) and there exists a positive constant  such that holds for every  ∈ R  and 0 <  < ∞.
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  2  with potential  which belongs to the reverse Hölder class.Theorem 9. Let  and  be integers,  ≥ 0, and  ≥ 2 +1 + 1. Suppose that  ∈ () /2 .Then for any positive integer  there exists a positive constant   such that In Theorem 9, the cases  = 0 and  = 1 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  =  + 1.We also prove the following theorem which states derivative estimates of the fundamental solution.
Theorem 10.Let , , and  be integers,  ≥ 0,  ≥ 2 +1 + 1, and 1 ≤  ≤ 2 +1 − 1. Suppose that  ∈ () In Theorem 10, the cases  = 0 and  = 1 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  2  which are needed to prove the case  = +1 in Theorem 9.In Section 4, we prove the case  =  + 1 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) Suppose  ∈ () /2 .Then there exist positive constants   and  0 such that, for ,  ∈ R  , Lemma 13 (Caccioppoli type inequality).Let , , and  be integers,  ≥ 2, 1 ≤  ≤ /2, and 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  = /2 in Lemma 13, we have the following.
where the summation is taken over all integers  and  satisfying  ≥ 0,  ≥ 0, and 1 ≤  +  ≤ .Let  be a positive real number which will be determined later.Then the righthand side of (32) is bounded by Then choosing  such that ( − 1)( + 2) = 1, we arrive at the desired estimate.

Estimates for 𝐻 2 𝑙
In this section, we assume that Theorem 9 is true for  =  and show estimates for  2  which is needed to prove the case  =  + 1 in Theorem 9.

Proof of Theorem 9 (Case 𝑚 = 𝑙 + 1)
In this section, we assume that Theorem 9 is true for  =  and prove the case  =  + 1 in Theorem 9.It follows easily from the following lemma.
Proof.By the case  = 2 in Lemma 19 and Corollary 21, we have Now we are ready to give the following.

Proof of Theorem 10
In this section, we prove Theorem 10 which states derivative estimates of the fundamental solution for  2  .We arrive at Theorem 10 combining Lemmas 25 with 22.

Proofs of Theorems 5 and 8
In this section, we prove Theorems 5 and 8.