Weighted L p estimates for the elliptic Schrödinger operator

In this paper we study weighted Lp estimates for the elliptic Schrödinger operator P = −∆ + V(x) with non-negative potentials V(x) on Rn (n ≥ 3) which belongs to certain reverse Hölder class.


Introduction
Shen [19] proved the L p boundedness with 1 < p ≤ 2 of the nontangential maximal function of ∇u for the L p -Neumann problem of the elliptic Schrödinger operator with V ∈ V ∞ (see Definition 1.1 ) in a domain Ω ⊂ R n .Moreover, Shen [20] has obtained the following L p estimates for (1.1) for 1 < p ≤ q, assuming that V ∈ V q for some q ≥ n/2.In this paper we consider weighted L p estimates for the elliptic Schrödinger operator (1.1) Definition 1.1.The function V(x) is said to belong to the reverse Hölder class V q for some 1 < q ≤ ∞ if V ∈ L q loc (R n ), V ≥ 0 almost everywhere and there exists a constant C such that for all balls B r of R n , B r V q (x) dx If q = ∞, then the left-hand side is the essential supremum on B r , i.e., sup In fact, if V ∈ V ∞ , it clearly implies V ∈ V q for every q > 1.
We use the Hardy-Littlewood maximal function which controls the local behavior of a function.
Definition 1.2.Let v be a locally integrable function.The Hardy-Littlewood maximal function Mv(x) is defined as where the supremum is taken over all cubes Q in R n containing x.
It is well known that the maximal functions satisfy strong p-p estimate for any 1 < p < ∞ and weak (1, 1) estimate (see [21]).
Definition 1.3.A q for some q > 1 is the class of the Muckenhoupt weights: w ∈ A q if w ∈ L 1 loc (R n ), w > 0 almost everywhere and there exists a constant C such that for all balls B r in where Ω ⊂ R n .Furthermore, the corresponding weighted Lebesgue space L q w (Ω) consists of all functions h which satisfy Remark 1.4.We remark that A q 1 ⊂ A q 2 for any 1 < q 1 ≤ q 2 < ∞ (see [21, p. 195]).
Lemma 1.5.If w ∈ A q with q > q 1 > 1, then we have Weighted L p estimates for the elliptic Schrödinger operator 3 Proof.From Hölder's inequality we have Since w ∈ A q with q > q 1 > 1, from Remark 1.4 we find that w ∈ A q/q 1 .Furthermore, we conclude that ) and w > 0 almost everywhere.This finishes our proof.
Lemma 1.8 (see [22,Theorem 3.5 in Chapter 9]).If w ∈ A q for some q > 1, then there exists a small positive constant 0 < 1 and a constant C 2 > 1 such that F. Yao Lemma 1.9.If w ∈ A q for some q > 1 and B r ⊂ B R ⊂ R n , then there exists σ > 0 such that w(B r ) Proof.We first conclude that by using Hölder's inequality.Thus, it follows from the lemma above that , which finishes our proof by selecting σ = 0 /(1 + 0 ).Now let us state the main results of this work: Theorem 1.10 and Theorem 1.11.We shall give the direct proofs of the main results via the maximal function approach which was employed by [1,5,7,13,15,16,17].
then we have then we have Remark 1.12.Assume that u ∈ C ∞ 0 (R n ) and V ∈ V q with 1 < p ≤ q and q ≥ n/2.The authors of [4] proved that for (1.3) and the general case.

Proofs of the main results
In this section we shall finish the proofs of the main results: Theorem 1.10 and Theorem 1.11.

Proof of Theorem 1.10
We first give the following Calderón-Zygmund decomposition, which is much influenced by [14].Lemma 2.1.Let D be a cube in R n and A, B ⊂ D be measurable sets.Assume that 0 < w(A) < µw(D) for 0 < µ < 1.Then there exists a sequence of disjoint cubes {Q k } k∈N satisfying ) disjoint cubes with the same size.Therefore, we obtain a sequence of disjoint cubes {Q k } k∈N which satisfies ( 2)-( 3) by repeating the process above.If x ∈ D \ {Q k } k∈N , then there is a sequence of cubes P i containing x with the diameters of P i converging to 0 and From elementary measure theory and the fact that w(x) > 0 almost everywhere we can conclude that for almost every x ∈ D \ {Q k } k∈N , x ∈ D \ A. That is say, (1) is true.
2. Let Q k be the predecessor (father) of Q k .Now we choose a disjoint predecessor subsequence Q k j still denoted by Q k such that k∈N Q k ⊂ k∈N Q k .Thus, from (1), (3) and the hypothesis (2.1) we deduce that which finishes our proof.
Next, we shall prove the following important result.
Lemma 2.2.Assume that 1 < q < p.For any µ, α ∈ (0, 1) there exist two constants 2) Proof. 1. From the hypothesis (2.2) there exists Let v 1 be the solution of where f is the zero extention of f from 4Q to R n .Then from the elementary L p -type estimates we have Therefore, from (2.4) we conclude that 4Q From the definition of f , we find that h 1 satisfies Moreover, it follows from W 2,∞ loc regularity that sup where M 1 > 1 only depends on n.

2.
The proof is totally similar to the proof of Lemma 2.8.Here we omit the details.

Proof of Theorem 1.11
We first recall the following result (see [21, p. 195]).
holds for any nonnegative function g and all cubes Q, where Furthermore, we have the following local boundedness property.
where C depends on n.
for some r 0 > 0. Recalling the elementary local boundedness property of the second-order elliptic equation (see [9,Theorem 9.20], or [10, Theorem 4.1]), we have sup for any r > 0. Then using the above inequality and Lemma 2.6 with r = 1 t , we find that sup V|h| dx.
This completes our proof.
Next, we shall prove the following important result.

F. Yao
Lemma 2.8.For any µ, α ∈ (0, 1) there exist two constants Proof. 1. From the hypothesis (2.11) there exists (2.12) Let v be the solution of where f is the zero extention of f from 4Q to R n .Then recalling the well-known L 1 estimate (see [3,8]), we have where N 2 := max {2N 1 , 3 n }.Actually, from (2.17) we find that Therefore, from (2.12) we find that by choosing δ small enough satisfying the last inequality.Thus we complete the proof.
Furthermore, we can directly obtain the following result from the lemma above.

F. Yao
Then Lemma 1.9 implies that w ({x which completes our proof by selecting µ = C 2 µ σ 1 .
Furthermore, we can obtain the following result.
Corollary 2.10.Let D be a cube in R n .Assume that w, µ, δ, N 3 satisfy the same conditions as those in Corollary 2.9.If w ({x Proof.We denote Then A, B ⊂ D and w (A) ≤ µw(D).Therefore, it follows from Lemma 2.1 that there exists a sequence of disjoint cubes {Q k } satisfying where Q k is the predecessor of Q k , then we obtain (2.21) with Q repacing by Q k .Furthermore, it follows from Corollary 2.9 that So, we get a contradiction with (2) and then know that w Q k ∩ B > 1 2 w Q k .Finally, we can use Lemma 2.1 again to get that w (A) ≤ 2µw (B) , which implies (2.22) is true.Thus, we finish the proof.
Corollary 2.11.Assume that µ ∈ (0, 1) with C 2 µ σ < 1 and w, δ, N 3 satisfy the same conditions as those in Corollary 2.9.For any λ > 0 we have (2.23) Proof.Without loss of generality, we may as well assume that λ = 1.Let where {Q i } is a sequence of disjoint same side-length cubes.Moreover, from the weak 1-1 estimate and L 1 estimate (see [3,8]) we conclude that We may as well assume that f ∈ C ∞ 0 (R n ) via an elementary approximation argument.So, we can obtain |{x ∈ Q by selecting |Q i | large enough for i ∈ N. Furthermore, from Lemma 1.9 we have w ({x Thus, by Corollary 2.10 we obtain w ({x which implies that the desired estimate (2.23) is true.This finishes our proof.