On characterization of Poisson integrals of Schrödinger operators with BMO traces
Abstract.
Let be a Schrödinger operator of the form acting on where the nonnegative potential belongs to the reverse Hölder class for some Let denote the BMO space on associated to the Schrödinger operator . In this article we will show that a function is the trace of the solution of where satisfies a Carleson condition
Conversely, this Carleson condition characterizes all the harmonic functions whose traces belong to the space . This result extends the analogous characterization founded by Fabes, Johnson and Neri in [11] for the classical BMO space of John and Nirenberg.
Key words and phrases:
Poisson integrals, Schrödinger operators, BMO space, Lipschitz space, Carleson measure, reverse Hölder inequality, Dirichlet problem.2010 Mathematics Subject Classification:
42B35, 42B37, 35J10, 47F05Contents
1. Introduction and statement of the main result
Consider the Laplace operator on the Euclidean space . A basic tool in harmonic analysis to study a (suitable) function on is to consider a harmonic function on which has the boundary value as . A standard choice for such a harmonic function is the Poisson integral and one recovers when letting . In other words, one obtains as the solution of the equation . This approach is intimately related to the study of singular integrals. For the classical case , , we refer the reader to Chapter 2 of the standard textbook [24].
At the endpoint space , the study of singular integrals has a natural substitution, the BMO space, i.e. the space of functions of bounded mean oscillation. A celebrated theorem of Fefferman and Stein [14] states that a BMO function is the trace of the solution of whenever satisfies
(1.1) 
where Expanding on this result, Fabes, Johnson and Neri [11] showed that condition (1.1) characterizes all the harmonic functions whose traces are in . The study of this topic has been widely extended to more general operators such as elliptic operators (instead of the Laplacian) and for domains other than such as Lipschitz domains. See for examples [4, 12, 13, 18].
The main aim of this article is to study a similar characterization to (1.1) for the Schrödinger operators with appropriate conditions on its potentials. Let us consider the Schrödinger operator
(1.2) 
As to the nonnegative potential , we assume that it is not identically zero and that for some , which by definition means that , and there exists a constant such that the reverse Hölder inequality
(1.3) 
holds for all balls in
The operator is a selfadjoint operator on . Hence generates the Poisson semigroup on . Since the potential is nonnegative, the semigroup kernels of the operators satisfy
for all and , where
is the kernel of the classical Poisson semigroup on . For , it is well known that the Poisson extension , is a solution to the equation
(1.4) 
with the boundary data on (see Remark 3.3 for below). The equation is understood in the weak sense, that is, is a weak solution of if it satisfies
In the sequel, we call such a function an harmonic function associated to the operator .
As mentioned above, we are interested in deriving the characterization of the solution to the equation in having boundary values with BMO data. Following [10], a locally integrable function belongs to BMO whenever there is constant so that
(1.5) 
for every ball , and
(1.6) 
for every ball with . Here and the critical radii above are determined by the function which takes the explicit form
(1.7) 
We define to be the smallest in the right hand sides of (1.5) and (1.6). Because of (1.6), this space is in fact a proper subspace of the classical BMO space of John and Nirenberg, and it turns out to be a suitable space in studying the case of the endpoint estimates for concerning the boundedness of some classical operators associated to such as the LittlewoodPaley square functions, fractional integrals and Riesz transforms (see [1, 2, 5, 6, 10, 17, 19]).
The following theorem is the main result of this article.
Theorem 1.1.
Suppose for some We denote by the class of all functions of the solution of in such that
(1.8) 
where Then we have

If , then there exists some such that , and
with some constant independent of and .

If , then the function satisfies estimates (1.8) with
We should mention that for the Schrödinger operator in (1.2), an important property of the class, proved in [15, Lemma 3], assures that the condition also implies for some and that the constant of is controlled in terms of the one of membership. This in particular implies for some strictly greater than However, in general the potential can be unbounded and does not belong to for any As a model example, we could take . Moreover, as noted in [21], if is any nonnegative polynomial, then satisfies the stronger condition
which implies for every with a uniform constant.
This article is organized as follows. In Section 2, we recall some preliminary results including the kernel estimates of the heat and Poisson semigroups of , the and spaces associated to the Schrödinger operators and certain properties of harmonic functions. In Section 3, we will prove our main result, Theorem 1.1. The proof of part (1) follows a similar method to that of [11], which depends heavily on three nontrivial results: Alaoglu’s Theorem on the weak compactness of the unit sphere in the dual of a Banach space, the duality theorem asserting that the space is the dual space of , and some specific properties of the Hardy space and the Carleson measure. For part (2), we will prove it by making use of the full gradient estimates on the kernel of the Poisson semigroup in the variables under the assumption on for some . This improves previously known results (see [6, 10, 17, 19]) which characterize the space in terms of Carleson measure which are only related to the gradient in the variable. In Section 4, we will extend the method for the space in Section 3 to obtain some generalizations to Lipschitztype spaces for .
Throughout the article, the letters “ ” and “ ” will denote (possibly different) constants that are independent of the essential variables.
2. Basic properties of the heat and Poisson semigroups of Schrödinger operators
In this section, we begin by recalling some basic properties of the critical radii function under the assumption (1.3) on (see Section 2, [10]).
Lemma 2.1.
Suppose for some There exist and such that for all
(2.1) 
In particular, when and .
It follows from Lemmas 1.2 and 1.8 in [21] that there is a constant such that for a nonnegative Schwartz class function there exists a constant such that
(2.2) 
where and
Let be the heat semigroup associated to :
(2.3) 
From the FeynmanKac formula, it is wellknown that the kernel of the semigroup satisfies the estimate
(2.4) 
for all and , where
(2.5) 
is the kernel of the classical heat semigroup on . This estimate can be improved in time when satisfies the reverse Hölder condition for some . The function arises naturally in this context.
Lemma 2.2 (see [10]).
Suppose for some For every there exist the constants and such that for ,
Lemma 2.3 (see [9]).
Suppose for some There exists a nonnegative Schwartz function on such that
where and
The Poisson semigroup associated to can be obtained from the heat semigroup (2.2) through Bochner’s subordination formula (see [23]):
(2.6) 
From (2), the semigroup kernels , associated to , satisfy the following estimates. For its proof, we refer to [19, Proposition 3.6].
Lemma 2.4.
Suppose for some For any and every , there exists a constant such that


For every ,

Recall that a Hardytype space associated to was introduced by J. Dziubański et al. in [8, 9, 10], defined by
(2.7) 
with
For the above class of potentials, admits an atomic characterization, where cancellation conditions are only required for atoms with small supports. It can be verified that for every , for fixed and with
(2.8) 
Indeed, from (ii) of Lemma 2.4 we have that for a fixed
which, in combination with the fact that , shows estimate (2.8).
Lemma 2.5.
Suppose for some Then the dual space of is , i.e.,
In the sequel, we may sometimes use capital letters to denote points in , e.g., and set
For simplicity we will denote by the full gradient in . We now recall a Moser type local boundedness estimate (see for instance, [17]) and include a proof here for the sake of selfcontainment.
Lemma 2.6.
Suppose for some Let be a weak solution of in the ball . Then for any , there exists a constant such that
Proof.
It is enough to show that is a subharmonic function. Since for any with , we have that (see for instance, [16, Lemma 3.3]). This gives
The desired result follows readily. ∎
Lemma 2.7.
Suppose for some Assume that is a weak solution of in . Also assume that there is a such that
(2.9) 
Then in .
Proof.
Fix an , we let such that on , and Following [22, Lemma 6.1], one writes
Then we have
where denotes the fundamental solution of in . Hence, for any ,
(2.10)  
where we have used the Hölder inequality and Caccioppoli’s inequality.
From the upper bound of in [21, Theorem 2.7], we have that for every and every ,
(2.11)  
Recall that the condition implies for some . It follows from Lemma 2.9 of [22] that if (see also Lemma 2.6). Then we have
This, in combination with (2.10) and (2.11), yields that for every ,
Letting , we obtain that and therefore, in the whole The proof is complete. ∎
Remark 2.8.
At the end of this section, we establish the following characterization of Poisson integrals of Schrödinger operators with functions in
Proposition 2.9.
Suppose for some If is a continuous weak solution of in and there exist a constant and a , such that
for all , then

when , is the Poisson integral of a function in

if , is the PoissonStieltjes integral of a finite Borel measure; if, in addition, is Cauchy in the norm as then is the Poisson integral of a function in
Proof.
The proof of Proposition 2.9 is standard (see for instance, Theorem 2.5, Chapter 2 in [24]). We give a brief argument of this proof for completeness and the convenience of the reader.
If and for all , then there exists a sequence such that , and a function such that converges weakly to as That is, for each
If there exists a finite Borel measure that is the weak limit of a sequence . That is, for each in ,
For any , we take for and also belongs to we have, in particular,
when , and
when
Since is continuous, . It is well known that if , then for almost every Set . The function satisfies . Define,
Then satisfies
where is an extension operator of on . Observe that if with then it can be verified that on . To apply Lemma 2.7, we need to verify (2.9). Indeed,
and so (2.9) holds. By Lemma 2.7, we have that , and then , that is,
Therefore,
when , and when The proof is complete. ∎
3. Proof of the Main Theorem
3.1. Existence of boundary values of harmonic functions
Lemma 3.1.
For every and for every , there exists a constant such that
hence . Therefore for all , exists everywhere in