Weighted Morrey spaces related to certain nonnegative potentials and Riesz transforms

Let $\mathcal L=-\Delta+V$ be a Schr\"odinger operator, where $\Delta$ is the Laplacian on $\mathbb R^d$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_q$ for $q\geq d$. The Riesz transform associated with the operator $\mathcal L=-\Delta+V$ is denoted by $\mathcal R=\nabla{(-\Delta+V)}^{-1/2}$ and the dual Riesz transform is denoted by $\mathcal R^{\ast}=(-\Delta+V)^{-1/2}\nabla$. In this paper, we first introduce some kinds of weighted Morrey spaces related to certain nonnegative potentials belonging to the reverse H\"older class $RH_q$ for $q\geq d$. Then we will establish the boundedness properties of the operators $\mathcal R$ and its adjoint $\mathcal R^{\ast}$ on these new spaces. Furthermore, weighted strong-type estimate and weighted endpoint estimate for the corresponding commutators $[b,\mathcal R]$ and $[b,\mathcal R^{\ast}]$ are also obtained. The classes of weights, the classes of symbol functions as well as weighted Morrey spaces discussed in this paper are larger than $A_p$, $\mathrm{BMO}(\mathbb R^d)$ and $L^{p,\kappa}(w)$ corresponding to the classical Riesz transforms ($V\equiv0$).

1. Introduction 1.1. The critical radius function ρ(x). Let d ≥ 3 be a positive integer and R d be the d-dimensional Euclidean space. A nonnegative locally integrable function V (x) on R d is said to belong to the reverse Hölder class RH q for some exponent 1 < q < ∞, if there exists a positive constant C > 0 such that the following reverse Hölder inequality holds for every ball B in R d . For given V ∈ RH q with q ≥ d, we introduce the critical radius function ρ(x) = ρ(x, V ) which is given by where B(x, r) denotes the open ball centered at x and with radius r. It is well known that 0 < ρ(x) < ∞ for any x ∈ R d under our assumption (see [9]). We need the following known result concerning the critical radius function. Lemma 1.1 ([9]). If V ∈ RH q with q ≥ d, then there exist two constants C > 0 and N 0 ≥ 1 such that for all x, y ∈ R d . As a straightforward consequence of (1.2), we have that for all k = 1, 2, 3, . . . , the following estimate is valid for any y ∈ B(x, r) with x ∈ R d and r > 0. Boundedness properties of R and its adjoint R * have been obtained by Shen in [9], where he showed that they are all bounded on L p (R d ) for any 1 < p < ∞ when V ∈ RH q with q ≥ d. Actually, R and its adjoint R * are standard Calderón-Zygmund operators in such a situation. The operators R and R * have singular kernels that will be denoted by K(x, y) and K * (x, y), respectively. For such kernels, we have the following key estimates, which can be found in [9] and [2,3].
Lemma 1.2. Let V ∈ RH q with q ≥ d. For any positive integer N, there exists a positive constant C N > 0 such that 1.3. A ρ,∞ p weights. A weight will always mean a nonnegative function which is locally integrable on R d . Given a Lebesgue measurable set E and a weight w, |E| will denote the Lebesgue measure of E and Given B = B(x 0 , r) and t > 0, we will write tB for the t-dilate ball, which is the ball with the same center x 0 and with radius tr. In [1] (see also [2,3]), Bongioanni, Harboure and Salinas introduced the following classes of weights that are given in terms of the critical radius function (1.1). Following the terminology of [1], for given 1 < p < ∞, we define where A ρ,θ p is the set of all weights w such that θ holds for every ball B = B(x 0 , r) ⊂ R d with x 0 ∈ R d and r > 0, where p ′ is the dual exponent of p such that 1/p + 1/p ′ = 1. For p = 1 we define holds for every ball B = B(x 0 , r) in R d . For θ > 0, let us introduce the maximal operator that is given in terms of the critical radius function (1.1).
Observe that a weight w belongs to the class A ρ,∞ 1 if and only if there exists a positive number θ > 0 such that M ρ,θ w ≤ Cw, where the constant C > 0 is independent of w. Since where A p denotes the classical Muckenhoupt's class (see [4,Chapter 7]), and hence A p ⊂ A ρ,∞ p . In addition, for some fixed θ > 0, whenever 1 ≤ p 1 < p 2 < ∞. Now, we present an important property of the classes of weights in A ρ,θ p with 1 ≤ p < ∞, which was given by Bongioanni et al. in [1,Lemma 5].
Given a weight w on R d , as usual, the weighted Lebesgue space L p (w) for 1 ≤ p < ∞ is defined to be the set of all functions f such that We also denote by W L 1 (w) the weighted weak Lebesgue space consisting of all measurable functions f for which Recently, Bongioanni et al. [1] obtained weighted strong-type and weaktype estimates for the operators R and R * defined in (1.4) and (1.5). Their results can be summarized as follows: ). Let 1 < p < ∞ and w ∈ A ρ,∞ p . If V ∈ RH q with q ≥ d, then the operators R and R * are all bounded on L p (w). Theorem 1.6 ([1]). Let p = 1 and w ∈ A ρ,∞ 1 . If V ∈ RH q with q ≥ d, then the operators R and R * are all bounded from L 1 (w) into W L 1 (w).
1.4. The space BMO ρ,∞ (R d ). We denote by T either R or R * . For a locally integrable function b on R d (usually called the symbol ), we will also consider the commutator operator Recently, Bongioanni et al. [3] introduced a new space BMO ρ,∞ (R d ) defined by where for 0 < θ < ∞ the space BMO ρ,θ (R d ) is defined to be the set of all locally integrable functions b satisfying for all x 0 ∈ R d and r > 0, b B(x 0 ,r) denotes the mean value of b on B(x 0 , r), that is, A norm for b ∈ BMO ρ,θ (R d ), denoted by b BMO ρ,θ , is given by the infimum of the constants satisfying (1.9), or equivalently, where the supremum is taken over all balls B(x 0 , r) with x 0 ∈ R d and r > 0. With the above definition in mind, one has for 0 < θ 1 < θ 2 < ∞, and hence BMO(R d ) ⊂ BMO ρ,∞ (R d ). Moreover, the classical BMO space [5] is properly contained in BMO ρ,∞ (R d ) (see [2,3] for some examples). We need the following key result for BMO ρ,θ (R d ), which was proved by Tang in [10].
Then there exist two positive constants C 1 and C 2 such that for any given ball B(x 0 , r) in R d and for any λ > 0, we have where θ * = (N 0 + 1)θ and N 0 is the constant appearing in Lemma 1.1.
As a consequence of Proposition 1.7 and Lemma 1.4, we have the following result: Then there exist positive constants C 1 , C 2 and η > 0 such that for any given ball B(x 0 , r) in R d and for any λ > 0, we have where θ * = (N 0 + 1)θ and N 0 is the constant appearing in Lemma 1.1.

1.5.
Orlicz spaces. In this subsection, let us give the definition and some basic facts about Orlicz spaces. For more information on this subject, the reader may consult the book [8]. Recall that a function A : [0, ∞) → [0, ∞) is called a Young function if it is continuous, convex and strictly increasing with An important example of Young function is A(t) = t · (1 + log + t) m with some 1 ≤ m < ∞. Given a Young function A and a function f defined on a ball B, we consider the A-average of a function f given by the following Luxemburg norm:  Such a functionĀ is also a Young function. It is well known that the following generalized Hölder inequality in Orlicz spaces holds for any given ball B ⊂ R d : In particular, for the Young function A(t) = t · (1 + log + t), the Luxemburg norm will be denoted by · L log L,B = · A,B . A simple computation shows that the complementary Young function of A(t) = t · (1 + log + t) is A(t) = exp(t) − 1. The corresponding Luxemburg norm will be denoted by · exp L,B = · Ā ,B . In this situation, we have We next define the weighted A-average of a function f over a ball B. Given a Young function A and a weight function w, let (see [8] for instance) Also, the complementary Young function of Φ is given byΦ(t) = e t −1 with corresponding Luxemburg norm denoted by · exp L(w),B . Given a weight w on R d , we can also show the weighted version of (1.12). That is, the following generalized Hölder inequality in the weighted setting holds for every ball B in R d . It is a simple but important observation that for any ball B in R d , This is because t ≤ t · (1 + log + t) for all t > 0. So we have In [2], Bongioanni et al.obtained weighted strong (p, p), 1 < p < ∞, and weak L log L estimates for the commutators of the Riesz transform and its adjoint associated with the Schrödinger operator L = −∆ + V , where V satisfies some reverse Hölder inequality. Their results can be summarized as follows: , then for any given λ > 0, there exists a positive constant where Φ(t) = t · (1 + log + t) and log + t := max{log t, 0}; that is, In this paper, firstly, we will define some kinds of weighted Morrey spaces related to certain nonnegative potentials. Secondly, we prove that the Riesz transform R and its adjoint R * are both bounded operators on these new spaces. Finally, we also obtain the weighted boundedness for the commu- Throughout this paper C denotes a positive constant not necessarily the same at each occurrence, and a subscript is added when we wish to make clear its dependence on the parameter in the subscript. We also use a ≈ b to denote the equivalence of a and b; that is, there exist two positive constants

our main results
In this section, we introduce some types of weighted Morrey spaces related to the potential V and then give our main results. Definition 2.1. Let 1 ≤ p < ∞, 0 ≤ κ < 1 and w be a weight. For given 0 < θ < ∞, the weighted Morrey space L p,κ ρ,θ (w) is defined to be the set of all L p locally integrable functions f on R d for which , is given by the infimum of the constants in (2.1), or equivalently, where the supremum is taken over all balls B in R d , x 0 and r denote the center and radius of B respectively. Define Note that this definition does not coincide with the one given in [7] (see also [11] for the unweighted case), but in view of the space BMO ρ,∞ (R d ) defined above it is more natural in our setting. Obviously, if we take θ = 0 or V ≡ 0, then this new space is just the weighted Morrey space L p,κ (w), which was first defined by Komori and Shirai in [6] (see also [12]). Definition 2.2. Let p = 1, 0 ≤ κ < 1 and w be a weight. For given 0 < θ < ∞, the weighted weak Morrey space W L 1,κ ρ,θ (w) is defined to be the set of all measurable functions f on R d for which Correspondingly, we define Clearly, if we take θ = 0 or V ≡ 0, then this space is just the weighted weak Morrey space W L 1,κ (w) (see [13]). According to the above definitions, one has ρ,θ (w) (or W L 1,κ ρ,θ (w)) could be viewed as an extension of weighted (or weak) Lebesgue space (when κ = θ = 0). Naturally, one may ask the question whether the above conclusions (i.e., Theorems 1.5 and 1.6 as well as Theorems 1.9 and 1.10) still hold if replacing the weighted Lebesgue spaces by the weighted Morrey spaces. In this work, we give a positive answer to this question. Our main results in this work are presented as follows: Theorem 2.4. Let p = 1, 0 < κ < 1 and w ∈ A ρ,∞ 1 . If V ∈ RH q with q ≥ d, then the operators R and R * are both bounded from L 1,κ ρ,∞ (w) into W L 1,κ ρ,∞ (w).
To deal with the commutators in the endpoint case, we need to consider a new kind of weighted Morrey spaces of L log L type.
Definition 2.6. Let p = 1, 0 ≤ κ < 1 and w be a weight. For given 0 < θ < ∞, the weighted Morrey space (L log L) 1,κ ρ,θ (w) is defined to be the set of all locally integrable functions f on R d for which Concerning the continuity properties of [b, R] and [b, R * ] in the weighted Morrey spaces of L log L type, we have , then for any given λ > 0 and any given ball B = B(x 0 , r) of R d , there exist some constants C > 0 and ϑ > 0 such that the following inequalities In this section, we will prove the conclusions of Theorems 2.3 and 2.4.
Proof of Theorem 2.3. We denote by T either R or R * . By definition, we only have to show that for any given ball B = B(x 0 , r) of R d , there is some ϑ > 0 such that holds for any f ∈ L p,κ ρ,∞ (w) with 1 < p < ∞ and 0 < κ < 1. Suppose that f ∈ L p,κ ρ,θ (w) for some θ > 0 and w ∈ A ρ,θ ′ p for some θ ′ > 0. We decompose f as where 2B is the ball centered at x 0 and radius 2r > 0, and χ 2B is the characteristic function of 2B. Then by the linearity of T , we write We now analyze each term separately. By Theorem 1.5, we get Since w ∈ A ρ,θ ′ p with 1 < p < ∞ and 0 < θ ′ < ∞, then we know that the following inequality is valid. In fact, for 1 < p < ∞, by Hölder's inequality and the definition of A ρ,θ ′ p , we have If we take (x) = χ B (x), then the above expression becomes which in turn implies (3.2). Therefore, where ϑ ′ := κ · θ ′ + θ. For the other term I 2 , notice that for any x ∈ B and y ∈ (2B) c , one has |x − y| ≈ |x 0 − y|. It then follows from Lemma 1.2 that for any x ∈ B(x 0 , r) and any positive integer N, In view of (1.3) in Lemma 1.1, we further obtain (3.5) Moreover, by using Hölder's inequality and A ρ,θ ′ p condition on w, we get Hence, Recall that w ∈ A ρ,θ ′ p with 0 < θ ′ < ∞ and 1 < p < ∞, then there exist two positive numbers δ, η > 0 such that (1.7) holds. This allows us to obtain Thus, by choosing N large enough so that N > θ + θ ′ + η(1 − κ)/p, we then have

H. WANG
Summing up the above estimates for I 1 and I 2 and letting ϑ = max ϑ ′ , N · N 0 N 0 +1 , we obtain our desired inequality (3.1). This completes the proof of Theorem 2.3.
Proof of Theorem 2.4. We denote by T either R or R * . To prove Theorem 2.4, by definition, it suffices to prove that for any given ball B = B(x 0 , r) of R d , there is some ϑ > 0 such that holds for any f ∈ L 1,κ ρ,∞ (w) with 0 < κ < 1. Now suppose that f ∈ L 1,κ ρ,θ (w) for some θ > 0 and w ∈ A ρ,θ ′ 1 for some θ ′ > 0. We decompose f as Then for any given λ > 0, by the linearity of T , we can write 1 We first give the estimate for the term I ′ 1 . By Theorem 1.6, we get Since w ∈ A ρ,θ ′ 1 with 0 < θ ′ < ∞, similar to the proof of (3.2), we can also show the following estimate as well.
By selecting N large enough so that N > θ + θ ′ + η ′ (1 − κ), we thus have Here N is an appropriate constant. Summing up the above estimates for I ′ 1 and I ′ 2 , and then taking the supremum over all λ > 0, we obtain our desired inequality (3.6). This finishes the proof of Theorem 2.4.

Proofs of Theorems 2.5 and 2.7
For the results involving commutators, we need the following properties of BMO ρ,∞ (R d ) functions, which are extensions of well-known properties of BMO(R d ) functions.

Making change of variables, then we get
which yields the desired inequality if we choose C = C 1 pΓ(p) and µ = θ * + η/p.
Proof. Recall the following identity (see Proposition 1.1.4 in [4]) Using this identity and Proposition 1.8, we obtain where λ * is given by If we take γ small enough so that 0 < γ < C 2 , then the conclusion follows immediately.
with 0 < θ < ∞, then for any positive integer k, there exists a positive constant C > 0 such that for every ball Proof. For any positive integer k, we have Since for any 1 ≤ j ≤ k + 1, the following estimate holds trivially, and hence We obtain the desired result. This completes the proof. Now, we are in a position to prove our main results in this section.
Proof of Theorem 2.5. We denote by [b, T ] either [b, R] or [b, R * ]. By definition, we only need to show that for any given ball B = B(x 0 , r) of R d , there is some ϑ > 0 such that holds for any f ∈ L p,κ ρ,∞ (w) with 1 < p < ∞ and 0 < κ < 1, whenever b belongs to BMO ρ,∞ (R d ). Suppose that f ∈ L p,κ ρ,θ (w) for some θ > 0, Then by the linearity of [b, T ], we write Now we give the estimates for J 1 , J 2 , respectively. According to Theorem 1.9, we have Moreover, in view of the inequality (3.2), we get where ϑ ′ := θ ′ · κ + θ. On the other hand, by the definition (1.8), we can see that for any x ∈ B(x 0 , r), So we can divide J 2 into two parts: From the pointwise estimate (3.5) and (4.1) in Lemma 4.1, it then follows that Following along the same lines as that of Theorem 2.3, we are able to show that The last inequality is obtained by using (1.7). For any x ∈ B(x 0 , r) and any positive integer N, similar to the proof of (3.4) and (3.5), we can also deduce that where in the last inequality we have used (1.3) in Lemma 1.1. Hence, by the above pointwise estimate for ζ(x), Moreover, for each integer k ≥ 1, By using Hölder's inequality, the first term of the expression (4.6) is bounded by Since w ∈ A ρ,θ ′ p with 0 < θ ′ < ∞ and 1 < p < ∞, then by the definition of A ρ,θ ′ p , it can be easily shown that w ∈ A ρ,θ ′ p if and only if w −p ′ /p ∈ A ρ,θ ′ p ′ , where 1/p + 1/p ′ = 1 (see [10]). If we denote v = w −p ′ /p , then v ∈ A ρ,θ ′ p ′ .
This fact together with Lemma 4.1 implies Therefore, the first term of the expression (4.6) can be bounded by a constant times Since b ∈ BMO ρ,θ ′′ (R d ) with 0 < θ ′′ < ∞, then by Lemma 4.3, Hölder's inequality and the A ρ,θ ′ p condition on w, the latter term of the expression (4.6) can be estimated by Thus, in view of (4.7), Notice that w ∈ A ρ,θ ′ p with 0 < θ ′ < ∞. A further application of (1.7) yields Combining the above estimates for J 3 and J 4 , we get By choosing N large enough so that N > θ + θ ′ + θ ′′ + µ + η(1 − κ)/p, we thus have Finally, collecting the above estimates for J 1 , J 2 , and letting ϑ = max ϑ ′ , µ+ N · N 0 N 0 +1 , we obtain the desired result (4.3). The proof of Theorem 2.5 is finished.
as desired. Next let us deal with the term J ′ 2 . Taking into account of (4.4), we can divide it into two parts, namely, Since b ∈ BMO ρ,θ ′′ (R d ) for some θ ′′ > 0, from the pointwise inequality (3.5) and Chebyshev's inequality, we then have where in the last inequality we have used (4.1) in Lemma 4.1. Moreover, it follows directly from the condition A ρ,θ ′ 1 that for each integer k ≥ 1, Notice also that trivially (4.9) t ≤ t · (1 + log + t) = Φ(t), for any t > 0.
This fact along with (1.14) implies that for each integer k ≥ 1, .
On the other hand, it follows from the pointwise inequality (4.5) and Chebyshev's inequality that where the last inequality follows from (4.9). Furthermore, by the definition of A ρ,θ ′ 1 , we compute (4.10) By using generalized Hölder inequality (1.13), the first term of the expression (4.10) is bounded by where in the last inequality we have used the fact that b − b B exp L(w),B ≤ C 1 + r ρ(x 0 ) η ′ , for any ball B = B(x 0 , r) ⊂ R n , which is equivalent to the inequality (4.2) in Lemma 4.2. By Lemma 4.3 and (1.14), the latter term of the expression (4.10) can be estimated by Consequently,