# Weighted Morrey Spaces Related to Certain Nonnegative Potentials and Riesz Transforms.

1. Introduction1.1. The Critical Radius Function [rho](x). Let d [greater than or equal to] 3 be a positive integer and let [R.sup.d] be the d-dimensional Euclidean space. A nonnegative locally integrable function V(x) on [R.sup.d] is said to belong to the reverse Holder class R[H.sub.q] for some exponent 1 < q < [infinity], if there exists a positive constant C > 0 such that the reverse Holder inequality

[mathematical expression not reproducible] (1)

holds for every ball B in [R.sup.d]. For given V [member of] R[H.sub.q] with q [greater than or equal to] d, we introduce the critical radius function [rho](x) = [rho](x, V) which is given by

[mathematical expression not reproducible], (2)

where B(x, r) denotes the open ball centered at x and with radius r. It is well known that 0 < [rho](x) < [infinity] for any x [member of] [R.sup.d] under our assumption (see [1]). We need the following known result concerning the critical radius function.

Lemma 1 (from [1]). If V [member of] R[H.sub.q] with q [greater than or equal to] d, then there exist two constants C > 0 and [N.sub.0] [greater than or equal to] 1 such that

[mathematical expression not reproducible] (3)

for all x,y [member of] [R.sup.d]. As a straightforward consequence of (3), we have that, for all k = 1,2,3,..., the estimate

[mathematical expression not reproducible] (4)

is valid for any y [member of] B(x, r) with x [member of] [R.sup.d] and r > 0.

1.2. Schrodinger Operators. On [R.sup.d], d [greater than or equal to] 3, we consider the Schrodinger operator

L := -[DELTA] + v; (5)

where V [member of] R[H.sub.q] for q [greater than or equal to] d. The Riesz transform associated with the Schrodinger operator L is defined by

R := [nabla][L.sup.-1/2], (6)

and the associated dual Riesz transform is defined by

[R.sup.*] := [L.sup.-1/2][nabla]. (7)

Boundedness properties of R and its adjoint [R.sup.*] have been obtained by Shen in [1], where he showed that they are all bounded on [L.sup.p]([R.sup.d]) for any 1 < p < [infinity] when V [member of] R[H.sub.q] with q [greater than or equal to] d. Actually, R and its adjoint [R.sup.*] are standard Calderon-Zygmund operators in such a situation. The operators R and [R.sup.*] have singular kernels that will be denoted by K(x,y) and [K.sup.*](x,y), respectively. For such kernels, we have the following key estimates, which can be found in [1-3].

Lemma 2. Let V [member of] R[H.sub.q] with q [greater than or equal to] d. For any positive integer N, there exists a positive constant [C.sub.N] > 0 such that

[mathematical expression not reproducible]. (8)

1.3. [A.sup.[rho],[infinity].sub.p] Weights. A weight will always mean a nonnegative function which is locally integrable on [R.sup.d]. Given a Lebesgue measurable set E and a weight w, [absolute value of E] will denote the Lebesgue measure of E and

w(E) = [[integral].sub.E] w(x)dx. (9)

Given B = B([x.sub.0], r) and t > 0, we will write tB for the t-dilate ball, which is the ball with the same center [x.sub.0] and with radius tr. In [4] (see also [2, 3]), Bongioanni, Harboure, and Salinas introduced the following classes of weights that are given in terms of critical radius function (2). Following the terminology of [4], for given 1< p < [infinity], we define

[mathematical expression not reproducible], (10)

where [A.sup.[rho],[theta].sub.p] is the set of all weights w such that

[mathematical expression not reproducible] (11)

holds for every ball B = B([x.sub.0], r) [subset] [R.sup.d] with [x.sub.0] [member of] [R.sup.d] and r > 0, where p' is the dual exponent of p such that 1/p + 1/p' = 1. For p = 1 we define

[mathematical expression not reproducible], (12)

where [A.sup.[rho][theta].sub.1] is the set of all weights w such that

[mathematical expression not reproducible] (13)

holds for every ball B = B([x.sub.0], r) in [R.sup.d]. For [theta] > 0, let us introduce the maximal operator that is given in terms of critical radius function (2).

[mathematical expression not reproducible]. (14)

Observe that a weight w belongs to the class [A.sup.[rho][theta].sub.1] if and only if there exists a positive number [theta] > 0 such that [M.sub.p,[theta]] w [less than or equal to] Cw, where the constant C >0 is independent of w. Since

[mathematical expression not reproducible] (15)

for 0 < [[theta].sub.1] < [[theta].sub.2] < [infinity], then, for given p with 1 [less than or equal to] p < [infinity], one has

[mathematical expression not reproducible], (16)

where [A.sub.p] denotes the classical Muckenhoupt class (see [5, Chapter 7]), and hence [A.sub.p] [subset] [A.sup.[rho],[infinity].sub.p]. In addition, for some fixed [theta] > 0,

[mathematical expression not reproducible] (17)

whenever 1 [less than or equal to] [p.sub.1] < [p.sub.2] < [infinity]. Now, we present an important property of the classes of weights in [A.sup.[rho],[theta].sub.p] with 1 [less than or equal to] p < [infinity], which was given by Bongioanni et al. in [4, Lemma 5].

Lemma 3 (from [4]). If we [member of] [A.sup.[rho],[theta].sub.p] with 0 [theta] < [infinity] and 1 [less than or equal to] p < [infinity], then there exist positive constants [epsilon], [eta] > 0, and C > 0 such that

[mathematical expression not reproducible] (18)

for every ball B = B([x.sub.0], r) in [R.sup.d].

As a direct consequence of Lemma 3, we have the following result.

Lemma 4. If w [member of] [A.sup.[rho],[theta].sub.p] with 0 < [theta] < [infinity] and 1 [less than or equal to] p < [infinity], then there exist two positive numbers [delta] > 0 and [eta] > 0 such that

[mathematical expression not reproducible] (19)

for any measurable subset E of a ball B = B([x.sub.0], r), where C > 0 is a constant which does not depend on E and B.

For any given ball B = B([x.sub.0],r) with [x.sub.0] [member of] [R.sup.d] and r > 0, suppose that E [subset] B; then by Holder's inequality with exponent 1 + [epsilon] and (18), we can deduce that

[mathematical expression not reproducible]. (20)

This gives (19) with [delta] = [epsilon]/(1 + [epsilon]).

Given a weight w on [R.sup.d], as usual, the weighted Lebesgue space L[rho](w) for 1 [less than or equal to] p < [infinity] is defined to be the set of all functions f such that

[mathematical expression not reproducible]. (21)

We also denote by W[L.sup.1] (w) the weighted weak Lebesgue space consisting of all measurable functions f for which

[mathematical expression not reproducible]. (22)

Recently, Bongioanni et al. [4] obtained weighted strong-type and weak-type estimates for the operators R and [R.sup.*] defined in (6) and (7). Their results can be summarized as follows.

Theorem 5 (from [4]). Let 1 < p < [infinity] and w [member of] [A.sup.[rho],[infinity].sub.p]. If V [member of] R[H.sub.q] with q [greater than or equal to] d, then the operators R and [R.sup.*] are all bounded on [L.sup.p](w).

Theorem 6 (from [4]). Let p = 1 and w [member of] [A.sup.[rho],[infinity].sub.1]. If V [member of] R[H.sub.q] with q [greater than or equal to] d, then the operators R and [R.sup.*] are all bounded from [L.sup.1](w) into W[L.sup.1](w).

1.4. The Space BM[O.sub.[rho],[infinity]]([R.sup.d]). We denote by T either R or [R.sup.*]. For a locally integrable function b on [R.sup.d] (usually called the symbol), we will also consider the commutator operator

[b, T] f (x) := b (x) x Tf (x) - T (bf) (x), v [member of] [R.sup.d]. (23)

Recently, Bongioanni et al. [3] introduced a new space BM[O.sub.[rho],[infinity]] ([R.sup.d]) defined by

[mathematical expression not reproducible], (24)

where for 0 < [theta] < [infinity] the space BM[O.sub.[rho],[theta]] ([R.sup.d]) is defined to be the set of all locally integrable functions b satisfying

[mathematical expression not reproducible], (25)

for all [mathematical expression not reproducible] denotes the mean value of b on B([x.sub.0], r); that is,

[mathematical expression not reproducible]. (26)

A norm for [mathematical expression not reproducible], is given by the infimum of the constants satisfying (25), or, equivalently,

[mathematical expression not reproducible], (27)

where the supremum is taken over all balls B([x.sub.0], r) with [x.sub.0] [member of] [R.sup.d] and r > 0. With the above definition in mind, one has

[mathematical expression not reproducible] (28)

for 0 < [[theta].sub.1] < [[theta].sub.2] < [infinity], and hence BMO([R.sup.d]) [subset] BM[O.sub.[rho],[infinity]] ([R.sup.d]). Moreover, the classical BMO space [6] is properly contained in BM[O.sub.[rho],[infinity]] ([R.sup.d]) (see [2, 3] for some examples). We need the following key result for the space BM[O.sub.[rho],[theta]]([R.sup.d]), which was proved by Tang in [7].

Proposition 7 (from [7]). Let b [member of] BM[O.sub.[rho],[theta]]([R.sup.d]) with 0 < [theta] < [infinity]. Then there exist two positive constants [C.sub.1] and [C.sub.2] such that, for any given ball B([x.sub.0], r) in [R.sup.d] and for any [lambda] > 0, we have

[mathematical expression not reproducible], (29)

where [[theta].sup.*] = ([N.sub.0] + 1)[theta] and [N.sub.0] is the constant appearing in Lemma 1.

As a consequence of Proposition 7 and Lemma 4, we have the following result.

Proposition 8. Let b [member of] BM[O.sub.[rho],[theta]] ([R.sup.d]) with 0 < [theta] < [infinity] and w [member of] [A.sup.[rho],[infinity].sub.p] with 1 [less than or equal to] p < [infinity]. Then there exist positive constants [C.sub.1], [C.sub.2], and [eta] > 0 such that, for any given ball B([x.sub.0], r) in [R.sup.d] and for any [lambda] > 0, we have

[mathematical expression not reproducible], (30)

where [[theta].sup.*] = ([N.sub.0] + 1)[theta] and [N.sub.0] is the constant appearing in Lemma 1.

1.5. Orlicz Spaces. In this subsection, let us give the definition of and some basic facts about Orlicz spaces. For more information on this subject, the reader may consult book [8]. Recall that a function A : [0, [infinity]) [right arrow] [0, [infinity]) is called a Young function if it is continuous, convex, and strictly increasing with

[mathematical expression not reproducible]. (31)

An important example of Young functions is A(t) = t x [(1 + [log.sup.+]f).sup.m] with some 1 [less than or equal to] m < [infinity]. 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:

[mathematical expression not reproducible]. (32)

Associated with each Young function A, one can define its complementary function [bar.A]as follows:

[mathematical expression not reproducible]. (33)

Such a function [bar.A] is also a Young function. It is well known that the following generalized Holder inequality in Orlicz spaces holds for any given ball B [subset] [R.sup.d]:

[mathematical expression not reproducible]. (34)

In particular, for the Young function A(t) = t x (1 + [log.sup.+]t), the Luxemburg norm will be denoted by [mathematical expression not reproducible]. A simple computation shows that the complementary Young function of A(f) = t x (1 + [log.sup.+]t) is [bar.A](t) [approximately equal to] [e.sup.t] - 1 (see [9, 10] for instance). The corresponding Luxemburg norm will be denoted by [mathematical expression not reproducible]. In this situation, we have

[mathematical expression not reproducible]. (35)

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)

[mathematical expression not reproducible]. (36)

When [mathematical expression not reproducible]. Also, the complementary Young function of [PHI] is given by [bar.[PHI]](t) [approximately equal to] [e.sup.t] - 1 with the corresponding Luxemburg norm denoted by [parallel] x [parallel] [sub.exp L(w),B]. Given a weight w on [R.sup.d], we can also show the weighted version of (35). That is, the generalized Holder inequality in the weighted setting (see [11] for instance)

[mathematical expression not reproducible] (37)

holds for every ball B in [R.sup.d]. It is a simple but important observation that, for any ball B in [R.sup.d],

[mathematical expression not reproducible]. (38)

This is because t [less than or equal to] x (1 + [log.sup.+]f) for all t > 0. So we have

[mathematical expression not reproducible]. (39)

In [2], Bongioanni et al. obtained weighted strong (p, p), 1 < p < [infinity], and weak L log L estimates for the commutators of the Riesz transform and its adjoint associated with the Schrodinger operator L = -[DELTA] + V, where V satisfies some reverse Holder inequality. Their results can be summarized as follows.

Theorem 9 (from [2]). Let 1 < p < [infinity] and w [member of] [A.sup.[rho],[infinity].sub.p]. If V [member of] R[H.sub.q] with q [greater than or equal to] d, then the commutator operators [b, R] and [b, [R.sup.*]] are all bounded on L[rho](w), whenever b belongs to BM[O.sub.[rho],[infinity]] ([R.sup.d]).

Theorem 10 (from [2]). Let p = 1 and w [member of] [A.sup.[rho],[infinity].sub.1]. If V [member of] R[H.sub.q] with q [greater than or equal to] d and b [member of] BM[O.sub.[rho],[infinity]] ([R.sup.d]), then, for any given [lambda] > 0, there exists a positive constant C >0 such that, for those functions f such that [PHI]([absolute value of f]) [member of] [L.sup.1](w),

[mathematical expression not reproducible], (40)

where [PHI](f) = t x (1 + [log.sup.+]f) and [log.sup.+]1 := max{log t, 0}; that is,

[mathematical expression not reproducible]. (41)

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.sup.*] are both bounded operators on these new spaces. Finally, we also obtain the weighted estimates for the commutators [b, R] and [b, [R.sup.*] ] defined in (23).

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 [approximately equal to] b to denote the equivalence of a and b; that is, there exist two positive constants [C.sub.1] and [C.sub.2] independent of a, b such that [C.sub.1]a [less than or equal to] b [less than or equal to] [C.sub.2]a.

2. 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 11. Let 1 [less than or equal to] p < [infinity], let 0 [less than or equal to] [kappa] < 1, and let w be a weight. For given 0 < [theta] < [infinity], the weighted Morrey space [L.sup.p,[kappa].sub.[rho],[theta]](w) is defined to be the set of all [L.sup.p] locally integrable functions f on [R.sup.d] for which

[mathematical expression not reproducible] (42)

for every ball [mathematical expression not reproducible], is given by the infimum of the constants in (42), or, equivalently,

[mathematical expression not reproducible], (43)

where the supremum is taken over all balls B in [R.sup.d] and [x.sub.0] and r denote the center and radius of B, respectively. Define

[mathematical expression not reproducible]. (44)

Note that this definition does not coincide with the one given in [12] (see also [13] for the unweighted case), but in view of the space BM[O.sub.[rho],[infinity]] (O([R.sup.d]) defined above it is more natural in our setting. Obviously, if we take [theta] = 0 or V [equivalent to] 0, then this new space is just the weighted Morrey space L[P.sup.p,[kappa]] (w), which was first defined by Komori and Shirai in [14] (see also [15]).

Definition 12. Let p = 1, let 0 [less than or equal to] [kappa] < 1, and let w be a weight. For given 0 < [theta] < [infinity], the weighted weak Morrey space W[L.sup.1,[kappa].sub.[rho],[theta]]e(w) is defined to be the set of all measurable functions f on [R.sup.d] for which

[mathematical expression not reproducible] (45)

for every ball B = B([x.sub.0], r) in [R.sup.d], or, equivalently,

[mathematical expression not reproducible] (46)

Correspondingly, we define

[mathematical expression not reproducible]. (47)

Clearly, if we take [theta] = 0 or V [equivalent to] 0, then this space is just the weighted weak Morrey space W[L.sup.1,[kappa]](w) (see [16]). According to the above definitions, one has

[mathematical expression not reproducible], (48)

for [mathematical expression not reproducible].

The space [L.sup.p,[kappa].sub.[rho],[theta]](w) (or W[L.sup.1,[kappa].sub.[rho],[theta]] (w)) could be viewed as an extension of the weighted (or weak) Lebesgue space (when [kappa] = [theta] = 0). Naturally, one may ask the question whether the above conclusions (i.e., Theorems 5 and 6 as well as Theorems 9 and 10) still hold if we replace 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 13. Let 1 < p < [infinity], 0 < [kappa] < 1, and w [member of] [A.sup.[rho],[infinity].sub.p]. If V [member of] R[H.sub.q] with q [greater than or equal to] d, then the operators R and [R.sup.*] map [L.sup.[kappa].sub.[rho],[infinity]] (w) into itself.

Theorem 14. Let p = 1, 0 < [kappa] < 1, and w [member of] [A.sup.[rho],[infinity].sub.1]. If V [member of] R[H.sub.q] with q [greater than or equal to] d, then the operators R and [R.sup.*] map [L.sup.1,[kappa].sub.[rho],[infinity]] (w) into W[L.sup.1,[kappa].sub.[rho],[infinity]](w).

Theorem 15. Let 1 < p < [infinity], 0 < [kappa] < 1, and w [member of] [A.sup.[rho],[infinity].sub.p]. If V [member of] R[H.sub.q] with q [greater than or equal to] d, then the commutator operators [b, R] and [b, [R.sup.*]] map [L.sup.p,[kappa].sub.[rho],[infinity]](w) into itself, whenever b [member of] BM[O.sub.[rho],[infinity]]([R.sup.d]).

To deal with the commutators in the endpoint case, we need to consider a new kind of weighted Morrey spaces of the L log L type.

Definition 16. Let p = 1, let 0 [less than or equal to] [kappa] < 1, and let w be a weight. For given 0 < [theta] < [infinity], the weighted Morrey space [(L log L).sup.1,[kappa].sub.[rho],[theta]] (w) is defined to be the set of all locally integrable functions f on [R.sup.d] for which

[mathematical expression not reproducible] (49)

for every ball B = B([x.sub.0], r) in [R.sup.d], or, equivalently,

[mathematical expression not reproducible]. (50)

Concerning the mapping properties of [b, R] and [b, [R.sup.*]] in the weighted Morrey spaces of the L log L type, we have the following.

Theorem 17. Let p =1, 0 < [kappa] < 1, and w [member of] [A.sup.[rho],[infinity].sub.1]. If V [member of] R[H.sub.q] with q [greater than or equal to] d and b [member of] BM[O.sub.[rho],[infinity]] ([R.sup.d]), then, for any given [lambda] > 0 and any given ball B = B([x.sub.0], r) of [R.sup.d], there exist some constants C >0 and [??] > 0 such that the inequalities

[mathematical expression not reproducible]. (51)

hold for those functions f such that [PHI]([absolute value of f]) [member of] [(L log L).sup.1,[kappa].sub.[rho],[theta]](w) with some fixed [theta] > 0, where [PHI](t) = t x (1 + [log.sup.+]t).

If we denote

[mathematical expression not reproducible], (52)

then Theorem 17 now tells us that the commutators [b, R] and [b, [R.sup.*]] map [(L log L).sup.1,[kappa].sub.[rho],[infinity]](w) into W[L.sup.1,[kappa].sub.[rho],[infinity]](w), when b is in BM[O.sub.[rho],[infinity]] ([R.sup.d]).

3. Proofs of Theorems 13 and 14

In this section, we will prove the conclusions of Theorems 13 and 14.

Proof of Theorem 13. We denote by T either R or [R.sup.*]. By definition, we only have to show that, for any given ball B = B([x.sub.0],r) of [R.sup.d], there is some [??] > 0 such that

[mathematical expression not reproducible] (53)

holds for any f [member of] [L.sup.p,[kappa].sub.[rho],[infinity]] (w) with 1 < p < [infinity] and 0 < [kappa] < 1. Suppose that f [member of] [L.sup.p,[kappa].sub.[rho],[theta]] (w) for some [theta] > 0 and w [member of] [A.sup.[rho],[theta]'.sub.p] for some [theta]' > 0. We decompose f, in the classical way, as

[mathematical expression not reproducible]m (54)

where 2B is the ball centered at [x.sub.0] and radius 2r > 0 and [[chi].sub.2B] is the characteristic function of 2B. Then by the linearity of T, we write

[mathematical expression not reproducible]. (55)

We now analyze each term separately. By Theorem 5, we get

[mathematical expression not reproducible]. (56)

Since w [member of] [A.sup.[rho],[theta]'.sub.p] with 1 < p [infinity] and 0 < [theta]' < [infinity], then we know that the inequality

[mathematical expression not reproducible] (57)

is valid. In fact, for 1 < p < [infinity], by Holders inequality and the definition of [A.sup.[rho],[theta]'.sub.p], we have

[mathematical expression not reproducible]. (58)

If we take [??](x) = [[chi].sub.B](x), then the above expression becomes

[mathematical expression not reproducible], (59)

which in turn implies (57). Therefore,

[mathematical expression not reproducible], (60)

where [??]' = [kappa] x [theta]' + [theta]. For the other term [I.sub.2], notice that, for any x [member of] Band y [member of] [(2B).sup.c], one has [absolute value of x-y] [approximately equal to] [absolute value of [x.sub.0] - y]. It then follows from Lemma 2 that, for any x [member of] B([x.sub.0],r) and any positive integer N,

[mathematical expression not reproducible]. (61)

In view of (4) in Lemma 1, we further obtain

[mathematical expression not reproducible]. (62)

Moreover, by using Holder's inequality and the [A.sup.[rho],[theta]'.sub.p] condition on w, we get

[mathematical expression not reproducible]. (63)

Hence,

[mathematical expression not reproducible]. 64)

Recall that w [member of] [A.sup.[rho],[theta]'.sub.p] with 0 < [theta]' < [infinity] and 1 < p < [infinity]; then there exist two positive numbers [delta], [eta] > 0 such that (19) holds. This allows us to obtain

[mathematical expression not reproducible]. (65)

Thus, by choosing N large enough so that N > [theta] + [theta]' + [eta](1 - [kappa])/p, we then have

[mathematical expression not reproducible]. (66)

Summing up the above estimates for [mathematical expression not reproducible], we obtain our desired inequality (53). This completes the proof of Theorem 13.

Proof of Theorem 14. We denote by T either R or [R.sup.*]. To prove Theorem 14, by definition, it is sufficient to prove that, for any given ball B = B([x.sub.0],r) of [R.sup.d], there is some [??] > 0 such that

[mathematical expression not reproducible] (67)

holds for any f [member of] [L.sup.1,[kappa].sub.[rho],[infinity]](w) with 0 < [kappa] < 1. Now suppose that f [member of] [L.sup.1,[kappa].sub.[rho],[theta]](w) for some [theta] > 0 and w [member of] [A.sup.[rho],[theta]'.sub.1 for some [theta]' > 0. We decompose f, in the classical way, as

[mathematical expression not reproducible]. (68)

Then for any given [lambda] > 0, by the linearity of T, we can write

[mathematical expression not reproducible]. (69)

We first give the estimate for the term [I[.sub.1]. By Theorem 6, we get

[mathematical expression not reproducible]. (70)

Since w [member of] [A.sup.[rho],[theta]'.sub.1] with 0 < [theta]' < [infinity], similar to the proof of (57), we can also show the following estimate as well:

w (2B ([x.sub.0], r)) [less than or equal to] C x [(1 + 2r/[rho]([x.sub.0)).sup.[theta]'] w(B([x.sub.0],r)). (71)

In fact, by the definition of [A.sup.[rho],[theta]'.sub.1], we can deduce that

[mathematical expression not reproducible]. (72)

If we choose [??](x) = [[chi].sub.B](x), then the above expression becomes

[mathematical expression not reproducible], (73)

which in turn implies (71). Therefore,

[mathematical expression not reproducible], (74)

where [??]' := [theta]' x [kappa] + [theta]. As for the other term [I'.sub.2], by using pointwise inequality (62) and Chebyshev's inequality, we deduce that

[mathematical expression not reproducible]. (75)

Moreover, by the [A.sup.[rho],[theta]'.sub.1] condition on w, we compute

[mathematical expression not reproducible]. (76)

Consequently,

[mathematical expression not reproducible]. (77)

Recall that w [member of] [A.sup.[rho],[theta]'.sub.1] with 0 < [theta]' < [infinity]; then there exist two positive numbers [delta]', [eta]' > 0 such that (19) holds. Therefore,

[mathematical expression not reproducible]. (78)

By selecting N large enough so that N > [theta] + [theta]' + [eta]'(1 - k), we thus have

[mathematical expression not reproducible]. (79)

Let [mathematical expression not reproducible]. Here N is an appropriate constant. Summing up the above estimates for [I'.sub.1] and [I'.sub.2] and then taking the supremum over all [lambda] > 0, we obtain our desired inequality (67). This finishes the proof of Theorem 14.

4. Proofs of Theorems 15 and 17

For the results involving commutators, we need the following properties of BM[O.sub.[rho],[infinity]]([R.sup.d]) functions, which are extensions of well-known properties of BMO([R.sup.d]) functions.

Lemma 18. If b [member of] BM[O.sub.[rho],[infinity]] ([R.sup.d]) and w [member of] [A.sup.[rho],[infinity].sub.p] with 1 [less than or equal to] p < [infinity], then there exist positive constants C > 0 and [mu] > 0 such that, for every ball B = B([x.sub.0], r) in [R.sup.d], we have

[mathematical expression not reproducible], (80)

where [b.sub.B] = (1/[absolute vale of B]) [[integral].sub.B] b(y)dy.

Proof. We may assume that b [member of] BM[O.sub.[rho],[theta]] ([R.sup.d]) with 0 < [theta] < [infinity]. According to Proposition 8, we can deduce that

[mathematical expression not reproducible]. (81)

Making change of variables, then we get

[mathematical expression not reproducible], (82)

which yields the desired inequality if we choose [mathematical expression not reproducible].

Lemma 19. If b [member of] BM[O.sub.[rho],[theta]] ([R.sup.d]) with 0 < [theta] < [infinity] and w [member of] [A.sup.[rho],[infinity].sub.1], then there exist positive constants C, [gamma] > 0 and [eta] > 0 such that, for every ball B = B([x.sub.0], r) in [R.sup.d], we have

[mathematical expression not reproducible], (83)

where [b.sub.B] = (1/[absolute value of B]) b(y) dy and [[theta].sup.*] = ([N.sub.0] + 1)[theta] and [N.sub.0] is the constant appearing in Lemma 1.

Proof. Recall the following identity (see Proposition 1.1.4 in [5]):

[mathematical expression not reproducible]. (84)

Using this identity and Proposition 8, we obtain

[mathematical expression not reproducible], (85)

where [[lambda].sup.*] is given by

[mathematical expression not reproducible]. (86)

If we take [gamma] small enough so that 0 < [gamma] < [C.sub.2], then the conclusion follows immediately.

Lemma 20. If be BM[O.sub.[rho],[theta]] ([R.sup.d]) with 0 < [theta] < [infinity], then, for any positive integer k, there exists a positive constant C > 0 such that, for every ball B = B([x.sub.0], r) in [R.sup.d], we have

[mathematical expression not reproducible]. (87)

Proof. For any positive integer k, we have

[mathematical expression not reproducible]. (88)

Since, for any 1 [less than or equal to] j [less than or equal to] k + 1, the estimate

[mathematical expression not reproducible] (89)

holds trivially, then

[mathematical expression not reproducible]. (90)

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 15. We denote by [b, T] either [b, R] or [b, [R.sup.*]]. By definition, we only need to show that, for any given ball B = B([x.sub.0], r) of [R.sup.d], there is some [??] > 0 such that

[mathematical expression not reproducible] (91)

holds for any f [member of] [L.sup.p,[kappa].sub.[rho],[infinity]] (w) with 1 < p < [infinity] and 0 < [kappa] < 1, whenever b belongs to BM[O.sub.[rho],[infinity]] ([R.sup.d]). Suppose that f [member of] [L.sup.p,[kappa].sub.[rho],[theta]] (w) for some [theta] > 0, w [member of] [A.sup.[rho],[theta].sub.p] for some [theta]' > 0, and b [member of] BM[O.sub.[rho],[theta]]"] ([R.sup.d]) for some [theta]" > 0. We decompose f as

[mathematical expression not reproducible]. (92)

Then, by the linearity of [b, T], we write

[mathematical expression not reproducible]. (93)

Now we give the estimates for [J.sub.1] and [J.sub.2], respectively. According to Theorem 9, we have

[mathematical expression not reproducible]. (94)

Moreover, in view of inequality (57), we get

[mathematical expression not reproducible], (95)

where [mathematical expression not reproducible]. On the other hand, by definition (23), we can see that, for any x [member of] B([x.sub.0], r),

[mathematical expression not reproducible]. (96)

So we can divide ]2 into two parts:

[mathematical expression not reproducible]. (97)

From pointwise estimate (62) and (80) in Lemma 18, it then follows that

[mathematical expression not reproducible]. (98)

Following along the same lines as that of Theorem 13, we are able to show that

[mathematical expression not reproducible]. (99)

The last inequality is obtained by using (19). For any x [member of] B([x.sub.0],r) and any positive integer N, similar to the proof of (61) and (62), we can also deduce that

[mathematical expression not reproducible], (100)

where in the last inequality we have used (4) in Lemma 1. Hence, by the above pointwise estimate for [zeta](x),

[mathematical expression not reproducible]. (101)

Moreover, for each integer k [greater than or equal to] 1,

[mathematical expression not reproducible]. (102)

By using Holder's inequality, the first term of expression (102) is bounded by

[mathematical expression not reproducible]. (103)

Since w [member of] [A.sup.[rho],[theta]'.sub.p] with 0 < [theta]' < [infinity] and 1 < p < ([infinity], then, by the definition of [A.sup.[rho],[theta]'.sub.p], it can be easily shown that w [member of] [A.sup.[rho],[theta]'.sub.p] if and only if [w.sup.-p'/p] [member of] [A.sup.[rho],[theta]'.sub.p], where 1/p + 1/p' = 1 (see [7,17,18] and the references therein). If we denote v = [w.sup.-p'/p], then v [member of] [A.sup.[rho],[theta]'.sub.p]. This fact together with Lemma 18 implies

[mathematical expression not reproducible]. (104)

Therefore, the first term of expression (102) can be bounded by a constant times

[mathematical expression not reproducible]. (105)

Since b [member of] BM[O.sub.[rho],[theta]"] ([R.sup.d]) with 0 < [theta]" < [infinity], then, by Lemma 20, Holder's inequality, and the [A.sup.[rho],[theta]'.sub.p] condition on w, the latter term of expression (102) can be estimated by

[mathematical expression not reproducible]. (106)

Consequently,

[mathematical expression not reproducible]. (107)

Thus, in view of (107),

[mathematical expression not reproducible]. (108)

Notice that w [member of] [A.sup.[rho],[theta]'.sub.p] with 0 < [theta]' < [infinity]. A further application of (19) yields

[mathematical expression not reproducible]. (109)

Combining the above estimates for [J.sub.3] and [J.sub.4], we get

[mathematical expression not reproducible]. (110)

By choosing N large enough so that N > [theta] + [theta]' + [theta]" + [mu] + [eta](1 - [kappa])/p, we thus have

[mathematical expression not reproducible]. (111)

Finally, collecting the above estimates for [J.sub.1] and [J.sub.2] and letting [mathematical expression not reproducible], we obtain the desired result (91). The proof of Theorem 15 is finished.

Proof of Theorem 17. We denote by [b, T] either [b, R] or [b, [R.sup.*]]. We are going to prove that, for any given [lambda] > 0 and any given ball B = B([x.sub.0], r) of [R.sup.d], there is some [??] > 0 such that the inequality

[mathematical expression not reproducible] (112)

holds for those functions f such that [PHI]([absolute value of f]) [member of] [(L log L).sup.1,[kappa].sub.[rho],[theta]](w) with some fixed [theta] > 0. Now assume that w [member of] [A.sup.[rho],[theta]'.sub.1] for some [theta]' > 0 and b [member of] BM[O.sub.[rho],[theta]"] ([R.sup.d]) for some [theta]" > 0. As before, we decompose f as

[mathematical expression not reproducible]. (113)

Then for any given [lambda] > 0, by the linearity of [b, T], we can write

[mathematical expression not reproducible]. (114)

Let us first estimate the term [J].sub.1]. By using Theorem 10, we get

[mathematical expression not reproducible]. (115)

A further application of (39) yields

[mathematical expression not reproducible], (116)

where the last inequality is due to (71). If we denote [??]' = k x [theta]' + [theta], then

[mathematical expression not reproducible] (117)

as desired. Next let us deal with the term [J'.sub.2]. Taking into account (96), we can divide it into two parts, namely,

[mathematical expression not reproducible], (118)

where

[mathematical expression not reproducible]. (119)

Since b [member of] BM[O.sub.[rho],[theta]"] ([R.sup.d]) for some [theta]" > 0, from pointwise inequality (62) and Chebyshev's inequality, we then have

[mathematical expression not reproducible], (120)

where in the last inequality we have used (80) in Lemma 18. Moreover, it follows directly from the condition [A.sup.[rho],[theta]'.sub.1] that, for each integer k [greater than or equal to] 1,

[mathematical expression not reproducible]. (121)

Notice also that trivially

t [less than or equal to] t x (1 + [log.sup.+]t) = [PHI] (t), for any t > 0. (122)

This fact along with (39) implies that, for each integer k [greater than or equal to] 1,

[mathematical expression not reproducible]. (123)

Consequently,

[mathematical expression not reproducible]. (124)

Since w [member of] [A.sup.[rho],[theta]'.sub.1] with 0 < [theta]' < [infinity], then there exist two positive numbers [delta]' and [eta]' > 0 such that (19) holds. Therefore,

[mathematical expression not reproducible]. (125)

On the other hand, it follows from pointwise inequality (100) and Chebyshev's inequality that

[mathematical expression not reproducible], (126)

where the last inequality follows from (122). Furthermore, by the definition of [A.sup.[rho][theta]''.sub.1], we compute

[mathematical expression not reproducible]. (127)

By using generalized Holder inequality (37), the first term of expression (127) is bounded by

[mathematical expression not reproducible], (128)

where in the last inequality we have used the fact that

[mathematical expression not reproducible], (129)

which is equivalent to inequality (83) in Lemma 19. By Lemma 20 and (39), the latter term of expression (127) can be estimated by

[mathematical expression not reproducible]. (130)

Consequently,

[mathematical expression not reproducible]. (131)

Hence, combining the above estimates for [J'.sub.3] and [J'.sub.4], we have

[mathematical expression not reproducible], (132)

Now N can be chosen sufficiently large such that N > [theta]' + [theta] + [theta]' + [theta]" + [eta]' (1 - [kappa]), and hence the above series is convergent. Finally,

[mathematical expression not reproducible]. (133)

Fix this N and set [mathematical expression not reproducible]. Thus, combining the above estimates for [J'.sub.1] and [J'.sub.2], inequality (112) is proved and then the proof of Theorem 17 is finished.

The higher order commutators formed by a BM[O.sub.[rho],[infinity]] ([R.sup.d]) function b and the operator R and its adjoint [R.sup.*] are usually defined by

[mathematical expression not reproducible]. (134)

Let T denote R or [R.sup.*]. Obviously, [[b, T].sub.1] = [b, T] which is just the linear commutator (23), and

[[b, T].sub.m] = [[b,[b, T].sub.m-1], m = 2,3,.... (135)

By induction on m, we are able to show that the conclusions of Theorems 15 and 17 also hold for the higher order commutators [[b, T].sub.m] with m [greater than or equal to] 2. The details are omitted here.

Theorem 21. Let 1 < p [infinity], 0 < [kappa] < 1, and w [member of] [A.sup.[rho],[infinity].sub.p]. If V [member of] R[H.sub.q] with q [greater than or equal to] d, then, for any positive integer m [greater than or equal to] 2, the higher order commutators [[b, R].sub.m] and [b, [[R.sup.*]].sub.m] map [L.sup.p,[kappa].sub.[rho],[infinity]] (w) into itself, whenever b [member of] BM[O.sub.[rho],[infinity]] ([R.sup.d]).

Theorem 22. Let p = 1, 0 < [kappa] < 1, and w [member of] [A.sup.[rho],[infinity].sub.1]. If V [member of] R[H.sub.q] with q [greater than or equal to] d and b [member of] BM[O.sub.[rho],[infinity]] ([R.sup.d]), then, for any given [lambda] > 0 and any given ball B = B([x.sub.0], r) of [R.sup.d], there exist some constants C > 0 and [??] > 0 such that the inequalities

[mathematical expression not reproducible], (136)

[mathematical expression not reproducible] (137)

hold for those functions f such that [[PHI].sub.m]([absolute value of f]) [member of] [(L log L).sup.1,[kappa].sub.[rho],[theta]] (w) with some fixed [theta] > 0, where [[PHI].sub.m](t) = t x [(1 + [log.sup.+] t).sup.m], m = 2,3, ....

Data Availability

No data were used to support this study.

Conflicts of Interest

The author declares that there are no conflicts of interests regarding the publication of this article.

https://doi.org/10.1155/2019/7057512

Acknowledgments

The author would like to thank Professor L. Tang for providing [7].

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Hua Wang [ID] (1,2)

(1) College of Mathematics and Econometrics, Hunan University, Changsha, 410082, China

(2) Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada

Correspondence should be addressed to Hua Wang; wanghua@pku.edu.cn

Received 31 December 2018; Accepted 11 February 2019; Published 24 April 2019

Academic Editor: Maria Alessandra Ragusa

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Title Annotation: | Research Article |
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Author: | Wang, Hua |

Publication: | Journal of Function Spaces |

Date: | May 31, 2019 |

Words: | 7014 |

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