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Compact Operators on the Bergman Spaces with Variable Exponents on the Unit Disc of C.

1. Introduction and Statement of Results

Variable Lebesgue spaces are a generalization of the Lebesgue spaces that allow the exponents to be a measurable function and thus the exponent may vary. These spaces have many properties similar to the normal Lebesgue spaces, but they also differ in surprising and subtle ways. For this reason, the variable Lebesgue spaces have an intrinsic interest, but they are also very important in applications to partial differential equations and variational integrals with nonstandard growth conditions. See [1] for more details on the variable Lebesgue spaces.

Let [DELTA] denote the unit disc in C and dA the normalized Lebesgue measure on [DELTA]. For 1 [less than or equal to] p < [infinity], the Bergman space [A.sup.p] = [A.sup.p] ([DELTA], dA) is the space of all analytic functions, f, on [DELTA] such that

[[parallel]f[parallel].sup.p.sub.p] = [[integral].sub.[DELTA]] [[absolute value of f (z)].sup.p] dA (z) < [infinity]. (1)

Let P be the Bergman projection from [L.sup.2] onto [A.sup.2]. Then P is an integral operator given by

P(f)(z) = [[integral].sub.[DELTA]] f(w)/[(1 - z[bar.w]).sup.2] dA (w), (2)

for each z [member of] [DELTA] and f [member of] [L.sup.2]. Here, the function K(z, w) = [K.sub.w](z) = 1/[(1 - z[bar.w]).sup.2] is the reproducing kernel for [A.sup.2]. For f [member of] [L.sup.[infinity]], the Toeplitz operator with symbol f is defined on [A.sup.p] by

[T.sub.f]g = P(fg), g [member of] [A.sup.p]. (3)

Toeplitz operators are amongst the most widely studied classes of concrete operators and have attracted a lot of interest in recent years. The behaviour of these operators on the Hardy spaces, Bergman spaces, and Fock spaces has been studied widely and a lot of results are available in the literature. The characterization of compactness has been studied in [2-8] just to cite a few.

Given [OMEGA] [subset] [R.sup.n], a measurable function p : [OMEGA] [right arrow] [1, [infinity]) will be called a variable exponent. If p is a variable exponent then we denote

[mathematical expression not reproducible]. (4)

Let P([OMEGA]) denote the set of all variable exponents for which [p.sub.+] < [infinity].

For a complex-valued measurable function [phi] :[OMEGA] [right arrow] C, we define the modular [[rho].sub.p(*)] by

[[rho].sub.p(*)] ([phi]) := [[integral].sub.[OMEGA]] [[absolute value of [OMEGA](x)].sup.p(x)] dx (5)

and the norm

[[parallel][phi][parallel].sub.p(*)] := inf {[lambda] > 0: [[rho].sub.p(*)] ([phi]/[lambda]] [less than or equal to] 1}. (6)

Let p(*) [member of] P([OMEGA]). Then the Lebesgue variable exponent space [L.sup.p(*)] is the set of all complex-valued measurable functions [phi]: [OMEGA] [right arrow] C for which [[rho].sub.p(*)] ([phi]) < [infinity]. If we equip [[L.sup.p(*)] with the norm given in (6), then [[L.sup.p(*)] becomes a Banach space. We note here that the condition [[rho].sub.p(*)] ([phi]) < [infinity] is not enough in general to define the variable exponent Lebesgue space (e.g., see chapter 2 of [1]).

It is known (e.g., see chapter 2 of [1]) that the dual of [[L.sup.p(*)] is [[L.sup.p'(*)], where 1/p(*) + 1/p'(*) = 1. A straightforward computation shows that

[(p'(*)).sub.+] = ([P.sub.-])', [(p'(*)).sub.-] = ([P.sub.+])'. (7)

For simplicity, we will omit one set of parenthesis and write the left-hand side of each equality as p'[(*).sub.+] and p'[(*).sub.-]. Throughout this work, we shall use p'(*) as the conjugate exponent of p(*) and if p is a constant in (1, [infinity]) we shall use p' as the conjugate exponent of p. In other words, to study these spaces, some regularity conditions are imposed on the exponents. A function p :[OMEGA] [right arrow] C is said to be log-Holder continuous on [OMEGA] if there exists a positive constant [C.sub.log] such that

[absolute value of p(x) - p(y)] [less than or equal to] [C.sub.log]/log (1/[absolute value of x - y]), (8)

for all x, y [member of] [OMEGA] with [absolute value of x-y] < 1/2. It follows that

[absolute value of p(x) - p(y)] [less than or equal to] 2l[C.sub.log]/log(2l/[absolute value of x - y]), (9)

for all x, y [member of] [OMEGA] with [absolute value of x-y] < l. We denote by [P.sup.log] ([OMEGA]) the exponents in P([OMEGA]) that are log-Holder continuous on [OMEGA]. For p(*) [member of] [P.sup.log] ([OMEGA]) and a given measurable function, f, define

[mathematical expression not reproducible]. (10)

Theorem 2.34 of [1] shows that there exist constants [C.sub.1] and [C.sub.2], depending on p(*), such that

[mathematical expression not reproducible]. (11)

The next result which establishes a relationship between the Lebesgue spaces with exponents [p.sub.-], [p.sub.+], and p(*) will be very useful in the rest of the work. It is Corollary 2.50 of [1].

Lemma 1. Suppose p(*) [member of] [P.sup.log] ([OMEGA]) and [absolute value of [OMEGA]] < [infinity]. Then there exist constants [c.sub.1] and [c.sub.2] such that

[mathematical expression not reproducible]. (12)

The study of variable exponent Bergman space, [A.sup.p(*)], which is the space of analytic functions in [L.sup.p(*)], has been introduced in [9]. There it was shown, amongst other things, that the Bergman projector P is bounded from [L.sup.p(*)] onto [A.sup.p(*)]. Also in [10], the authors studied Carleson measures in such spaces.

In this paper, we will extend the results in [3, 7] on boundedness and compactness of operators for the Bergman spaces with constant exponents to the Bergman spaces with variable exponents.

For z [member of] [DELTA], let [[phi].sub.z] be the analytic map of [DELTA] onto [DELTA] given by [[phi].sub.z](w) = (z-w)/(1-[bar.z]w). We define the operator [U.sub.z] on [A.sup.2] by

[U.sub.z] f = (f * [[phi].sub.z]) [[phi]'.sub.z], z [member of] [DELTA]. (13)

Then [U.sub.z] is a unitary operator on [A.sup.2]. We shall show later that [U.sub.z] is bounded on [A.sup.p(*)]. For S, a bounded operator on [A.sup.p(*)], we define [S.sub.z] by [S.sub.z] = [U.sub.Z]S[U.sub.Z].

If S is a bounded operator on [A.sup.p(*)], then the Berezin transform of S is the function [??] on [DELTA] defined by

[??](z) = <S[k.sub.z], [k.sub.z]>, (14)

where [k.sub.z] (w) = (1-[[absolute value of z].sup.2]) [K.sub.z] is the normalized Bergman kernel which also belongs to [A.sup.p(*)] and <,> is the inner product of [A.sup.2]. We let

[mathematical expression not reproducible] (15)

and set

[mathematical expression not reproducible]. (16)

Our first result gives some conditions for the boundedness of Toeplitz operators with [L.sup.1] symbols on the variable Bergman spaces.

Theorem 2. Suppose p(*) [member of] [P.sup.log] ([DELTA]), 1 < [p.sub.0] [less than or equal to] [p.sub.-] [less than or equal to] [p.sub.+] < [infinity], and [p.sub.1] = min([p.sub.0], [p'.sub.0]). Suppose f [member of] [L.sup.1] is such that

[mathematical expression not reproducible] (17)

for all q > ([p.sub.1] + 1)/([p.sub.1]-1). Then [T.sub.f] is bounded on [A.sup.p(*)].

We note here that this result was proved in [3] in the Bergman spaces [A.sup.p], where p is a constant. We also have the following result on compactness.

Theorem 3. Suppose p(*) [member of] [P.sup.log] (DELTA]), 1 < [p.sub.0] [less than or equal to] [p.sub.-] [less than or equal to] [p.sub.+] < [infinity], and [p.sub.1] = min ([p.sub.0], [p'.sub.0]). If S is a bounded operator on [A.sup.p(*)] such that

[mathematical expression not reproducible] (18)

for some q > ([p.sub.1] + 1)/([p.sub.1]-1), then the following are equivalent:

(1) S is compact on [A.sup.p(*)],

(2) [??] (z) [right arrow] 0 as z [right arrow] [partial derivative] [DELTA],

(3) for every s [member of] [1, q), [[parallel][S.sub.z]1[parallel].sub.s] [right arrow] 0 as z [right arrow] [partial derivative][DELTA],

(4) [[parallel][S.sub.z] 1[parallel].sub.1] [right arrow] 0 as z [right arrow] [partial derivative] [DELTA].

This theorem is well known in the Bergman spaces with constant exponents; for example, see [3, 7]. However, the techniques here are different from those used in either of the papers for both the proof of boundedness and compactness. This is because their proofs depend on the use of Schur's test which does not hold in the variable Lebesgue space. However, using the Muckenhoupt weights we were able to develop some Schur-like tests from where we obtain the theory that builds upon the Rubio de Francia theory of extrapolation from the theory of weighted norm inequalities. The advantage of this approach is that it quickly yields to sufficient conditions for these operators to be bounded on variable Lebesgue spaces. Through such techniques, we are also able to obtain some norm estimates for bounded operators on the space [A.sup.p(*)]

Similar to the work of Miao and Zheng [7], we consider the case of the algebra of Toeplitz operators generated by symbols in the class BT. To be precise, we have the following.

Theorem 4. Suppose p(*)] [member of] [P.sup.log] ([DELTA]) and S is a finite sum of finite products of Toeplitz operators with symbols in the class BT. Then S is compact on [A.sup.p(*)] if and only if [??](z) [right arrow] 0 as z [right arrow] [partial derivative][DELTA].

This paper is organized as follows: in Section 2, we will study some basic concepts on the Muckenhoupt weights. Section 3 deals with the variable Bergman spaces and the proof of Theorem 2. In Section 4, we study some norm estimates on these spaces and in Section 5 we give the proof of the compactness results.

2. Muckenhoupt Weights [A.sub.1]

Definition 5. Let [OMEGA] be a set. Then the function d : [OMEGA] x [OMEGA] [right arrow] [R.sup.+] is said to be a pseudodistance on [OMEGA] if it satisfies the following:

(1) d (x, y) = 0 if and only if x = y;

(2) d (x, y) = d(y, x);

(3) there exists a positive constant K [greater than or equal to] 1, such that, for all x, y, z [member of] [OMEGA],

d (x, y) [less than or equal to] K (d (x, z) + d (z, y)). (19)

For x [member of] [OMEGA] and r > 0, the set B (x, r) = {y [member of] [OMEGA]: d (x, y) < r} is called a pseudoball with centre x and radius r. If [mu] is a measure on [OMEGA], then the triple ([OMEGA], d, [mu]) is called a homogeneous space if [OMEGA] is endowed with the topology generated by the collection {B(x, r) : x [member of] [OMEGA], r > 0} (that is, the topology generated by the pseudoballs) and [mu] satisfies the doubling property; there exists a constant [delta] such that, for all x [member of] [OMEGA] and r > 0, we have

0 < [mu] (B (x, 2r)) [less than or equal to] [delta][mu] (B (x, r)) < [infinity]. (20)

We now turn our attention to the case when [OMEGA] = [DELTA]. By lemma 2.2 of [11], it is shown that the distance function d given on [DELTA] by

[mathematical expression not reproducible], (21)

is a pseudodistance on [DELTA], where [[DELTA].sup.*] = [DELTA] {0}. It is known (see [12]) that, at the boundary of [DELTA], d becomes the Koranyi distance. Also by Lemma 2 of [12], we have that for any pseudoball B (w, r), w [not equal to] 0, and r [member of] (0, 2) we have that

[absolute value of B (w, r)] [approximately equal to] [r.sup.2], (22)

where [absolute value of B] denotes the Lebesgue area measure of set B. Also observe that the pseudoball B (0, 1) = [DELTA]. It is known that (see [12]) ([DELTA], d, dA) is a homogeneous space.

Let f be a locally integrable function in [DELTA]. Then the Hardy-Littlewood maximal function relative to the pseudodistance d is given by

[mathematical expression not reproducible], (23)

where the supremum is taken over all pseudoballs containing z.

Suppose 0 < [omega](z) < [infinity] almost everywhere on [DELTA]. Then we say that [omega] is in the Muckenhoupt weight [A.sub.1] if

[mathematical expression not reproducible]. (24)

There are two equivalent definitions which are useful in practice. First, [omega] [member of] [A.sub.1], if for almost every z [member of] [DELTA],

[mathematical expression not reproducible]. (25)

It follows that if [omega] [member of] [A.sub.1] then

[mathematical expression not reproducible], (26)

and thus

[mathematical expression not reproducible]. (27)

Alternatively, w [member of] [A.sub.1] if for every pseudoball B we have that

[mathematical expression not reproducible]. (28)

For more details on the Muckenhoupt weights, see chapter 9 of [13] or chapter 4 of [1].

We will need some results on extrapolation. The following proposition is Theorem 5.24 of [1].

Proposition 6. Let [OMEGA] [member of] [R.sup.n] and suppose there is some [p.sub.0] [greater than or equal to] 1 and the family F such that for all [omega] [member of] [A.sub.1],

[mathematical expression not reproducible]. (29)

Given p(*) [member of] [P.sup.log] ([OMEGA]), if 1 [less than or equal to] [p.sub.0] [less than or equal to] [p.sub.-] [less than or equal to] [p.sub.+] < [infinity] and the maximal operator is bounded on [mathematical expression not reproducible], then

[[parallel]F[parallel].sub.p(*)] [less than or equal to] [C.sub.p(*)] [[parallel]G[parallel].sub.p(*)], (F, G) [member of] F, (30)

where [C.sub.p(*)] = C[C.sub.0] and C is some positive constant depending on the dimension of [OMEGA].

The following is Theorem 3.16 of [1].

Proposition 7. Let p [member of] [P.sup.log] ([OMEGA]). Then the Hardy-Littlewood maximal operator function is bounded in [L.sup.p(*)] ([OMEGA]) and we have

[[parallel]Mf[parallel].sub.p(*)] [less than or equal to] C [[parallel]f[parallel].sub.p(*)]. (31)

3. Variable Exponent Bergman Spaces

Given p(*) [member of] [P.sup.log]([DELTA]), we define the variable exponent Bergman space [A.sup.p(*)] as the space of all analytic functions on [DELTA] that belong to the variable exponent Lebesgue space [L.sup.p(*)] with respect to the area measure dA on the unit disc. With this definition [A.sup.p(*)] is a closed subspace of [A.sup.p(*)]. By Theorem 4.4 of [9], the Bergman projection, P, given by (2) is bounded from [L.sup.p(*)] onto [A.sup.p(*)] for any p(*) [member of] [P.sup.log] ([DELTA]). It is, thus, necessary to study the behaviour of Toeplitz operators on such spaces. Similar to the definition of Toeplitz operators on the Bergman spaces with constant exponent, we define the Toeplitz operator with symbol f [member of] [L.sup.[infinity]] on [A.sup.p(*)] by

[T.sub.f]g = P (fg), g [member of [A.sup.p(*)]. (32)

Lemma 8. The operator [U.sub.z] is bounded on [A.sup.p(*)] for p(*) > 1.

Proof. Let 1 < [p.sub.0] [less than or equal to] [p.sub.-] [less than or equal to] [p.sub.+] < [infinity] and [omega] [member of] [A.sub.1]. Then

[mathematical expression not reproducible] (33)

since [omega] ([xi]) [less than or equal to] M [omega] ([xi]) for almost every [xi] [member of] [DELTA]. Now, for 0 < [member of] < [omega] ([DELTA]) there is a pseudoball B containing [xi] such that

M[omega] ([xi]) [less than or equal to] [omega](B)/[absolute value of B] + [member of] [less than or equal to] [omega] ([DELTA])/[absolute value of B] + [omega] ([DELTA]). (34)

It follows that

[mathematical expression not reproducible]. (35)

Also

[mathematical expression not reproducible], (36)

where the last inequality comes from (27). This shows that

[mathematical expression not reproducible]. (37)

It follows that the family [mathematical expression not reproducible] satisfies inequality (29). Also, by Proposition 7 the maximal function Mf belongs to [mathematical expression not reproducible]. Thus by Proposition 6 [U.sub.z] is bounded on [A.sup.p(*)].

Remark 9. We just want to give an alternative argument to obtain the estimate (35), and this argument has different effects and may be useful in applications.

We recall that if [omega] is locally integrable in [DELTA], then

[mathematical expression not reproducible]. (38)

The proof of this statement can easily be adapted from that of Theorem 1.3 of [14]. We use this statement as follows:

Let to [omega] [member of] [A.sub.1] and [p.sub.0] > 1. Then for any [epsilon] > 0, we can find R > 0 such that

[mathematical expression not reproducible], (39)

for all r [member of] (0, R). Now, if we fix such R then for 0 < r < R we have

[mathematical expression not reproducible], (40)

where the constant [C.sub.r] does not depend on [omega] [member of] [A.sub.1], and thus (35) holds. We may also use this same argument in some parts of the proofs of Theorem 2 and Proposition 17 by replacing similar statements that give rise to the estimates (34) in the proof of Lemma 8.

Let

Tf(z) = [[integral].sub.[DELTA]] f(w) K (z, w) dA (w), z [member of] [DELTA], (41)

where K: [DELTA] x [DELTA] [right arrow] is a kernel function. We give a Schur-type lemma that will be useful in our work.

Lemma 10. Let [omega] [member of] [A.sub.1], p > 1, and 1/p + 1/p' = 1 If there exist positive constants [C.sub.1] and [C.sub.2] that depends on [mathematical expression not reproducible] and not [omega], and a nonnegative measurable function, h, such that

[[integral].sub.[DELTA]] [absolute value of K ([xi], z)] h [(z).sup.p'] dA (z) [less than or equal to] [C.sub.1] h [([xi]).sub.p'] (42)

for almost every [xi] [member of] [DELTA] and

[omega] [([DELTA]).sup.-1] [[integral].sub.[DELTA]] [absolute value of K ([xi],z)] h [([xi]).sup.p] [omega] ([xi]) dA (z) [less than or equal to] [C.sub.2] h [(z).sup.p] (43)

for almost every z [member of] [DELTA], then

[mathematical expression not reproducible]. (44)

Proof. Using Holder's inequality, we have

[mathematical expression not reproducible], (45)

where the second inequality comes from (42) and the third inequality is from (27). Also, Fubini's theorem gives

[mathematical expression not reproducible], (46)

where we have used (43) to get the last inequality.

Proposition 11. Let p(*) [member of] [P.sup.log] ([DELTA]), p [member of] (1, [infinity]) be such that 1 < p [less than or equal to] [p.sub.-] [less than or equal to] [p.sub.+] < [infinity]. Suppose the function K : [DELTA] x [DELTA] [right arrow] C satisfies the hypothesis of Lemma 10. Then there is a constant C such that

[[parallel]Tf[parallel].sub.p(*)] [less than or equal to] C[C.sub.1][C.sub.2] [[parallel]f[parallel].sub.p(*)], (47)

for all f [member of] [L.sup.p(*)].

Proof. Let [omega] [member of] [A.sub.1]. Then by Lemma 10, we obtain (29) for the family [mathematical expression not reproducible], where 1 < [p.sub.0] [less than or equal to] [p.sub.-] [less than or equal to] [p.sub.+] < [infinity]. Also by Proposition 7, the maximal function Mf belongs to [mathematical expression not reproducible] and thus by Proposition 6 we get the required estimates.

In the application of Lemma 10, we may assume that [ess inf.sub.z[member of][DELTA]] [omega](z) [greater than or equal to] 1 for [omega] [member of] [A.sub.1], as the following lemma shows.

Lemma 12. Let [omega] [member of] [A.sub.1] be such that [ess inf.sub.z[member of][DELTA]][omega](z) < 1 and [mu] = [delta] [omega] for any [delta] > 1 such that [ess inf.sub.z[member of][DELTA]] [mu](z) [greater than or equal to] 1. If the hypothesis of Lemma 10 holds for the weight [mu], then the conclusion of Lemma 10 holds for the weight [omega].

Proof. By (27), we have that [mathematical expression not reproducible] for almost every z [member of] [DELTA]. Now by Proposition 9.1.5 of [13], we have that [mathematical expression not reproducible]. It follows that [mathematical expression not reproducible] for almost every z [member of] [DELTA]. Thus we have that the constants [mathematical expression not reproducible] are independent of [mu] and, hence, independent of [omega]. Now since the hypothesis of Lemma 10 holds for the weight [mu], we have that

[mathematical expression not reproducible]. (48)

Thus,

[mathematical expression not reproducible]. (49)

It follows that

[mathematical expression not reproducible], (50)

which gives the result.

The next lemma will be used frequently and is well known; see, for example, Lemma 3.10 of [15] for the proof.

Lemma 13. Suppose a < 1 and a + b < 2. Then

[mathematical expression not reproducible]. (51)

Proof of Theorem 2. Let g [member of] [A.sup.p(*)]. Then

[mathematical expression not reproducible]. (52)

Now, let h(z) = [(1 - [[absolute value of z].sup.2]).sup.[epsilon]] for

max (2/q[p.sub.0]], 2/q [p'.sub.0]) < [epsilon] < min (1/q'[p'.sub.0], 1/q'[p.sub.0]). (53)

Then using the identity

[mathematical expression not reproducible] (54)

we have

[mathematical expression not reproducible] (55)

Also,

[mathematical expression not reproducible] (56)

provided that [epsilon] < 1/[p'.sub.0] q' and [epsilon][p'.sub.0]q' + 2q' - 2[epsilon][p'.sub.0]q' < 2. That is, 2/[p'.sub.0]q < [epsilon] < 1/[p'.sub.0]q' which holds from the choice of [epsilon].

Now, observe that

[T.sub.[bar.f]][K.sub.z] ([xi]) = < [T.sub.[bar.f]][K.sub.z], [K.sub.[xi]]> = <[K.sub.z][T.sub.f][K.sub.[xi]]> = [bar.[T.sub.f][K.sub.[xi]](z)]. (57)

Thus, for each [omega] [member of] [A.sub.1], we have

[mathematical expression not reproducible] (58)

since [omega](z) [less than or equal to] M[omega](z) for almost every z [member of] [DELTA]. For any 0 < [[epsilon].sub.1] < [omega]([DELTA]), there is a pseudoball B containing z such that

[mathematical expression not reproducible]. (59)

Substitute this in (58) to obtain

[mathematical expression not reproducible]. (60)

Using the identity (54), we have

[mathematical expression not reproducible], (61)

where the last equality is from the change of variable z = [[phi].sub.[xi]]. By Holder's inequality, we have that

[mathematical expression not reproducible], (62)

if [epsilon] < 1/ [p.sub.0]q' and [epsilon][p.sub.0]q' + 2q'-2[epsilon][p.sub.0]q' < 2; that is, 2/q[p.sub.0] < [epsilon] < 1/[p.sub.0]q'. Now if [epsilon] > 0 is chosen to satisfy (53) we see that the hypothesis of Lemma 10 is satisfied and thus for every [omega] [member of] [A.sub.1] we have

[mathematical expression not reproducible], (63)

where the constant C does not depend on w but on [mathematical expression not reproducible]. Finally we apply Proposition 11 to [mathematical expression not reproducible] to obtain the result.

For f [epsilon] BT it is easy to show that

[mathematical expression not reproducible] (64)

for any q > 0. Thus, from Theorem 2, we immediately have the following.

Corollary 14. Suppose f [epsilon] BT. Then

(1) [T.sub.f] is bounded on [A.sup.p(*)],

(2) [mathematical expression not reproducible] is bounded on [A.sup.p(*)] and

[mathematical expression not reproducible]. (65)

Proof. (1) is an immediate consequence of Theorem 2.

(2) follows from the fact that [mathematical expression not reproducible], which is given by assertion (1).

We also have the following estimate for operators in the Toeplitz algebra. To be precise, we have the following.

Lemma 15. Let [mathematical expression not reproducible]. Then

[mathematical expression not reproducible] (66)

for any P [member of] [P.sup.log] ([DELTA]).

Proof. By assertion (2) of Corollary 14, we have that

[mathematical expression not reproducible]. (67)

Also, since each [bar.[f.sub.j]] [memner of] BT and [[parallel][bar.[f.sub.j]][parallel].sub.BT] = [[parallel][f.sub.j][parallel].sub.BT], we have that

[mathematical expression not reproducible]. (68)

This completes the proof of the lemma.

4. Norm Estimates

Lemma 16. Let 1 < [p.sub.0] < [infinity] and [p.sub.1] = min ([p.sub.0], [p'.sub.0]) and suppose that S is a bounded operator on [A.sup.p(*)] and q > ([p.sub.1] + 1)/([p.sub.1] - 1). If [epsilon] > 0 satisfies (53), then for all [omega] [member of] [A.sub.1] we have the following:

[mathematical expression not reproducible] (69)

for all z [member of] [DELTA] and

[mathematical expression not reproducible] (70)

for all [xi] [member of] [DELTA], where the constant [C.sub.2] does not depend on [omega].

Proof. Fix z [member of] [DELTA]. Then

[mathematical expression not reproducible], (71)

where the second equality comes from the definition of [S.sub.z] and the third equality from the definition of [U.sub.z]. Thus,

[mathematical expression not reproducible], (72)

where [mathematical expression not reproducible]. By the choice of [epsilon], we have that [sup.sub.z[member of][DELTA]] [[parallel][f.sub.z][parallel].sub.q'] < [infinity] and (69) holds.

To prove (70), replace S by [S.sup.*] in (69), interchange [xi] and [xi] in (69), and then use the equation

[S.sup.*] [K.sub.[xi]] (z) = <[S.sup.*] [K.sub.[xi]] [K.sub.z]> = <[K.sub.[xi]], S[K.sub.z]> = [bar.S[K.sub.z]([xi])]. (73)

Finally, we use the same argument as in the proof of Theorem 2 to obtain that there is a pseudoball B containing z such that [omega](z) [less than or equal to] [less than or equal to] M[omega](z) [less than or equal to] [omega] ([DELTA]) ([[absolute value of B].sup.-1] + 1) and thus

[mathematical expression not reproducible]. (74)

A similar argument as the one used to obtain the estimate (69) will give us (70).

Proposition 17. Let p [member of] [P.sup.log]([DELTA]), 1 < [p.sub.0] [less than or equal to] [p.sub.-] [less than or equal to] [p.sub.+] < [infinity], and [p.sub.1] = min([p.sub.0], [p'.sub.0]) and suppose that S is abounded operator on [A.sup.p(*)] If

[mathematical expression not reproducible] (75)

for some q > ([p.sub.1]+1)/([p.sub.1]-1), then there is a constant C such that

[[parallel]S[parallel].sub.(*)] [less than or equal to] C[C.sub.1][C.sub.2]. (76)

Proof. For f [member of] [A.sup.p(*)] and w [member of] [DELTA], we have

[mathematical expression not reproducible], (77)

where the last equation follows from (69). Given that [epsilon] > 0 that satisfies (53), we have by (69) that

[mathematical expression not reproducible]. (78)

In a similar manner, we use (73) and (70) to get that

[mathematical expression not reproducible]. (79)

for all [omega] [member of] [A.sub.1] where the constant C depends on [mathematical expression not reproducible] and not on [omega]. We now apply Proposition 11 to get the required result.

Lemma 18. (a) [[parallel][K.sub.z]parallel].sub.p(*)] is equivalent to ([1-[[absolute value of z].sup.2]).sup.-2+2/p(z)] for all z [member of] [DELTA].

(b) [K.sub.z]/[[parallel][K.sub.z][parallel].sub.p(*)] [right arrow] 0 weakly in [A.sup.p(*)] as z [right arrow] [partial derivative][DELTA].

Proof. The assertion (a) is just Theorem 3.5 of [10].

(b) If f [member of] [A.sub.p'(*)], then assertion (a) implies

[mathematical expression not reproducible]. (80)

Thus if f is a bounded function in [A.sub.p'(*)], then <f, [K.sub.z]/[parallel][K.sub.z][parallel].sub.p(*)]> [right arrow] 0 as z [right arrow] [partial derivative][DELTA]. The assertion follows from the fact that polynomials are dense in [A.sup.p'(*)].

5. Compact Operators on [A.sup.p(*)]

Theorem 19. Let p [member of] [P.sup.log] ([DELTA]) be such that 1 < [p.sub.-] [less than or equal to] [p.sub.+] < [infinity] and suppose that

[mathematical expression not reproducible]. (81)

Then the operator T given by (41) is compact on [L.sup.p(*)] ([DELTA].

Proof. Firstly we observe that if (81) holds and g [member of] [L.sup.p'(*)], then the function w [??] F([omega]), where

F (w) = [[integral].sub.[DELTA]] K (z, w) g(z) dA(z) (82)

belongs to [L.sup.p'(*)] ([DELTA]). Indeed,

[mathematical expression not reproducible]. (83)

It follows from Lemma 1 that [mathematical expression not reproducible]. Thus for g [member of] [L.sup.p'(*)], we see that F [member of] [L.sup.p'(*)]. Now, suppose ([[phi].sub.n]) is a bounded sequence in [L.sup.p(*)] such that [[phi].sub.n] [right arrow] [phi] weakly in [L.sup.p(*)]. For [epsilon] > 0 and any g [member of] [L.sup.p'(*)], we can find that N [member of] N such that for n [greater than or equal to] N we have

[[absolute value of [[integral].sub.[DELTA]] ([[phi].sub.n] (w) - [phi] (w)) [bar.g] (w) dA (w)] < [epsilon]. (84)

We will show that [[parallel]T[[phi].sub.n] - T[phi][parallel].sub.p(*)] [right arrow] 0 as n [right arrow] [infinity]. Now, given [epsilon], we fix N such that (84) holds. It follows for any n [greater than or equal to] N and (11) that

[mathematical expression not reproducible]. (85)

Thus [[parallel]T[[phi].sub.n] - T[phi][parallel].sub.p(*)] [right arrow] 0 as n [right arrow] [infinity]. Finally, it is shown in Corollary 2.81 of [1] that the variable Lebesgue space [L.sup.p(*)] is reflexive if and only if 1 < [p.sub.-] [less than or equal to] [p.sub.+] < [infinity]. We thus conclude that T is compact, since [L.sup.p(*)] is reflexive.

We will need the power series formula for the Berezin transform of the bounded operator S on [A.sup.2]. From the definition of the reproducing kernel, we get that

[k.sub.z] (w) = (1 - [[absolute value of z].sup.2]) [[infinity].summation over (n=0)] (n + 1) [w.sup.n] [[bar.z].sup.n] (86)

for z, w [member of] [DELTA]. To compute [??](z) = <[Sk.sub.z], [k.sub.z]), we first compute [Sk.sub.z] by applying S to both sides of (86) and then take the inner product with [k.sub.z], again using (86), to obtain

[mathematical expression not reproducible]. (87)

Lemma 20. Suppose S is a bounded operator on [A.sup.p(*)] such that

[mathematical expression not reproducible] (88)

for some q > 1. Then [??](z) [right arrow] 0 as z [right arrow] [partial derivative][DELTA] if and only if for every t [member of] [1, q), [[parallel][S.sub.z]1[parallel].sub.t] [right arrow] 0 as z [member of] [partial derivative][DELTA].

Proof. Suppose for every t [member of] [1, q), [[parallel][S.sub.z]1[parallel].sub.t] [right arrow] 0 as z [right arrow] [partial derivative][DELTA]. In particular, [[parallel][S.sub.z]1[parallel].sub.1] [right arrow] as z [member of] [partial derivative][DELTA]. Thus

[absolute value of [??](z)] = [absolute value value of <[S.sub.z]1, 1>] [less than or equal to] [[parallel][S.sub.z]1[parallel].sub.1] [right arrow] 0 (89)

as z [member of] [partial derivative][DELTA].

Suppose [??](z) [right arrow] 0 as z [right arrow] [partial derivative][DELTA]. Fix t [member of] [1, q). We will show that [[parallel][S.sub.z]1[parallel].sub.t] [right arrow] 0 as z [member of] [partial derivative][DELTA].

For z [member of] [DELTA], j, m = 0, 1, ..., an easy computation shows that

[mathematical expression not reproducible]. (90)

Since

[mathematical expression not reproducible], (91)

we have that

[mathematical expression not reproducible]. (92)

It follows from (90) and (92) and Holder's inequality that

[mathematical expression not reproducible]. (93)

Now, let

[mathematical expression not reproducible]. (94)

Then [S.sub.1] [subset] [S.sub.2] and thus inf [S.sub.1] [greater than or equal to] inf [S.sub.2]. This shows that

[[parallel][[phi].sup.j.sub.z][K.sub.z][parallel].sub.p(*)] [less than or equal to] [parallel][K.sub.z][parallel].sub.p(*)] (95)

for any p(*) [member of] [P.sup.log]([DELTA]). This and Lemma 18(a) show that

[mathematical expression not reproducible]; (96)

that is, <[S.sub.z] [w.sup.j], [w.sup.m]) is uniformly bounded in z [member of] [DELTA] and j, m = 0, 1, ....

Now, we will show that for every nonnegative integer n

<[S.sub.z] 1, [w.sup.n]> [right arrow] 0 as z [right arrow] [partial derivative][DELTA]. (97)

If this is not true, then there is a sequence [z.sub.k] [member of] [DELTA] such that

[mathematical expression not reproducible] (98)

for some nonzero constant [a.sub.0n] and n [greater than or equal to] 1. Since <[S.sub.z][w.sup.j], [w.sup.m]> is uniformly bounded, we may assume without loss of generality that for each j and m

<[S.sub.z] [w.sup.j], [w.sup.m]> [right arrow] [a.sub.jm] as z [right arrow] [partial derivative][DELTA]. (99)

for some constant [a.sub.jm]. For z, [xi] [member of] [DELTA], we have

[mathematical expression not reproducible], (100)

where the second equality comes from (87). Also, note that the power series in (100) converges uniformly for each [xi] [member of] [DELTA].

For each [xi] [member of] [DELTA], we know that [mathematical expression not reproducible]. Thus [??]([[phi].sub.z]([xi])) [right arrow] 0 as [z.sub.k] [right arrow] [partial derivative][DELTA] for each [xi] [member of] [DELTA]. Replacing z by [z.sub.k] in (100) and taking the limit as [z.sub.k] [right arrow] [partial derivative][DELTA], we get

[(1 - [[absolute value of [xi]].sup.2].sup.2] [[infinity].summation over (j, m=0)] (j + 1) (m + 1) [a.sub.jm] [[bar.[xi]].sup.j] [[xi].sup.m] = 0 (101)

for each [xi] [member of] [DELTA]. If

f([xi]) = [[infinity].summation over (j, m=0)] (j + 1) (m + 1) [a.sub.jm] [[bar.[xi]].sup.j] [[xi].sup.m], (102)

then f([xi]) = 0 for all [xi] [member of] [DELTA]. This gives

[[partial derivative].sup.m]/[partial derivative][[xi].sub.m] [[partial derivative].sup.j]/[partial derivative] [[bar.[xi]].sub.j] f] (0) = 0 (103)

for each j and m. On the other hand, we have

[[partial derivative].sup.m]/[partial derivative][[xi].sub.m] [[partial derivative].sup.j]/[partial derivative] [[bar.[xi]].sub.j] f] (0) = ((j + 1)! (m + 1)!) [a.sub.jm] (104)

for each j and m. In particular, [mathematical expression not reproducible] which is a contradiction. Hence, we obtain

[mathematical expression not reproducible]. (105)

For [xi] [member of] [DELTA], we have

([S.sub.z]1) ([xi]) = [[infinity].summation over (n=0)] (n + 1) <[S.sub.z]1, [[omega].sup.n]> [[xi].sup.n]. (106)

It is clear that for each fixed [xi] [member of] [DELTA], the power series above converges uniformly for z [member of] [DELTA]. This gives

[mathematical expression not reproducible] (107)

for each [xi] [member of] [DELTA]. It follows that

[mathematical expression not reproducible] (108)

for each [xi] [member of] [DELTA]. If s = q/t then s > 1 and

[mathematical expression not reproducible]. (109)

This shows that [{[[absolute value of ([S.sub.z]1)].sup.t]}.sub.z[member of][DELTA]] is uniformly integrable. By Vitali's Theorem or Exercise 11 on pages 133-134 of [16], we have that

[mathematical expression not reproducible]. (110)

This completes the proof of the lemma.

Proof of Theorem 3. If S is compact on [A.sup.p(*)], then by Lemma 18(b),

[mathematical expression not reproducible] (111)

as z [right arrow] [delta] [DELTA]. Now by Lemma 18(a), we see that [??](z) is equivalent to ([SK.sub.z]/[[parallel][K.sub.z][parallel].sub.p(*)], [K.sub.z]/[[parallel][K.sub.z][parallel].sub.p'(*)]> for z [member of] [DELTA]. Thus [??](z) [right arrow] 0 as z [right arrow] [partial derivative][DELTA].

Suppose [??](z) [right arrow] 0 as z [right arrow] [partial derivative][DELTA]. By Lemma 20, we have that [[parallel][S.sub.z] 1][parallel].sub.t] [right arrow] 0 as z [right arrow] [partial derivative][DELTA] for every t [member of] [1, q). We will show that S is compact on [A.sup.(*)]. Fix t [member of] (1, q) in the rest of the proof.

For f [member of] [A.sup.p(*)], we have that

(Sf) (w) = [[integral].sub.[DELTA]] f (z) ([SK.sub.z]) (w) dA (z). (112)

For 0 < r < 1, we define an operator [S.sub.[r]] on [A.sup.p(*)] by

([S.sub.[r]] f) (w) = [[integral].sub.r[DELTA]] f (z) ([SK.sub.z]) (w) dA (z). (113)

Then [S.sub.[r]] is an integral operator with kernel ([SK.sub.z]) (w) [[chi].sub.r[DELTA]] (z). We will first show that the operator [S.sub.[r]] is compact on [A.sup.p(*)]. By Theorem 19, we only need to show that

[mathematical expression not reproducible]. (114)

But

[mathematical expression not reproducible]. (115)

This shows that

[mathematical expression not reproducible], (116)

and thus [S.sub.[r]] is compact on [A.sup.p(*)]. Hence, to prove that S is compact, we only need to show that [[parallel]S-[S.sub.[r]][parallel].sub.p(*)] [roght arrow] 0 as r [right arrow] [1.sup.-].

If r [member of] (0, 1) then S-[S.sub.[r]] is the integral operator with kernel

([SK.sub.z]) (w) [[chi].sub.[DELTA]/r[DELTA]] (z) (117)

as can be seen from (77) and (113). The proof of Proposition 17 indicates that

[parallel]S-[S.sub.[r]][parallel].sub.p(*)] [less than or equal to] C[C.sub.1][C.sub.2], (118)

where

[mathematical expression not reproducible]. (119)

We have shown that [C.sub.1] [right arrow] 0 as r [right arrow] [1.sup.-] and the hypothesis of the theorem shows that [C.sub.2] < [infinity]. Thus, [parallel]S-[S.sub.[r]][parallel].sub.p(*)] right arrow] 0 as r [right arrow] [1.sup.-], which completes the proof.

Proof of Theorem 4. Suppose S is a finite sum of operators of the form [mathematical expression not reproducible]. By Corollary 14 and Lemma 15, we have that S is bounded on [A.sup.p(*)] and

[mathematical expression not reproducible] (120)

for all q > 0. The conclusion then follows from Theorem 3.

https://doi.org/10.1155/2018/1417989

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this paper.

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Dieudonne Agbor (iD)

Department of Mathematics, Faculty of Science, University of Buea, P.O. Box 63, Buea, Cameroon

Correspondence should be addressed to Dieudonne Agbor; dieu_agb@yahoo.co.uk

Received 20 October 2017; Accepted 6 December 2017; Published 30 January 2018

Academic Editor: Nageswari Shanmugalingam
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