Complex Anal. Oper. Theory DOI 10.1007/s11785-016-0573-0

Complex Analysis and Operator Theory

Carleson Measures for Variable Exponent Bergman Spaces Gerardo R. Chacón1 · Humberto Rafeiro2 · Juan Camilo Vallejo2

Received: 18 November 2015 / Accepted: 15 June 2016 © Springer International Publishing 2016

Abstract In this article we define and characterize Carleson measures for the setting of variable exponent Bergman spaces. We also estimate the norm of the reproducing kernels under this context. Keywords Bergman spaces · Variable exponent Lebesgue spaces · Carleson measures Mathematics Subject Classification Primary 32A36; Secondary 46E30

1 Introduction Variable Lebesgue spaces generalize Lebesgue spaces allowing the exponent to be a measurable function and thus the exponent may vary. It seems that the first occurrence in the literature is in the 1931 paper of Orlicz [22]. The seminal work on this field is the 1991 paper of Kováˇcik and Rákosník [19] where many basic properties of Lebesgue and Sobolev spaces were shown. To see a more detailed history of such spaces see,

Communicated by Amol Sasane.

B

Gerardo R. Chacón [email protected] Humberto Rafeiro [email protected] Juan Camilo Vallejo [email protected]

1

Department of Science, Technology, and Mathematics, Gallaudet University, 800 Florida Ave. NE, Washington, DC 20002, USA

2

Departamento de Matemáticas, Pontificia Universidad Javeriana, Bogotá, Colombia

G. R. Chacón et al.

e.g., [14, Sect. 1.1]. These variable exponent function spaces have a wide variety of applications, e.g., in the modeling of electrorheological fluids [3,4,25,26] as well as thermorheological fluids [5], in the study of image processing [1,2,7,8,11,12,27] and in differential equations with non-standard growth [18,21]. For details on variable Lebesgue spaces one can refer to [13,14,19,23]. Let D denote the unit disc in the complex plane and d A the Lebesgue measure on D normalized in such a way that A(D) = 1. For a given 1  p < ∞ define the Bergman space A p (D) as the space of all analytic functions on D that satisfies: p

ˆ

 f  A p (D) :=

| f (z)| p dA(z) < +∞. D

The theory of Bergman spaces was introduced by Bergman in [6] and since the 1990s it has gained a lot of attention mainly due to some major breakthroughs at the time. For details on the theory of Bergman spaces we refer to the books [15,28,29] and the references therein. Recently in [9] the first two authors of this article introduced variable exponent Bergman spaces A p(·) and studied such properties as the density of polynomials and boundedness of some projection transforms, it should be stressed that the standard approaches based on certain change of variables are no more useful in this context due to the anisotropic nature of the space regarding the exponent. In this article, we continue the study of variable exponent Bergman spaces by giving a better estimate of evaluation functionals and using that to characterize Carleson measures under this contexts. The article is distributed as follows: In Sect. 2 we give some basic notions about variable exponent spaces that are going to be needed for the rest of the article. In Sect. 3, we introduce the definition of variable exponent Bergman spaces and find an estimate for the norm of evaluation functionals that improves the one obtained in [9]. In Sect. 4 we introduce the notion of Carleson measures in this setting and show a geometrical characterization under the additional hypothesis that p(·) ∈ P log (D). We also show an example for which such characterization does not hold in the general case. For the rest of the paper, we will use the notation a  b if there exists a constant C > 0 such that a  Cb. Similarly, we use a  b if we have a  b  a. The symbol | · | stands for the Lebesgue measure and also for the absolute value, which will be clear from the context.

2 Basic Notions 2.1 On Lebesgue Spaces with Variable Exponent The basics on variable Lebesgue spaces may be found in the monographs [13,14,23], but we recall here some necessary definitions and propositions without explicitly citing the exact references from the aforementioned books. Given  ⊂ Rd , a measurable + := function p :  −→ [1, ∞), will be called a variable exponent. Denote p

Carleson Measures for Variable Exponent Bergman Spaces − ess supx∈ p(x) and p := ess inf x∈ p(x). When clear from the context, we use the + − + and p − = p . Define the set of all variable exponents with abbreviations p = p + p < ∞ as P(). For a complex-valued measurable function ϕ :  −→ C we define the modular ρ p(·),μ by

ˆ ρ p(·),μ (ϕ) :=

|ϕ(x)| p(x) dμ(x) 

and the Luxemburg-Nakano norm by  ϕ   ϕ L p(·) (,μ) := inf λ > 0 : ρ p(·),μ 1 . λ

(1)

When dealing with the Lebesgue measure L or the normalized Lebesgue measure A on D we will drop the subscript μ in the modular ρ p(·) and in the norm · L p(·) () , and sometimes we will write the norm simply as · p(·) . Definition 2.1 Let p(·) ∈ P(). The variable Lebesgue space L p(·) (, μ) is introduced as the set of all complex-valued measurable functions ϕ :  −→ C for which the modular is finite, i.e. ρ p(·),μ (ϕ) < ∞. Equipped with the Luxemburg-Nakano norm (1) this is a Banach space. 

Proposition 2.2 Let p(·) ∈ P(), then the dual space of L p(·) () is L p (·) (), where 1/ p  (x) + 1/ p(x) = 1. We need to impose some regularity in the variable exponent in order to have some “fruitful” theory (e.g. the boundedness of the maximal operator). Definition 2.3 A function p :  −→ R is said to be log-Hölder continuous or satisfy the Dini-Lipschitz condition on  if there exists a positive constant Clog such that | p(x) − p(y)| 

Clog , log (1/|x − y|)

(2)

for all x, y ∈  such that |x − y| < 1/2, from which we obtain | p(x) − p(y)| ≤

2Clog   2 log |x−y|

(3)

for all x, y ∈  such that |x − y| < . We will write p(·) ∈ P log (D) when p(·) is log-Hölder continuous. It is known that condition (2) is equivalent to the following characterization −

+

|B| p B − p B ≤ C,

(4)

G. R. Chacón et al.

for all balls, where | · | stands for the normalized Lebesgue measure. Notice that for z, w ∈ B, as a consequence of the previous inequality, we get that |B| p(z)  |B| p(w) . We will use this relation several times in the article. The following Jensen type inequality was proved in [24] in the context of spaces of homogeneous type (SHT). Lemma 2.4 Let X be a SHT. Suppose that p(·) ∈ P log (X ) Then ⎞ p(z)

⎛

⎟ | f (y)|dμ(y)⎠

⎜ ⎝

⎛

⎞

⎜ ⎝

⎟ | f (y)| p(y) dμ(y) + 1⎠

B(x,r )

(5)

B(x,r )

for all z ∈ B(x, r ) provided that  f  L p(·) (X )  1. 2.2 Variable Exponent Bergman Spaces We now introduce Bergman spaces in the variable exponent framework. Definition 2.5 Given a measurable function p ∈ P(D) we define the variable exponent Bergman space A p(·) (D) as the space of all analytic functions on D that belong to the variable exponent Lebesgue space L p(·) (D) with respect to the area measure dA on the unit disc, i.e. ⎧ ⎫ ˆ ⎨ ⎬ A p(·) (D) = f is an analytic function and | f (z)| p(z) dA(z) < ∞ . ⎩ ⎭ D

With this definition, it is easy to show that A p(·) (D) is a closed subspace of L p(·) (D). As a matter of fact, for every z ∈ D, the evaluation functional γz : A p(·) (D) −→ C defined as (6) γz ( f ) := f (z) is bounded. Therefore convergence in A p(·) (D) implies uniform convergence on compact subsets of D and a standard argument show that in consequence A p(·) (D) is a Banach space (and hence closed in L p(·) (D)).

3 Norm Estimates Before estimating the norm of the evaluation functional, we will need some considerations about some test functions and some preliminary estimates (cf. [28]): Proposition 3.1 For any −1 < α < ∞ and any real β, let ˆ Iα,β (z) := D

(1 − |w|2 )α dA(w), z ∈ D. |1 − zw|2+α+β

Carleson Measures for Variable Exponent Bergman Spaces

Then

⎧ ⎪ ⎨ 1, 1 Iα,β (z) ∼ log 1−|z|2 , ⎪ 1 ⎩ , (1−|z|2 )β

if β < 0, if β = 0, if β > 0,

(7)

as |z| → 1. In the case p ≡ 2, we have that the corresponding Bergman space is a reproducing kernel space with kernel: ka (z) =

1 , (1 − az) ¯ 2

a, z ∈ D.

(8)

Reproducing kernels play an important role in the classical theory of Bergman spaces. We will estimate the norm of the reproducing kernel in the case of variable exponents. First, we will introduce some useful notation. Let ϕa (z) denote the disk automorphism ϕa (z) =

a−z 1 − az ¯

and for each a ∈ D, notice that ϕa (z) =

|a|2 − 1 . (1 − az) ¯ 2

The function ϕa appears as the Jacobian of the change of variables with respect to the disk automorphism, it is also the kernel defining the Berezin transform on Bergman spaces. We will also use it as a tool for the norm estimate of the reproducing kernels. Given two points z and w in the unit disc, its pseudo-hyperbolic distance is defined as ρ(z, w) := |ϕz (w)|. Given 0 < r < 1, the pseudo-hyperbolic disk with center at a ∈ D and radius r is defined as the set Dr (a) := {z ∈ D : |ϕa (z)| < r }. It is well known that pseudo-hyperbolic disks are Euclidean disks with different center and radius. We will need the following properties of the pseudo-hyperbolic disks that can be found for example in [28, Lemma 2.12]. Proposition 3.2 Let 0 < r, s < 1 and ka be given by (8). Then the following estimates hold: 1. |Dr (a)|  |Ds (a)|  (1 − |a|2 )2

G. R. Chacón et al.

2. If z ∈ Dr (a), then |ka (z)|  (1 − |a|2 )−2 ; 3. If z, w ∈ Dr (a), then 1 − |z|2  1 − |w|2 ; where the constants involved depend only on r and s. A consequence of Eq. (4) is that if p(·) ∈ P log (D) and z, w ∈ Dr (a), then |Dr (a)| p(z)  |Dr (a)| p(w) . Moreover, using the previous proposition, we have that (1 − |a|2 )2 p(z)  (1 − |a|2 )2 p(w)

(9)

Now we are ready to prove the following upper bound. Proposition 3.3 Let p(·) ∈ P log (D). Then there exists a constant C > 0 such that for every a ∈ D, 2/ p(a)

ka

 A p(·) (D) ≤

C . (1 − |a|2 )2/ p(a)

Proof Let us first estimate the following integral ˆ Ia :=

|ϕa (z)|2 p(z)/ p(a) dA(z).

D

Notice that by a standard change of variables with respect to the automorphism ϕa we get that ˆ Ia =

|ϕa (ϕa (z))|2 p(ϕa (z))/ p(a) |ϕa (z)|2 dA(z)

D

but noticing that |ϕa (ϕa (z))| = |ϕa (z)|−1 we get that ˆ Ia =

|ϕa (z)|2−2 p(ϕa (z))/ p(a) dA(z).

D

We will now divide th integral Ia into two parts: ˆ I1 =

2− |ϕa (z)|

2 p(ϕa (z)) p(a)

|ϕa (z)|≥1

ˆ dA(z),

I2 =

|ϕa (z)|

a (z)) 2− 2 p(ϕ p(a)

dA(z)

|ϕa (z)|<1

Let us estimate I1 : Notice that in this case t → |ϕa (z)|t is an increasing function, consequently |ϕa (z)|2−2 p(ϕa (z))/ p(a) ≤ |ϕa (z)|2 and by (7) we get that I1  1.

Carleson Measures for Variable Exponent Bergman Spaces

On the other hand, to estimate I2 notice that, for ϕa (z) = 0, we have   2 1 | p(a) − p(ϕa (z))| log    ϕ (z) p(a) a   2 · 2 · Clog 2 1  log  ≤ 4 p(a) log |ϕa (z)| |a−ϕa (z)|   2 · 2 · Clog 2 1  log  = 4 p(a) log |ϕa (z)| 

     log |ϕa (z)|2−2 p(ϕa (z))/ p(a)  =

|ϕa (z)||z||1−az|

≤

8Clog p−

where the first inequality is due to the log-Hölder condition (3) and the last inequality is due to the fact that 1 |ϕa (z)|

≤

4 |z||ϕa (z)||1 − az|

since |z||1 − az| < 2. Hence, the integrand in I2 is bounded by a constant depending only on p(·). Thus, Ia is bounded and consequently there exists C > 0 depending only on p(·) such that |ϕa |2/ p(a)  p(·) ≤ C. The previous estimates imply the bound 2/ p(a)

ka

 p(·) ≤

C (1 − |a|2 )2/ p(a)  

which finishes the proof.

The next theorem gives an estimate for the norm of the functional γz , which improves previous results obtained in [9]. Theorem 3.4 Let p(·) ∈ P log (D). Then for every z ∈ D we have that γz  

1 . (1 − |z|2 )2/ p(z)

Proof Let f ∈ A p(·) (D) be such that  f  A p(·) (D) = 1. Then ρ( f ) ≤  f  A p(·) (D) . For any z ∈ D we have that by the mean value theorem and the change of variables formula that if 0 < r < 1, then

G. R. Chacón et al.

| f (z)| = | f (ϕz (0))| ≤  

1 r2

1 r2

ˆ | f (w)| Dr (z)

1 (1 − |z|2 )2

ˆ | f (ϕz (w))|dA(w) Dr (0)

(1 − |z|2 )2 dA(w) |1 − wz| ¯ 4

ˆ | f (w)|dA(w). Dr (z)

Therefore, using Lemma 2.4 we have | f (z)|

p(z)

1  (1 − |z|2 )2

ˆ | f (w)| p(w) dA(w) + 1 Dr (z)

1  f  A p(·) (D) +  f  A p(·) (D) (1 − |z|2 )2 1   f  A p(·) (D) . (1 − |z|2 )2 

Now for a general f , define g = f  f −1 and apply the previous result. A p(·) (D) For the other inequality, let us take 2

2/ p(a)

f a := (1 − |a|2 ) p(a) ka

,

and by Proposition 3.3 we conclude that  f a  A p (D) ≤ C. Noticing that f a (a) =

1 , (1 − |a|2 )2/ p(a)

γz  

1 . (1 − |z|2 )2/ p(z)

we obtain the estimate

  The previous estimate allows to get an estimate for the norm of the functions ka , a ∈ D. In [9] it is shown that every element φ ∈ (A p(·) (D))∗ is associated to a function  1 + p1(z) = 1, ∀z ∈ D in such a way that g ∈ A p (·) (D), where p(z) ˆ φ( f ) = and φ  g A p (·) (D) .

D

f (w)g(w)dA(w)

Carleson Measures for Variable Exponent Bergman Spaces

On the other hand, it is well-known that the functions ka , a ∈ D satisfy the reproducing property ˆ f (a) =

D

f (w)ka (w)dA(w)

for every f ∈ A1 (D). Thus, γa   ka  A p (·) (D) . This result is presented in the following theorem. Theorem 3.5 Let p(·) ∈ P log (D). Then for every z ∈ D we have that k z  A p(·) (D) 

1 . (1 − |z|2 )2(1−1/ p(z))

4 Carleson Measures We are ready to define Carleson measures on variable exponent Bergman spaces. Definition 4.1 Given a positive Borel measure μ defined on D, we say that μ is a Carleson measure for the variable exponent Bergman space A p(·) (D) if there exists a constant C > 0 such that  f  L p(·) (D,μ)  C f  A p(·) (D) . In other words, μ is a Carleson measure on A p(·) (D) if A p(·) (D) is continuously embedded in L p(·) (D, μ), i.e. A p(·) (D) → L p(·) (D, μ). Lennart Carleson geometrically characterized Carleson measures on Hardy spaces H p and used this result in his proof of the Corona theorem and some interpolation problems (cf. [17]). After that, Carleson measures had gained interest and a geometric characterization is usually important in spaces of analytic functions. In the case of the classical Bergman spaces A p (D), Carleson measures are characterized in terms of the measure on pseudo-hyperbolic disks. We will need the following well-known geometric observation (see, for example [15]). Lemma 4.2 For each 0 < r < 1, there exists a sequence {ak }k∈N of points in D and an integer N such that ∞ 

Dr (ak ) = D

k=1

and no point z ∈ D belongs to more than N of the dilated pseudo-hyperbolic disks D R (ak ), where R = 1+r 2 . The result for classical Bergman spaces in the unit disk is the following (see for example [15]).

G. R. Chacón et al.

Theorem 4.3 Let μ be a finite, positive Borel measure on D. The following statements are equivalent: (i) μ is a Carleson measure for A p (D); (ii) For every 0 < r < 1, there exists a constant C > 0 such that μ(Dr (a))  C|Dr (a)| for every a ∈ D; (iii) There exists 0 < r < 1 and a constant C > 0 such that μ(Dr (a))  C|Dr (a)| for every a ∈ D. Moreover, the geometric condition on pseudo hyperbolic discs, can be replaced by a condition over a different type of sets, cf. [15]. Proposition 4.4 Let μ be a finite, positive Borel measure on D, the following conditions are equivalent. (i) For every 0 < r < 1, there exists a constant C > 0 such that for every a ∈ D, μ(Dr (a)) ≤ C|Dr (a)|. (ii) For every 0 < h < 1, there exists a constant C > 0 such that for every θ0 ∈ [0, 2π ), μ(Sh,θ0 ) ≤ Ch 2 , where Sh,θ0 := {r eiθ : 1 − h < r < 1, |θ − θ0 | < 2π h}. The sets of the form Sh,θ0 are called Carleson squares and its area is comparable to h2. One interesting observation is that the characterization of Carleson measures does not depend on the exponent p. In the context of variable exponent Bergman spaces, since we cannot use the relation L p(·) (D) ⊂ L p− (D) + L p+ (D) in the case of analytic functions, we need to work directly in the space A p(·) (D). First, we show that the characterization is not true in general, however, when p(·) ∈ P log (D), then we can get such a result. The following example is inspired by [20]. Example 4.5 Consider a uniformly discrete sequence {z n } ⊂ (0, 1), i.e. a sequence such that there exists δ > 0 with the property that Dδ (z j ) ∩ Dδ (z k ) = ∅ for j = k. Such a choice is always possible. Moreover (see [16, Proposition 1]) such a sequence satisfies: 1 − z n+1 ≤ r (1 − z n ) for some 0 < r < 1. This implies that n(1 − z n )2 → 0 and taking k ∈ N such that for all n ≥ k, n(n + 1)−1 > r 2 , we have that n(1 − z n )2 ≥

n(1 − z n+1 )2 ≥ (n + 1)(1 − z n+1 )2 . r2

Thus, the following measure is well-defined: μ=

∞    n(1 − z n )2 − (n + 1)(1 − z n+1 )2 δz n n=k

Carleson Measures for Variable Exponent Bergman Spaces

where δz n denotes the one point mass measure at z n . Now since {z n } is increasing then if N ≥ k, μ(S1−z N ,0 ) = N (1 − z N )2 and consequently, Proposition 4.4(ii) does not hold. On the other hand, let us define p : D −→ (1, ∞) as  2n, p(z) = 0,

if z ∈ Dδ (z n ) otherwise.

Now, suppose that f ∈ A p(·) (D), and  f  A p(·) (D) = 1, then by the mean value theorem, 1 (1 − z n2 )2

| f (z n )| 

ˆ | f (w)|dA(w) Dδ (z n )

and using Jensen’s inequality, there exists a constant C > 0 not depending on n such that | f (z n )|

2n

C 2n ≤ (1 − z n2 )2 ≤ ≤

C 2n (1 − z n2 )2

ˆ | f (w)|2n dA(w) Dδ (z n )

ρ p(·) ( f )

C 2n . (1 − z n2 )2

Thus, | f (z n )| 

For the general case, take g =

f  f  A p(·) (D)

| f (z n )| 

1 . (1 − z n2 )1/n to conclude that  f  A p(·) (D) (1 − z n2 )1/n

.

G. R. Chacón et al.

With this inequality in hand, we have that  ρ p(·),μ



f

=

3  f  A p(·) (D)



∞ 

| f (z n )| p(z n )

n=1

3 p(z n )  f  A p(·)n (D)

p(z )

  n(1−z n )2 −(n+1)(1−z n+1 )2

∞  n 32n n=1

< 1, which shows that μ is a Carleson measure for A p(·) (D). We now show a characterization of Carleson measures. It is interesting to notice that, under the additional condition that p(·) ∈ P log (D), the geometric characterization does not depend on p(·) either. Theorem 4.6 Let μ be a finite, positive Borel measure on D and p(·) ∈ P log (D). Then μ is a Carleson measure for A p(·) (D) if and only if there exist constants C > 0 and 0 < r < 1 such that for every a ∈ D, μ(Dr (a)) ≤ C|Dr (a)|. Proof First, suppose that for every f ∈ A p(·) , we have that  f  L p(·) (μ)   f  A p(·) . 2/ p(a)

In particular, given a ∈ D the inequality holds for f a := (1 − |a|2 )2/ p(a) ka . By Proposition 3.3 there exists a constant C1 > 0 such that C1  f a  ≤ 1. Now, given 0 < r < 1, let C2 > 0 be such that for any z in the pseudo-hyperbolic disk: Dr (a) := {z ∈ D : |ϕa (z)| < r } we have that |1 − az| ¯ ≤ C2 (1 − |a|2 ). Then, ˆ

4

|C2p(a) f a (z)| p(z) (1 − |a|2 )2 p(z)/ p(a) p(z)

χ Dr (a)  L p(·) (D,μ)

D

ˆ ≥

4

|C2p(a) f a (z)| p(z) (1 − |a|2 )2 p(z)/ p(a) p(z)

Dr (a)

ˆ

≥

χ Dr (a)  L p(·) (D,μ) 1

dμ(z) p(z) χ Dr (a)  L p(·) (D,μ)   χ Dr (a) = ρ p(·),μ χ Dr (a)  L p(·) (D,μ) Dr (a)

= 1.

dμ(z)

dμ(z)

Carleson Measures for Variable Exponent Bergman Spaces

This shows that  f a  L p(·) (D,μ) 

χ Dr (a)  L p(·) (D,μ) (1 − |a|2 )2/ p(a)

.

Thus, χ Dr (a)  L p(·) (D,μ) (1 − |a|2 )2/ p(a)

  f a  L p(·) (D,μ)   f a  A p(·) (D)  1

and consequently there exists a constant C > 0, depending only on p(·), such that χ Dr (a)  L p(·) (D,μ) ≤ C(1 − |a|2 )2/ p(a) .   Consequently, we have that C −1 (1 − |a|2 )−2/ p(a) χ Dr (a)  L p(·) (D,μ) ≤ 1 and hence ρ p(·),μ (C −1 (1 − |a|2 )−2/ p(a) χ Dr (a) ) ≤ 1. This implies that ˆ Dr (a)

dμ(z) C p(z) (1 − |a|2 )2 p(z)/ p(a)

≤1

but since p(·) ∈ P log (D), then we use Eq. (9) to conclude that for every z ∈ Dr (a), (1 − |a|2 )2 p(z)/ p(a)  (1 − |a|2 )2 and consequently by Proposition 3.2, μ(Dr (a))  |Dr (a)|. This concludes the first part of the proof. For the second part, suppose that there exist constants C > 0 and 0 < r < 1 such that μ(Dr (a))  C|Dr (a)| for every a ∈ D. Take z ∈ Dr (a) and let R = 1+r 2 . Notice √ that if r ≥ 2 − 1 and w ∈ Dr (z) then ρ(w, a) ≤

2r ρ(w, z) + ρ(z, a) ≤ < R. 1 + ρ(w, z)ρ(z, a) 1 + r2

Consequently, by the mean value √ theorem, Proposition 3.2 and the change of variables formula, we have that if r ≥ 2 − 1 then for any function f ∈ A p(·) (D) such that

G. R. Chacón et al.

 f  A p(·) (D) = 1 it holds: | f (z)| 

ˆ

1 r2

| f (w)|

ˆ

Dr (z)

(1 − |z|2 )2 dA(w) |1 − wz| ¯ 4

| f (w)| dA(w) (1 − |a|2 )2 D R (a) ˆ 1  | f (w)|dA(w) |D R (a)| 

D R (a)

where the constant just depends on r and R but not on a or f . Consequently, using the Jensen type inequality (5) we have that | f (z)| p(z) 

and hence, ˆ | f (z)| p(z) dμ(z)  Dr (a)

1 |D R (a)|

1 |D R (a)|

μ(Dr (a)) |Dr (a)|

| f (w)| p(w) dA(w) + 1 D R (a)

ˆ

ˆ

| f (w)| p(w) dA(w)dμ(z) + μ(Dr (a)) Dr (a) D R (a)

μ(Dr (a))  |Dr (a)| If

ˆ

ˆ

| f (w)| p(w) dA(w) + μ(Dr (a)). D R (a)

 C, then we have that

ˆ

ˆ | f (z)|

p(z)

dμ(z) 

Dr (a)

| f (w)| p(w) dA(w) + μ(Dr (a)). D R (a)

Now, consider the sequence {ak }k∈N ⊂ D given by Lemma 4.2 and notice that ˆ | f (z)|

p(z)

dμ(z) 

D



∞ ˆ  k=1 D (a ) r k ˆ ∞ 

| f (z)| p(z) dμ(z)

| f (w)| p(w) dA(w) +

k=1 D (a ) R k

ˆ

N

∞  k=1

| f (w)| p(w) dA(w) + N μ(D) D

 N (1 + μ(D)).

μ(Dr (ak ))

Carleson Measures for Variable Exponent Bergman Spaces

This shows that if  f  A p(·) = 1 and r ≥

√ 2 − 1, then

 f  L p(·) (μ)   f  A p(·) . Now, if r <

(10)

√ 2 − 1, it follows from proposition 3.2 and the monotony of μ that

    μ(D√2−1 (a)) ≤ μ(Dr (a)) ≤ C |Dr (a)|  D√2−1 (a) and consequently Eq. (10) holds. The case  f  A p(·) = 1 follows by the homogeneity of the norms. Remark 4.7 There is the related concept of vanishing Carleson measures which was not discussed in this article since it is studied in [10]. In such article, the authors also investigate the boundedness and compactness of Toeplitz operators in variable exponent Bergman spaces. Acknowledgments The authors would like to thank the anonymous referee for pointing out an error in the proof of Theorem 4.6 and for her valuable suggestions. H.R. was partially supported by the Research Project Espacios de Bergman con Exponente Variable, ID-PPTA: 5992 of the Faculty of Sciences of the Pontificia Universidad Javeriana, Bogotá, Colombia. J.C.V. was supported by the Grant Joven Investigador of Colciencias (Gobierno de Colombia).

References 1. Aboulaich, R., Boujena, S., El Guarmah, E.: Sur un modèle non-linéaire pour le débruitage de l’image. C. R. Math. Acad. Sci. Paris 345(8), 425–429 (2007) 2. Aboulaich, R., Meskine, D., Souissi, A.: New diffusion models in image processing. Comput. Math. Appl. 56(4), 874–882 (2008) 3. Acerbi, E., Mingione, G.: Regularity results for electrorheological fluids, the stationary case. C. R. Math. Acad. Sci. Paris 334(9), 817–822 (2002) 4. Acerbi, E., Mingione, G.: Regularity results for stationary electrorheological fluids. Arch. Ration. Mech. Anal. 164(3), 213–259 (2002) 5. Antontsev, S.N., Rodrigues, J.F.: On stationary thermorheological viscous flows. Ann. Univ. Ferrara, Sez. VII Sci. Mat. 52(1), 19–36 (2006) 6. Bergman, S.: The kernel function and conformal mapping, 2nd edn. American Mathematical Society, Providence (1970) 7. Blomgren, P., Chan, T., Mulet, P., Wong, C.K.: Total variation image restoration, numerical methods and extensions. In: Proceedings of the 1997 IEEE International Conference on Image Processing, vol. III, pp. 384–387 (1997) 8. Bollt, E.M., Chartrand, R., Esedo¯glu, S., Schultz, P., Vixie, K.R.: Graduated adaptive image denoising, local compromise between total variation and isotropic diffusion. Adv. Comput. Math. 31(1–3), 61–85 (2009) 9. Chacón, G.R., Rafeiro, H.: Variable exponent Bergman spaces. Nonlinear Anal. 105, 41–49 (2014) 10. Chacón, G.R., Rafeiro, H.: Toeplitz operators on variable exponent Bergman spaces. Mediterr. J. Math. doi:10.1007/s00009-016-0701-0 11. Chen, Y., Guo, W., Zeng, Q., Liu, Y.: A nonstandard smoothing in reconstruction of apparent diffusion coefficient profiles from diffusion weighted images. Inverse Probl. Imaging 2(2), 205–224 (2008) 12. Chen, Y., Levine, S., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66(4), 1383–1406 (2006). (electronic) 13. Cruz-Uribe, D., Fiorenza, A.: Variable lebesgue spaces: foundations and harmonic analysis. Birkhäuser, Boston (2013)

G. R. Chacón et al. 14. Diening, L., Harjulehto, P., Hästö, P., R˚užiˇcka, M.: Lebesgue and sobolev spaces with variable exponents. lecture notes in mathematics. Springer, Berlin (2011) 15. Duren, P., Schuster, A.: Bergman spaces mathematical surveys and monographs, vol. 100. American Mathematical Society, USA (2004) 16. Duren, P., Schuster, A., Vukoti´c, D.: On uniformly discrete sequences in the disk. Operator Theory 156, 131–150 (2005) 17. Garnett, J.W.: Bounded analytic functions. Springer, Berlin (2007) 18. Harjulehto, P., Hästö, P., Lê, Ú.V., Nuortio, M.: Overview of differential equations with non-standard growth. Nonlinear Anal. 72(12), 4551–4574 (2010) 19. Kovˇcik, O., Rkosnk, J.: On spaces L p(x) and W k, p(x) . Czechoslov. Math. J. 41(4), 592–618 (1991) 20. Lefèvre, P., Li, D., Queffélec, H., Rodríguez-Piazza, L.: Composition operators on Hardy-Orlicz spaces. Mem. Am. Math. Soc. 207, 974 (2010) 21. Mingione, G.: Regularity of minima, an invitation to the dark side of the calculus of variations. Appl. Math. 51(4), 355–426 (2006) 22. Orlicz, W.: Über konjugierte Exponentenfolgen (German). Studia Math. 3, 200–211 (1931) 23. Rafeiro, H., Rojas, E.: Espacios de Lebesgue con exponente variable: Un espacio de Banach de funciones medibles (Spanish). IVIC–Instituto Venezolano de Investigaciones Científicas, pp. XI+136 (2014) 24. Rafeiro, H., Samko, S.: Variable exponent Campanato spaces. J. Math. Sci. (N. Y.) 172(1), 143–164 (2011) 25. R˚užiˇcka, M.: Electrorheological fluids, modeling and mathematical theory. Lecture Notes in Mathematics. Springer, Berlin (2000) 26. R˚užiˇcka, M.: Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math. 49(6), 565–609 (2004) 27. Wunderli, T.: On time flows of minimizers of general convex functionals of linear growth with variable exponent in BV space and stability of pseudo-solutions. J. Math. Anal. Appl. 364(2), 5915–5998 (2010) 28. Hedenmalm, H., Korenblum, B., Zhu, K.: Theory of Bergman Spaces. Springer, New York (2000) 29. Zhu, K.: Operator theory in function spaces. mathematical surveys and monographs. American Mathematical Society, USA (2007)