On Characterizations of Weighted Harmonic Bloch Mappings and Its Carleson Measure Criteria

Abstract

For , several characterizations of the -Bloch spaces of harmonic mappings are given. We also obtain several similar characterizations for the closed separable subspace. As an application, we give relations between and Carleson’s measure.

1. Introduction

Let denote a simply connected region in the complex plane ; a harmonic mapping is a complex-valued function defined on such that the Laplace equation satisfied where is the complex second partial derivative of the harmonic mapping .

It is known that in the literature, a harmonic mapping can be written in the form , where and are analytic functions. This form is unique if we fix such that .

Let be the well-known open unit disk in and and denote the class of analytic functions and harmonic mappings on , respectively.

In the last few decades, the Banach spaces of analytic functions on have been gaining a great deal of attention, but for the harmonic extensions of analytic spaces, it is still limited. Besides [1] by F. Colonna, papers such as [2] for the study of the operator theory on some spaces of harmonic mappings, [3] for characterizations of Bloch-type spaces of harmonic mappings, [4] for composition operators on some Banach spaces of harmonic mappings, [5] for the study of harmonic Bloch and Besov spaces, [6] for the study harmonic Zygmund spaces, [7] for the study of harmonic -Bloch mappings and [8] for the study of harmonic Lipschitz-type spaces. For , -Bloch space for harmonic mapping is defined such that where

The mapping defines a norm which yields a Banach space structure on . This space is an extension to harmonic mappings of the classical -Bloch space introduced by Zhu in [9], see also [10]. We recall that belongs to if and only if with norm . Thus, representing as with and , we see that and . Therefore,

Consequently, if and only if the functions such that with are in the classical -Bloch space. When , the space is the (analytic) Bloch space and the corresponding harmonic extension denoted by . The elements of were first introduced in [1].

The little harmonic -Bloch space is defined such that

is a separable closed subspace of (see [2]); for more information about and , see [2, 3, 11] and [1].

For , the conformal automorphism is given by and Green’s function with logarithmic singularity at the fixed point is defined by

For and , the pseudohyperbolic disk with the pseudohyperbolic center and pseudohyperbolic radius is given by

The pseudohyperbolic disk is a Euclidean disk with Euclidean center and Euclidean radius (see [12]). Now, we let denote the normalized Lebesgue area measure on , since is a Lebesgue measureable set; then, the Euclidean area of is given by

Thus, by directed computation, we have the following fact:

Fact 1. Let ; then, for all , For any , the hyperbolic distance between and is given by Meanwhile, for , the hyperbolic disk is given by Throughout this paper, we say that two quantities and , depending on the harmonic mappings , are equivalent denoted by , if there exists a constant such that Moreover, if , we have .

In this work, we expand the study carried out in [13, 14] and [15] for harmonic mappings.

2. Some Integral Criteria for Harmonic -Bloch Mappings

The following lemma needed in the prove of the main theorem of this section (see Lemma 3 in [15]).

Lemma 1. Assume that and ; then, where is the normalized Lebesgue area measure on .

The following theorem is the main theorem of this section.

Theorem 2. Assume that and let . Then for , the following quantities are equivalent: (A)(B)(C)(D)(E)

Proof. Let . Then, is a subharmonic function. Thus, Now, assume that ; then, Also, suppose that ; then, we obtain From [16], we know that Also, from [2], we know that .
Using the inequality, Therefore, Thus, we have where is a constant. Therefore, .
The equivalence follows directly from Fact 1, for all and .
. Since , so for every , we obtain Hence, .
It is clear that for all ; thus, we can infer that .
Now, making a change of variables in quantity , we have From Lemma 1, we know that Thus, we deduce that .
Suppose , for ; we have Therefore, Since is finite by Lemma 1, we have Next, since for any , so Then, for , we obtain Thus, Combining (28) and (31), we have . This completes the proof.

Remark 3. The above characterizations of harmonic -Bloch functions is an extension of Theorem 1 proved by Zhao in [15]. We extend the known characterizations of -Bloch space for analytic function to the harmonic setting using the conditions of the analytic functions and and their subharmonicity on the unit disk. Therefore, we used the proof technique as in the proof of Theorem 1 in [15].

3. Some Integral Criteria for Little Harmonic -Bloch Mappings

We will give some concerned integral-type criteria for the little harmonic -Bloch space and the little harmonic Besov space.

Theorem 4. Suppose that and let . For any harmonic function , the following statements are equivalent: ()()()()()

Proof. Suppose that ; then, By making the change of variables , we have Let such that , for . Set ; for any , we have . Then, This means that Since , then , and the convergence is uniform for any . So for the given , there is such that Thus, we obtain Then, we have By Lemma 1, for all , we know that .
Now combining (35) and (38), we have where is a constant depending on and the harmonic function . That is, . Thus, we deduce that .
Since for all , we have .
Next, follows immediately from the inequality For and , we have that ; then, we get
By the inequality we obtain .
Let ; that is, . Then, for any , there is such that , for every .
Now, we set For each , we know . Thus, Since for every , and the uniformly convergence for . Then, for any and , there is such that , for every .
Hence, Again by Lemma 1, for all and we know that . This means that By combining equations (43) and (45), we have where is a constant depending on and the harmonic function . That is, . Thus, we deduce that . That ends the proof of Theorem 4.

Remark 5. When is analytic function, is the little -Bloch space , and Theorem 4 is proved by Zhao in [15].

4. Carleson’s Measures and Harmonic -Bloch Mappings

Let be a positive measure on . For a subarc , we let be the Carleson box based on ; that is,

For , we let .

Let ; then, a positive Borel measure on is called a -Carleson measure if

Note that gives the classical Carleson measure. We say that is a vanishing -Carleson measure if

As is well known, the Berezin transform of a positive Borel measure on is bounded if and only if is a Carleson measure (see, for example, [13, 14]). Then, for any , we say is a Carleson measure if where is the normalized Lebesgue area measure on . Moreover, we say is a compact Carleson measure if

For all , a positive measure on is a bounded -Carleson measure if and only if

Moreover, is a compact -Carleson measure if and only if

For all , we denote by the weighted harmonic Bergman space, where is the set of all for which where .

Theorem 6. Assume that , and let . Then, the following are equivalent: (a)(b)(c),where is a positive constant

Proof. Firstly, . Suppose that , and let defined as Then, and . By Theorem 3.6 in [2], we obtain Now, setting , we have which means that That is, holds.
Secondly, . Suppose that Then for any , when , then Finally, . For some , let Then, For any , since such that with , has the Taylor series Hence, by a simple calculation for a Taylor series of , which converges uniformly on , we have Now, we let ; then, Similarly, for , we see that which means that So, we have .

Theorem 7. Assume that , and let and . Then, the following are equivalent: ()() is -Carleson measure() is -Carleson measure

Proof. First of all, . Suppose that , and let the integral below: Since when , At the same time, Consequently, . So, we see that holds.
Next, . This is unmistakable, since .
Finally, . Assuming that is -Carleson measure, then Moreover, Hence, So, we have .

Now, we give new characteristics of the little harmonic -Bloch space .

Theorem 8. Assume that , and let . Then, the following are equivalent: (i)(ii)(iii),where is a positive constant

Proof. Firstly, . Suppose that , as in the proof of Theorem 6; let defined as Then, and . By Theorem 3.6 in [2], we obtain For and , we obtain Meanwhile, since for any and , we have By combining equations (77) and (78), we deduce (ii).
The other direction comes easily from inequality Secondly, . For any , since is the closure in of the polynomials, there exist a polynomial such that (see [17]) Also since is the closure in of the polynomials, there exist a polynomial where , such that Hence, Furthermore, Now, set , where in (77); we get so that for , So, we see that (iii) holds.
On the other hand, suppose that (iii) is true. For any , we see that So, we have .

Next, we give relation between and the compact Carleson measure.

Theorem 9. Assume that , and let and . Then, the following are equivalent: (i)(ii) is a compact -Carleson measure(iii) is a compact -Carleson measure

Proof. First of all, . Suppose that , and let . Then, there is such that for , and Also, By combining equations (87) and (88), we see that (ii) holds.
Next, . This is very clear since .
After that, . Assuming that is a compact -Carleson measure, as in the proof of Theorem 7, we have So, we have .

Data Availability

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Conflicts of Interest

The authors declare no conflict of interest.