Characterizations of Bloch-Type Spaces of Harmonic Mappings

Abstract

We study the Banach space () of the harmonic mappings on the open unit disk satisfying the condition where and denote the first complex partial derivatives of . We show that several properties that are valid for the space of analytic functions known as the -Bloch space extend to . In particular, we prove that for the mappings in can be characterized in terms of a Lipschitz condition relative to the metric defined by . When , the harmonic -Bloch space can be viewed as the harmonic growth space of order , while for , is the space of harmonic mappings that are Lipschitz of order .

1. Introduction

Given a region in the complex plane , a harmonic mapping with domain is a complex-valued function defined on satisfying the Laplace equation having denoted by the mixed complex second partial derivatives of .

It is well known that a harmonic mapping admits a representation of the form , where and are analytic functions. This representation is unique if, fixing a base point , the function is chosen so that .

In the last several decades, much research has been carried out on the study of Banach spaces of analytic functions on the open unit disk in the complex plane. Since analytic functions are clearly harmonic, a natural question is whether such spaces are subspaces of some Banach space of harmonic mappings on in such a way that the norm on the larger space agrees with the norm of when restricting to the elements of .

A space that has been thoroughly studied in complex function theory is the classical Bloch space defined as the set of analytic functions on such that

In [1], Theorem 10, the second author observed that the functions in the Bloch space are precisely the analytic Lipschitz maps when regarded as functions between the metric spaces (where denotes the hyperbolic distance) and (see also Theorem 5.5 of [2]). Moreover, the correspondence is a seminorm on and is equal to the Lipschitz number, namely, for , where we recall

This result motivated the following notion of Bloch harmonic mapping in [3].

A harmonic mapping on is called Bloch if there exists a constant such that where for , with analytic on , the Lipschitz number was shown in [3] to be equal to the quantitythereby extending to harmonic mappings the corresponding result valid for analytic functions.

In this work, we expand the research done in [3] by focusing on the study of the harmonic extensions of the -Bloch spaces (for ) introduced by Zhu in [4].

Research on the study of harmonic mappings has been conducted extensively in the last two decades by mostly analyzing the function theoretic aspects. The study of the extensions of classical Banach spaces of analytic functions is still relatively limited. Besides [3], references in the unit disk setting include [5] for the study of harmonic Bloch and Besov spaces, [6] for the study of harmonic -Bloch mappings, [7] on planar harmonic Lipschitz and Hardy classes, and [8] for the study of harmonic Lipschitz-type spaces. In the setting of the unit ball in , see [9] for the study of the harmonic Bloch spaces, [10, 11] for the study of the harmonic Bergman spaces and [12] for extensions of the main results in [5]. For a general reference on harmonic mappings in the plane we refer the interested reader to [13].

After giving in Section 2 some preliminaries and background on the (analytic) -Bloch spaces, we introduce in Section 3 the harmonic -Bloch spaces and study their properties. In particular, we show that as done by the second author in [3] for the harmonic Bloch space (case ), the -Bloch space can be characterized by Lipschitz-type conditions similar to the conditions obtained by Zhu in [4] for other positive values of .

In Section 4, we show that for , as for the analytic counterparts described by Zhu in [4], such spaces can be divided into two classes: the space of Lipschitz harmonic mappings of order for , and the harmonic weighted Banach space of harmonic mappings with weight given by the Bergman weight for .

Finally, in Section 5, we give some properties of the harmonic growth spaces that are useful to extend a characterization of functions in the Zygmund space to the harmonic space counterpart. This topic is treated in [14].

2. Preliminaries and Background

Let denote the class of analytic functions on , and for , let be the open disk centered at 0 of radius .

In [4], for , Zhu introduced the -Bloch space as the collection of functions such that The correspondence is a seminorm and is a Banach space under the norm For , is with the classical Bloch space . Thus the -Bloch space can be considered as the space of functions such that is in the growth space , defined as the collection of functions satisfying the growth condition The subspace of consisting of the functions satisfying the condition known as the little -Bloch space, is the closure in of the polynomials and hence separable. Again, this subspace can be viewed as the collection of functions in whose derivative is in the little growth space , whose members satisfy the “little oh" version of (10) as .

Zhu proved that for a positive number , the -Bloch space can be identified with one of two families of spaces, depending on whether or . Specifically, (see Proposition 9 in [4]) if , then an analytic function on belongs to if and only if In particular, the space is contained in the disk algebra.

In Proposition 7 of [4] it was shown that for , the space (respectively, ) is the growth space (respectively, the little growth space ) and the corresponding norms are equivalent, where for the norm of a function is defined as For more information on the growth spaces we refer the interested reader to [15].

The following result proved by Zhu in [4] shows that, for any integer , the elements of the -Bloch space can be characterized in terms of their derivative and a certain Bergman weight dependent of and .

Theorem 1 ([4], Proposition 8). Let , be an integer, and . Then
(i) if and only if (ii) if and only if

Given , we recall the Poisson integral representation of bounded real-valued harmonic functions on the disk with continuous boundary values (see, e.g., [16], (2.13), p. 260), extended in the obvious way to harmonic mappings.

Theorem 2. For , a complex-valued continuous function on and harmonic on , admits the Poisson integral representation: for .

This representation yields the following integral formulas for the complex partial derivatives of a harmonic mapping on with continuous extension to .

Theorem 3. Let be a complex-valued continuous function on and harmonic on . Then for ,

Proof. Let us evaluate the partial derivatives with respect to and of the Poisson kernel: The results follow by applying (19) and (20) after differentiating (16) with respect to and under the integral sign.

Remark 4. In the special case when the function is constant on the unit circle, the harmonic extension to is constant as well, so its complex partial derivatives are identically 0.
With the goal of characterizing functions in by a Lipschitz type condition, Zhu proved the following result.

Proposition 5 ([4], Proposition 16). For , and , let Then defines a distance on .

In [4], Zhu observed that for , the metric is precisely the hyperbolic metric . To the best of our knowledge, an explicit formula of for the case has not been determined.

The following result shows that the ratio of the distance between two points in to their Euclidean distance yields in the limit the reciprocal of the Bergman weight of order .

Theorem 6 ([4], Theorem 17). For any and ,

This leads to the following Lipschitz-type characterization of analytic functions in .

Theorem 7 ([4], Theorem 18). For and the following statements are equivalent: (1).(2)There exists a constant such that for all Moreover, for each ,

In the next section, after giving the notions of harmonic growth space and of harmonic -Bloch space, we follow a similar framework that will then allow us to extend Theorem 7 to harmonic mappings.

3. Harmonic Growth Spaces and -Bloch Spaces

For , we define the harmonic growth space as the collection of all harmonic mappings on such that

Theorem 8. The mapping defines a Banach space structure on that extends the corresponding structure on .

Proof. It is straightforward to verify that is a norm. The space is clearly a subspace of and the respective norms coincide. To prove completeness, assume is a Cauchy sequence in . Fix and . Choose such that for all . Then for all , Thus, is a Cauchy sequence in , which is complete, and we may define . Then is continuous on and since harmonic mappings satisfy the mean value property, for each and and , Passing to the limit as , it follows that satisfies the mean value property. By Theorem 2.11 in [16] extended to complex-valued functions, is harmonic.
Since Cauchy sequences are bounded, is finite. Fixing , we have Therefore, and .
To prove that converges to in norm, note that for , and , Fixing and letting , we get Taking the supremum over all , we obtain , as desired.

Define the little harmonic growth space as the subspace of whose elements satisfy

It is evident from the definition that if (respectively, ), then (respectively, ). We shall prove that the converse holds as well.

For , we now define the harmonic -Bloch space and in Section 4, we shall prove that in analogy to the analytic case, for , as sets, and the harmonic growth space are equal, whereas for , the space is the Lipschitz space of harmonic mappings of order .

Definition 9. For , we define the harmonic -Bloch space as the collection of all harmonic mappings on such that Clearly, when this definition agrees with expression (7).
The harmonic little -Bloch space is defined as the subspace of consisting of the mappings such that

Observe that if with , then and . Thus This implies thatWe deduce the following result.

Proposition 10. Let and let be a harmonic mapping, with . Then (respectively, ) if and only if and are in (respectively, and are in .

In particular, since for , the space is contained in the disk algebra, the space is contained in the space of complex-valued harmonic functions in which are continuous on .

It is straightforward to verify that the mapping defines a Banach space structure on , which extends to harmonic mappings the norm on .

A natural question that arises is whether Theorems 6 and 7 extend to harmonic mappings. We shall show that this is indeed the case. We make use of the following result whose proof is elementary.

Proposition 11. For , the function defined by is a distance on .

Theorem 12. For any and ,

Proof. By Theorem 17 of [4], for , From the definitions of and , it is immediate to see that , for all . Hence, fixing , we have Therefore, to prove the result, it suffices to show that for each , Fix and let with . Replacing with , we may assume , so that . Represent as with analytic and . Proceeding as in the proof of Theorem 17 of [4], fix and let . Then for , whereBy Proposition 5 of [4], there exists some positive constant only dependent on such that for . Therefore, since the assumption implies that and noting that by (36), , using (46) we obtain Hence, Taking the supremum over all such mappings , we obtain for . Letting , we obtain as desired.

We can now prove one of our main results in this section.

Theorem 13. Let be a harmonic mapping on and . Then if and only if there exists a constant such that for all Moreover, for each ,

Proof. Assume and let denote the right-hand side of (52). If is constant, then , and we are done. So assume is nonconstant so that . Then, the mapping defined by is in and has seminorm 1, so, fixing distinct points , by the definition of , we have Dividing by and taking the supremum over all distinct points and , we obtain
Next, note that fixing , For with and fixed, as . Taking the supremum over , we obtain Hence, by Theorem 12, from (55) we obtain Taking the supremum over all , we conclude that , completing the proof.

In [17], the norm of the point-evaluation functional on the Bloch space was calculated precisely. Specifically, it was shown that for , the quantity where we recall that is the distance for .

We now provide a point-evaluation estimate for the harmonic -Bloch space in terms of the metric valid for all . For the case we obtain an extension of (59) to the harmonic Bloch space .

Theorem 14. For and , In the case when , for all ,

Proof. Let and fix . The inequality is clear for . So assume . Then, by (52), we have proving the estimate.
When , by (59), we have which, combined with (62), yields the conclusion.

We now prove that for each , the map is lower semicontinuous on the space .

Theorem 15. Let be a sequence in converging uniformly on compact subsets of to some function . If the sequence is bounded, then and

Proof. Let be as in the statement of the theorem. Applying the mean value property of harmonic mappings to each and then passing to the limit as , we see that also satisfies the mean value property and hence it is likewise harmonic. Set Then, there exists a subsequence converging to . We wish to show that for all , Fix . Since the above inequality is trivial if , assume and fix . Since , there exists such that for all Therefore, for , and having shown in Theorem 13 that the seminorm in equals the Lipschitz number with respect to the distance and the Euclidean distance in , we have Since is arbitrary, we conclude that for all .
This implies that the Lipschitz number of is no greater than . In particular, and , completing the proof.

4. Characterizations of the -Bloch Harmonic Mappings

We now introduce the space of harmonic Lipschitz mappings and prove that it is a complex Banach space.

For , let denote the collection of harmonic mappings on satisfying the condition Define

Theorem 16. For , under the above norm, is a Banach space.

Proof. It is immediate to verify that is a normed linear space. To prove completeness, suppose is a Cauchy sequence in . Fix and . Choose such that for all . Then for all , Thus, is a Cauchy sequence in , which is complete, and so exists. Arguing as in the proof of Theorem 8, we see that is harmonic.
Since Cauchy sequences are bounded, Then for fixed in , we have Taking the supremum over all pairs of distinct points , we obtain and .
Lastly, to show that , note that with fixed, for , with , and , Letting , we get whence .

Theorem 17. For , as sets, , and the respective norms are equivalent.

Proof. First, assume and let such that with . Then , so by Proposition 9 in [4], and belong to the Lipschitz space of analytic functions of order . Thus, for , Taking the supremum over all pairs of distinct points , we obtain Hence . Since the space and the space of analytic functions on which are Lipschitz of order have equivalent norms, where for , using the equalities and and (36), we deduce that for some positive constant .
Conversely, assume . Fix . By Theorem 3, making a change of variable and using Remark 4, and since , we have By the definition of , and expanding as a power series in about 0, for , we have From (80) and (81), noting that for , we deduceIf , then , so so from (83), we obtain Next assume . From (83), making the substitution , and noting that , we have where . Noting that it follows from (86) that for , Therefore, combining the results in the two cases and , we obtain This proves that and Combining (79) and (90), we obtain the equivalence of the norms and .

We next show that, in analogy to the analytic case, for the space can be identified with the harmonic growth space .

Theorem 18. For , as sets, , and the respective norms are equivalent. The little subspaces and are equal as well.

Proof. First, assume and let such that and . Then , so by Proposition 7 in [4], and belong to the growth space . Thus, which is finite. Therefore, . The inclusion follows as well.
Moreover, from (91), since the norms in and are equivalent, and , for some positive constant , we have Conversely, suppose . Fix and set . So . Using (17) and (18), we have Since , we have Since by our choice of , , we have By these facts and multiplying (93) by , it follows that where we used the identity since the mean of the Poisson kernel for the disk of radius over the circle of radius R is 1. Therefore, .
Finally, since , from (96) it follows that which, combined with (92), proves that the norms of and are equivalent.
Lastly, assume . Fix and let be chosen so that whenever . Choosing with and defining in terms of as done previously, from (93) arguing as above, we see that Hence, .

Given , , and harmonic on , let us define

The following result shows that, in analogy to the analytic case, for any integer , the elements of the harmonic -Bloch space and its subspace can be characterized in terms of the derivatives of and and the Bergman weight , whose exponent is a certain function of and .

Theorem 19. Suppose , is an integer, and is a harmonic mapping on . Then (i) if and only if .(ii) if and only if

Proof. To prove (i), assume . By Theorem 1, there exists such that for each analytic on ,
Let such that for some . Then by Theorem 1 and the inequality (36) In particular, if , then is finite.
Conversely, assume . Then and . Thus, and are finite. Therefore, again by Theorem 1, , and thus by Proposition 10, .
To prove (ii), assume first that . Then , so by Theorem 1, proving that (102) holds.
Conversely, suppose (102) holds. Then and the same property holds for . Therefore, , hence .

5. Remarks on the Harmonic Growth Spaces

An immediate consequence of the identification of the harmonic -Bloch space with the harmonic growth space of order is the following result.

Corollary 20. Given and , (respectively, ) if and only if (respectively, ).

Proof. We only need to prove the necessity. Assume . Then by Theorem 18, . So by Proposition 10, . Therefore, by Proposition 7 of [4], . The proof for the little spaces case is similar.

We now apply this result to prove a property of the harmonic growth spaces and their “little” subspaces.

Proposition 21. Let and .
(a) If is bounded and (respectively, ), then (respectively, ).
(b) If is bounded away from 0 near the unit circle and (respectively, ), then (respectively, ).

Proof. Assume is bounded and . Then by Corollary 20, , so by the triangle inequality, which is finite.
Next, assume . Then , so by the triangle inequality and the boundedness of , Therefore, , proving (a).
Next, assume , where is bounded away from zero. Then, again by Corollary 20, the functions and are in , and there exist constants and such that whenever .
We now show that . Since is analytic, it is bounded on each compact subset of , and thus On the other hand,Therefore, combining (108) and (109), we have proving that . The same argument shows that . Therefore, by Corollary 20, .
Next, assume . Then, by Corollary 20, the functions and are in . Fix and choose and such that and whenever . Then, for , proving that . Arguing similarly, we see that . Therefore, again by Corollary 20, we have , completing the proof of (b).

The following result is an immediate consequence of Proposition 21.

Corollary 22. Let , , and a polynomial in with no zeros on the unit circle. Then, (respectively, ) if and only if (respectively, ).

The results of Proposition 21 and its corollary do not hold if is unbounded or is not bounded away from 0 as approaches the unit circle. For example, if , thenwhich is in , but . On the other hand, if we choose and , then while, .

Data Availability

Data is not needed since this research does not involve data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.