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

For α > 0 , several characterizations of the α -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 B α H and Carleson ’ s measure.


Introduction
Let Ω denote a simply connected region in the complex plane ℂ; a harmonic mapping is a complex-valued function h defined on Ω such that the Laplace equation satisfied where h w w is the complex second partial derivative of the harmonic mapping h.
It is known that in the literature, a harmonic mapping h can be written in the form f + g, where f and g are analytic functions.This form is unique if we fix w 0 such that gðw 0 Þ = 0.
Let D = fw ∈ ℂ : jwj < 1g be the well-known open unit disk in ℂ and HolðDÞ and HarðDÞ denote the class of analytic functions and harmonic mappings on D, respectively.
In the last few decades, the Banach spaces of analytic functions on D 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 ν-Bloch mappings and [8] for the study of harmonic Lipschitz-type spaces.For α > 0, α-Bloch space for harmonic mapping is defined such that where The mapping h ↦ khk B α H ≔ jhð0Þj + B α h defines a norm which yields a Banach space structure on B α H .This space is an extension to harmonic mappings of the classical α-Bloch space B α introduced by Zhu in [9], see also [10].We recall that f ∈ HolðDÞ belongs to B α if and only if with norm k f k B α = j f ð0Þj + B α f .Thus, representing h ∈ HarðDÞ as f + g with f , g ∈ HolðDÞ and gð0Þ = 0, we see that h w = f ′ and h w = g ′ .Therefore, Consequently, h ∈ B α H if and only if the functions f , g ∈ HolðDÞ such that h = f + g with gð0Þ = 0 are in the classical α-Bloch space.When α = 1, the space B α is the (analytic) Bloch space B and the corresponding harmonic extension denoted by B H .The elements of B H were first introduced in [1].
The little harmonic α-Bloch space B α H,0 is defined such that [2]); for more information about B α H and B α H,0 , see [2,3,11] and [1].For b ∈ D, the conformal automorphism is given by and Green's function with logarithmic singularity at the fixed point b is defined by For b ∈ D and 0 < δ < 1, the pseudohyperbolic disk Dðb, δÞ with the pseudohyperbolic center b and pseudohyperbolic radius δ is given by The pseudohyperbolic disk Dðb, δÞ is a Euclidean disk with Euclidean center ð1 − jbj 2 Þδ/ð1 − δ 2 jbj 2 Þ and Euclidean radius ð1 − δ 2 Þb/ð1 − δ 2 jbj 2 Þ (see [12]).Now, we let λ denote the normalized Lebesgue area measure on D, since Dðb, δÞ ⊂ D is a Lebesgue measureable set; then, the Euclidean area of Dðb, δÞ is given by Thus, by directed computation, we have the following fact: Fact 1.Let δ ∈ ð0, 1Þ; then, for all w ∈ Dðb, δÞ, For any w, b ∈ D, the hyperbolic distance between w and b is given by Meanwhile, for R ∈ ð0,∞Þ, the hyperbolic disk is given by Throughout this paper, we say that two quantities Q 1 ðhÞ and Q 2 ðhÞ, depending on the harmonic mappings h, are equivalent denoted by Q 1 ðhÞ ≈ Q 2 ðhÞ, if there exists a constant C > 0 such that In this work, we expand the study carried out in [13,14] and [15] for harmonic mappings.

Some Integral Criteria for Harmonic α-Bloch Mappings
The following lemma needed in the prove of the main theorem of this section (see Lemma 3 in [15]).
The following theorem is the main theorem of this section.
Remark 3. The above characterizations of harmonic α-Bloch functions is an extension of Theorem 1 proved by Zhao in [15].We extend the known characterizations of α-Bloch space for analytic function to the harmonic setting using the conditions of the analytic functions f and g and their subharmonicity on the unit disk.Therefore, we used the proof technique as in the proof of Theorem 1 in [15].

Remark 5.
When h is analytic function, B α H,0 is the little α-Bloch space B α 0 , and Theorem 4 is proved by Zhao in [15].

Carleson's Measures and Harmonic α-Bloch Mappings
Let μ be a positive measure on For a subarc J ⊆ ∂D, we let SðJÞ be the Carleson box based on J; that is, For J = ∂D, we let SðJÞ = D. Let s > 0; then, a positive Borel measure μ on D is called a s-Carleson measure if Note that s = 1 gives the classical Carleson measure.We say that μ is a vanishing s-Carleson measure if As is well known, the Berezin transform of a positive Borel measure μ on D is bounded if and only if μ is a Carleson measure (see, for example, [13,14]).Then, for any δ ∈ ð0, 1Þ, we say dμ is a Carleson measure if where λ is the normalized Lebesgue area measure on D.Moreover, we say dμ is a compact Carleson measure if For all α ∈ ð0,∞Þ, a positive measure μ on D is a bounded α-Carleson measure if and only if For all α ∈ ð−1,∞Þ, p ∈ ½0,∞Þ, we denote by A p,α H ðDÞ the weighted harmonic Bergman space, where A p,α H ðDÞ is the set of all h ∈ H arðDÞ for which where dλ α ðwÞ = ð1 − jwj 2 Þ α dλðwÞ.
Then, F b ð0Þ = 0 and [2], we obtain Now, setting w = φ b ðζÞ, we have which means that That is, ðbÞ holds.Secondly, ðbÞ ⇒ ðcÞ.Suppose that Then for any r ≥ 0, Journal of Function Spaces Finally, ðcÞ ⇒ ðaÞ.For some C > 0, let Then, For any h ∈ HarðDÞ, since f , g ∈ HolðDÞ such that h = f + g with gð0Þ = 0, h has the Taylor series Hence, by a simple calculation for a Taylor series of F b ðwÞ, which converges uniformly on D δ = fw : jwj < δg, δ ∈ ð0, 1Þ, we have Now, we let δ ⟶ 1; then, Similarly, for δ ⟶ 1, we see that which means that Since log 1/jwj ≲ 1 − jwj 2 when jwj > 1/4, At the same time, . So, we see that ð bÞ holds.
The other direction comes easily from inequality Secondly, ðiÞ ⇔ ðiiiÞ.For any f ∈ HolðDÞ, since B α 0 is the closure in B α of the polynomials, there exist a polynomial p such that (see [17]) Also since B α H,0 is the closure in B α H of the polynomials, there exist a polynomial P = p 1 + p 2 where p 1 , p 2 ∈ B α , such that Hence, Furthermore, Now, set δ = 1 − ε, where ε ∈ ð0, 1Þ in (77); we get δ 1 ∈ ð0, 1Þ so that for D \ Dð0, δ 1 Þ, So, we see that (iii) holds.
After that, ðiiiÞ ⇒ ðiÞ.Assuming that dμ 2 is a compact p-Carleson measure, as in the proof of Theorem 7, we have So, we have h ∈ B α H,0 .